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Spin effects in quantum point contacts Ren, Yuan 2011

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Spin Effects in Quantum Point Contacts  by Yuan Ren B. Sc., Nanjing University, 2003 M. Eng., Nanjing University, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2011 c Yuan Ren, 2011 ⃝  Abstract Quantum point contacts (QPCs) are narrow constrictions between large reservoirs of two-dimensional electron gas, with conductance quantized in units of G = 2e2 /h at zero magnetic field. Despite decades of investigation, some conductance features of QPCs remain mysterious, such as an extra conductance plateau at 0.7(2e2 /h) (0.7 structure) and a zero-bias peak (ZBP) in nonlinear conductance. In this thesis, we present experimental studies of transport anomalies in QPCs, aiming at shedding more light on these features. Conductance measurements are performed for ZBPs in a much wider range than in most previous work, focused especially on the low- and high-conductance regimes. The Kondo model and a model of subband motion are compared with experimental results, but both of them fall short of explaining the data. The subband-motion model is not spin-dependent, so it conflicts with the spin-related nature of ZBPs as confirmed by measurements of nuclear spin polarization in QPCs in an in-plane magnetic field. However, the motion of subbands and the spin dependence of these motions are clearly shown by thermopower spectroscopy. These results may help understand the origin of ZBPs and 0.7 structure.  ii  Preface Semiconductor heterostructures studied in this thesis were provided by Prof. Werner Wegscheider in University of Regensburg (now in ETH Zurich). All measured devices were fabricated by Dr. Sergey Frolov and Mr. Wing Wa Yu. Literature review, experiment design, experiment performance, data analysis and manuscript preparation were done by Yuan Ren under the supervision of Dr. Joshua Folk. A version of Chapter 2 has been published. Y. Ren, W. W. Yu, S. M. Frolov, J. A. Folk, and W. Wegscheider, Zero-bias anomaly of quantum point contacts in the low-conductance limit, Phys. Rev. B 82 045313 (2010). A version of Chapter 3 has been published. Y. Ren, W. Yu, S. M. Frolov, J. A. Folk, and W. Wegscheider, Nuclear Polarization in Quantum Point Contacts in an In-Plane Magnetic Field, Phys. Rev. B 81 125330 (2010). A version of Chapter 4 is in preparation for publication.  iii  Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  viii  1 Electron spin transport in quantum point contacts . . . . . . . . . .  1  1.1  Ballistic transport . . . . . . . . . . . . . . . . . . . . . . . . . .  3  1.2  GaAs/AlGaAs heterostructures and device fabrication . . . . . . .  5  1.3  Thermopower . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  1.4  Nonlinear conductance . . . . . . . . . . . . . . . . . . . . . . .  11  1.5  Saddle point model . . . . . . . . . . . . . . . . . . . . . . . . .  14  1.6  Interaction effects . . . . . . . . . . . . . . . . . . . . . . . . . .  17  1.6.1  0.7 structure . . . . . . . . . . . . . . . . . . . . . . . . .  18  1.6.2  Enhanced thermopower . . . . . . . . . . . . . . . . . .  19  1.6.3  Zero-bias anomaly . . . . . . . . . . . . . . . . . . . . .  21  Theoretical models . . . . . . . . . . . . . . . . . . . . . . . . .  22  1.7.1  Spontaneous spin polarization . . . . . . . . . . . . . . .  22  1.7.2  Kondo effect . . . . . . . . . . . . . . . . . . . . . . . .  24  1.7.3  Subband motion . . . . . . . . . . . . . . . . . . . . . .  27  Organization of this thesis . . . . . . . . . . . . . . . . . . . . .  29  1.7  1.8  iv  2 Zero-bias anomaly in low and high-conductance regimes . . . . . .  31  2.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  31  2.2  Devices and measurement setup . . . . . . . . . . . . . . . . . .  34  2.3  ZBPs in low-conductance regime . . . . . . . . . . . . . . . . . .  34  2.3.1  Magnetic field dependence . . . . . . . . . . . . . . . . .  36  2.3.2  Temperature dependence . . . . . . . . . . . . . . . . . .  38  2.4  ZBPs in high-conductance regime . . . . . . . . . . . . . . . . .  40  2.5  Comparison with theoretical models . . . . . . . . . . . . . . . .  42  3 Nuclear polarization in an in-plane magnetic field . . . . . . . . . .  47  3.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  47  3.2  Hyperfine interaction . . . . . . . . . . . . . . . . . . . . . . . .  48  3.3  Dynamic nuclear polarization in quantum-Hall regime . . . . . .  49  3.4  Evidence for nuclear spin polarization in an in-plane magnetic field  50  3.5  Theoretical analysis and discussion . . . . . . . . . . . . . . . . .  53  4 Imaging many-body effects via differential thermopower spectroscopy 59 4.1  Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  59  4.2  Differential thermopower . . . . . . . . . . . . . . . . . . . . . .  61  4.3  Devices and measurement setup . . . . . . . . . . . . . . . . . .  64  4.4  Differential thermopower at zero field . . . . . . . . . . . . . . .  66  4.5  Differential thermopower at finite field . . . . . . . . . . . . . . .  68  4.6  Temperature dependence . . . . . . . . . . . . . . . . . . . . . .  71  4.7  Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  73  5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  75  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  78  Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  86  A  Universal thermopower fluctuations . . . . . . . . . . . . . . . .  86  B  Measurements of electron temperature . . . . . . . . . . . . . . .  88  v  List of Figures Figure 1.1  Conductance quantization in a quantum point contact . . . . .  Figure 1.2  Schematic 3D view of a GaAs/AlGaAs heterostructure and a  2  quantum point contact . . . . . . . . . . . . . . . . . . . . .  6  Figure 1.3  Experimental thermopower and electrical conductance of a QPC  8  Figure 1.4  Nonlinear conductance and transconductance of a QPC . . . .  Figure 1.5  Schematic of the subband energy diagram as a function of Vg  11  and Vdc . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  13  Figure 1.6  Saddle point potential and simulation results . . . . . . . . .  15  Figure 1.7  Temperature and magnetic dependence of the 0.7 structure . .  18  Figure 1.8  Enhanced thermopower around 0.7 structure . . . . . . . . . .  20  Figure 1.9  Zero-bias anomaly in nonlinear conductance of QPCs . . . . .  21  Figure 1.10 Energy of spin up and down subbands as a function of chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . .  23  Figure 1.11 The spin up and down density distribution near the gate voltage corresponding to the 0.7 structure . . . . . . . . . . . . . . .  25  Figure 1.12 Effects of subband motion on differential conductance . . . .  28  Figure 2.1  Comparison between experimental ZBPs and simulated results  32  Figure 2.2  ZBPs in the low-conductance regime . . . . . . . . . . . . . .  35  Figure 2.3  Evolution of the low-conductance ZBPs in an in-plane field . .  37  Figure 2.4  Evolution of the ZBPs with temperature . . . . . . . . . . . .  39  Figure 2.5  ZBPs in the high-conductance regime . . . . . . . . . . . . .  41  Figure 2.6  Evolution of the high-conductance ZBPs in an in-plane field .  42  Figure 2.7  Energy profiles for symmetric and asymmetric Kondo-type localized states . . . . . . . . . . . . . . . . . . . . . . . . . . vi  43  Figure 3.1  Hysteresis in differential conductance curves of QPCs in the quantum-hall regime . . . . . . . . . . . . . . . . . . . . . .  Figure 3.2  49  Hysteresis in differential conductance curves of QPCs in an in-plane field . . . . . . . . . . . . . . . . . . . . . . . . . .  51  Figure 3.3  Relaxation of conductance as a function of time . . . . . . . .  52  Figure 3.4  Examples for more complicated hysteresis and relaxation effects 55  Figure 3.5  Implications of nuclear polarization on ZBPs . . . . . . . . .  57  Figure 4.1  Simulated S × G as a function of Vg and Vdc . . . . . . . . . .  63  Figure 4.2  Micrograph of the device and schematic of measurement setup  65  Figure 4.3  Results of conductance and thermopower spectroscopies at zero magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . .  67  Figure 4.4  Magnetic field dependence of thermopower spectroscopy . . .  69  Figure 4.5  Negative thermovoltage and its evolution with magnetic field .  71  Figure 4.6  Temperature dependence of thermopower spectroscopy for G < 2e2 /h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  72  Figure A.1  Universal thermopower fluctuations as a function of B⊥ . . . .  87  Figure B.1  UCF measurements for determining electron temperature in the channel . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  89  Acknowledgments First and foremost, I would like to thank my advisor Joshua Folk, for taking me into his research group, for his superb guidance and constant support over the past five years, and for his immense knowledge of physics and endless creative ideas. Josh is not only a great supervisor for research but also a mentor for life. I have been fortunate to be his student, and will cherish what he taught me as treasuries of life. I would like to extend thanks to my labmates: Sergey Frolov, Ananth Venkatesan, Wing Wa Yu, Mark Lundeberg, George Kamps, Chong-Yo Lo, Hadi Ebrahimnejad, Silvia Folk, Sasha Rossokhaty, Julien Renard, Dennis Huang, Aryan Navabi, Josh Vandenham, Sifang Chen, Ali Khademi, Greg Mcmurtrie and Matthias Studer. This thesis would not have been finished without their generous help. My sincere thanks also goes to my supervisory committee members, Ian Affleck, Konrad Walus and Jeff Young, for their encouragement during my PhD study and their insightful comments on my thesis. Last but not least, I am deeply indebted to my parents and my wife, for their love, trust, and all the supports they have been giving me.  viii  Chapter 1  Electron spin transport in quantum point contacts One of the most fundamental systems in mesoscopic physics is the quantum point contact (QPC). In general, QPCs are narrow constrictions between two large reservoirs and are essentially short one-dimensional (1D) channels in which electron transport is quantized and ballistic. The first QPCs were fabricated in 1988 in a two-dimensional electron gas (2DEG) of GaAs/AlGaAs heterostructure [1, 2]. Since then, there has been ongoing interest in this structure due to its rich physics and various applications [3–5]. A signature of QPCs is the quantization in differential conductance, G ≡ dI/dV . At zero magnetic field, the differential conductance presents plateaus at multiplies of 2e2 /h, where e is the electron charge and h is the Planck constant. Upon application of a large magnetic field that is parallel to the plane of the sample, the spin degeneracy is lifted and new plateaus appear between those already observed in zero field (Fig. 1.1). This phenomenon is one of the cornerstones of mesoscopic physics, manifesting effects of both quantum mechanics and ballistic transport. Within the single-particle picture, conductance quantization, along with many other characteristics of QPCs, can be well explained. However, later experiments have revealed several transport anomalies that could not be accounted for by a single-particle model. These features include an extra conductance plateau at G = 0.7 × 2e2 /h (referred to as “0.7 structure”) [6], a 1  6  B=0T B=8T G (e2/h)  4  2  Vdc Vac Vg  0 -400  -300 Vg (mV)  -200  Figure 1.1: Conductance quantization in a quantum point contact. The differential conductance, G, shows steps at multiples of 2e2 /h at low temperature (T = 50mK) and zero magnetic field (B = 0T, solid). At B = 8T (dashed), spin degeneracy is lifted and conductance quantization becomes 1e2 /h. Inset: Scanning electron micrograph (SEM) of the QPC and circuit setup for conductance measurement. The dark area is the wafer surface and bright areas are the metal gates patterned with electron-beam lithography.  narrow peak around zero bias in nonlinear conductance (referred to as “zero-bias anomaly”) [7], and conductance plateaus around 0.25 and 0.85 × 2e2 /h at finite source-drain bias (referred to as “0.25 plateau” and “0.85 plateau”, respectively) [8, 9]. It is generally believed that these phenomena arise from many-particle interactions, and are very likely related to electron spin. Many explanations have been proposed, such as spontaneous spin polarization [6, 10, 11], electron-phonon scattering [12], spin-incoherent Luttinger liquid [13], Wigner crystallization [5, 14], and Kondo screening of a quasi-localized state [7, 15, 16]. However, there remains no consensus on which interpretation is correct, and the subject is still widely de-  2  bated. In this thesis, we present experimental results of three projects, aiming at shedding more light on spin-related transport anomalies such as the 0.7 structure, the zero-bias anomaly, etc. In the rest of this introductory chapter, more specific descriptions of electron spin transport in QPCs are given. In Sec. 1.1, we begin by discussing conductance quantization of QPCs in terms of ballistic transport. This is followed by a description of QPC fabrication in Sec. 1.2. From Secs. 1.3 to 1.4, we come back to transport properties of QPCs by discussing thermopower and nonlinear conductance. In Sec. 1.5, we introduce the saddle-point model, a singleparticle description of QPCs. However, some transport phenomena of QPCs are believed to arise from many-particle interactions. Sections 1.6 and 1.7 describe respectively the experimental results and explanations for some of these phenomena. In Sec. 1.8, we provide an outline for the rest of this thesis. Chapters 2-4 present our measurement results of transport anomalies in QPCs.  1.1  Ballistic transport  In a macroscopic wire whose length (L) is much longer than the electron mean free path le , electrons move in random paths due to the numerous collisions they experience. This process is called diffusive transport, and the conductance of the wire is given by G = σ A/L,  (1.1)  where σ is the material dependent conductivity and A is the cross-sectional area. For a mesoscopic device, the dimension ranges from nanometers to a few micrometers, which is smaller than le in a high-mobility 2DEG. In this case, electron motion is no longer randomized since carriers typically pass through the wire without suffering collisions. Such a process is called “ballistic transport”, and Eq. (1.1) breaks down. To calculate ballistic conductance, a new theory is needed. The initial approach for describing ballistic transport was put forward by Landauer [17] and later generalized by B¨uttiker [18] for the mesoscopic systems. The basic idea can be summarized in a short sentence: conduction is transmission. More specifically, the Landauer-B¨uttiker formalism expresses the current through a conductor in terms of the transmission probability that an electron passes through 3  it:  Nm  I = 2e ∑  ∫  dE Nn (E)vn (E)tn (E)[ fL (E) − fR (E)],  (1.2)  n=1  where the pre-factor 2 comes from spin degeneracy; the summation is over all Nm modes propagating through the conductor; Nn (E) is the density of states; vn (E) is the group velocity; t(E) is the transmission probability through the conductor; and fi (E) are the Fermi functions for the right (i = R) or left (i = L) lead, i.e. 1 , 1 + exp[(E − µi )/kB Ti ]  fi (E) =  (1.3)  with chemical potential µi and temperature Ti . Here we are interested in transport properties of QPCs which are essentially 1D conducting channels. In 1D the energy dependence of the density of states exactly cancels that of the velocity, Nn (E)vn (E) = 1/h, therefore Eq. (1.2) becomes 2e Nm I= ∑ h n=1  ∫  dE tn (E)[ fL (E) − fR (E)].  (1.4)  When the bias V applied between the source and the drain is small, electron transport is in the linear response regime. In this case, the difference in the Fermi functions is given in the first order approximation: 2e2V I≈ h  ∫  Nm  ∑  n=1  [ ] ∂ f (E) dE tn (E) − . ∂E  In this thesis, devices are measured at temperature T  (1.5)  1K, where thermal broad-  ening is much smaller than other energy scales. So we can readily assume T = 0, and Eq. (1.5) is reduced to I=  2e2V h  Nm  ∑ tn (EF ).  (1.6)  n=1  The differential conductance of a QPC is given by G≡  2e2 Nm ∂I = ∑ tn (EF ). ∂V h n=1 4  (1.7)  In the limit of perfect transmission, tn (EF ) = 1, and Eq. (1.7) becomes G=  2e2 Nm . h  (1.8)  Equation (1.8) indicates that, in the absence of a magnetic field, each 1D transmitting mode contributes a conductance quantum 2e2 /h = (12.9kΩ)−1 to the total conductance. Therefore, differential conductance of a QPC is quantized in integer multiples of 2e2 /h [1]. At high magnetic field, spin degeneracy is lifted, and the pre-factor 2 in Eq. (1.8) drops out. As a result, each 1D mode only contributes 1e2 /h to the total conductance at high field. The above results contradicts with Eq. (1.1), which predicts infinite conductance for a perfect conductor without impurity scattering. The dilemma was resolved by Imry who identified G in Eq. (1.8) as a contact conductance between the 1D channel and macroscopic leads [19]. Actually, Eq. (1.8) represents the result of a two-probe measurement that includes not only the conductance of the QPC, Gw , but also the contact conductance in series, Gc . The universal conductance quantum comes from electron scattering at the interface between the QPC and the 2D reser−1 voirs. Therefore, the total conductance, G−1 = G−1 w + Gc , is a finite number even  though Gw is infinity as predicted by Eq. (1.8). This interpretation has been confirmed by the four-terminal measurements performed by de Picciotto et al. , who found that the conductance of a clean quantum wire is indeed infinity (G−1 w = 0) and the finite conductance originate from the contacts alone [20]. For a historical review, see papers by Landauer [21] and by Stone and Szafer [22].  1.2  GaAs/AlGaAs heterostructures and device fabrication  Quantum point contacts measured in this thesis are made in a 2DEG of GaAs/Al0.3 Ga0.7 As semiconductor heterostructure grown by Werner Wegscheider in University of Regensburg1 using molecular beam epitaxy (MBE). The reasons for choosing 2DEGs with GaAs/AlGaAs heterstructures as the host material are twofold. First, the electron mean free path in the 2DEG is very long (le > 10µ m), so it is easy to make 1 In  February 2009, Prof. Wegscheider moved from University of Regensburg to ETH Zurich.  5  Ohmic contact EF  GaAs Cap AlGaAs Si-doped AlGaAs  Gate  AlGaAs 2DEG GaAs substrate Quantum point contact  Energy  Figure 1.2: Schematic 3D view of a GaAs/AlGaAs heterostructure (left) and its band diagram (right). The heterostructure is comprised of GaAs substrate/AlGaAs (75nm)/Si-doped AlGaAs (15nm)/AlGaAs (14nm)/GaAs cap (5nm). At low temperature, a two-dimensional electron gas (blue plane) is formed at the GaAs/AlGaAs interface. On the surface of the wafer, two ohmic contacts and two gates are shown. The ohmic contacts are thermally annealed so that metals diffuse down and make contact with the 2DEG. The gates are negatively biased to raise the conduction band above the Fermi level, so electrons gas underneath the gates is depleted (depleted region shown in black). Two nearby gates can give rise to a quantum point contact in the 2DEG.  ballistic devices with length much shorter than le . Second, for effects of quantum mechanics to be detectable, the size of the device must be comparable with the electron Fermi wavelength, λF . For a 2DEG in GaAs/AlGaAs heterostructures,  λF ∼ 50nm, a hundred times larger than that of a metal. With the advance of electron beam lithography, it is now possible to define constrictions that are as narrow as tens of nanometers and exhibit effects of quantum confinement. Figure 1.2 shows the layering of GaAs/AlGaAs heterostructures, which starts with a GaAs substrate in the bottom, then a layer of AlGaAs, and finally a thin  6  (∼ 5nm) cap layer of GaAs to prevent oxidation. At the interface between the GaAs and AlGaAs layers, a triangular potential well is formed due to the 0.3eV bandgap difference between these two materials. For our samples, a 15nm AlGaAs layer located 75nm above the GaAs/AlGaAs interface is doped with Si donor so as to raise Fermi level above the potential dip and inject electrons into the quantum well at the interface. At low temperatures, these electrons can move freely in the plane of GaAs/AlGaAs interface but their motion in z-direction (perpendicular to the interface) is confined within a narrow region ∼ 10nm. The confinement in z-direction gives rise to a large energy spacing between the first and the second subbands, and usually only the first subband is populated at low temperatures. We thus ignore the electron motion in z-direction altogether and simply consider the electron gas as two-dimensional. The 2DEG is extremely clean with few defects because of near perfect matching of lattice constants between GaAs and AlGaAs. In addition, the Si-doped layer is separated from the interface by a 75nm intrinsic AlGaAs spacer to avoid defects near the 2DEG plane. These features make the electron mobility of 2DEG in our samples get to a high value of 4.44 × 106 cm2 /Vs. At temperature T = 1.5K, electron density ns = 1.11 × 1011 cm2 and the mean free path of electrons is as long as 20µ m. Although the 2DEG is 110nm below the surface, it can be electrically contacted by Ni-Ge-Au ohmic contacts. These ohmic contacts are made by deposition of Ni, Ge and Au on the surface of heterostucture, followed by thermal annealing at 410◦ C for 3 minutes to facilitate diffusion of metals down to the 2DEG and making contact with it. Quantum point contacts are created by confining a 2DEG to prescribed areas. This is usually done by applying negative voltages to Cr/Au gates patterned on the surface of heterostructures using electron-beam lithography. A negative voltage on the gates raises potential of the 2DEG underneath, and depletes the electron density when the bottom of quantum well is brought above the Fermi level. Figure 1.2 shows a perspective view of a QPC defined by two metal gates on the surface. The 2DEG plane is shown in blue, and the electron gas exists everywhere except the regions underneath the negatively biased gates. These depleted regions (black) force the electrons to flow from source to drain through a narrow channel between 7  8 Conductance Thermopower Calculations (arbitrary unit)  4 4 2  2 0  Vth (μ V)  G (e2/h)  6  6  -250  -200  -150 -100 Vg (mV)  -50  0  Figure 1.3: Experimental electrical conductance G (black) and thermopower S (grey) versus gate voltage Vg at T = 50mK. Actually it is the thermovoltage Vth which is measured and shown, but S is directly related to Vth as S = −Vth /∆T , and they follow the same functional form. Thermopower shows peaks between conductance plateaus and becomes zero on the plateaus. Dashed line is calculated based on the Mott formula in Eq. (1.17).  the gates. Varying the gate voltage can precisely control the width of the channel and consequently the number of subbands that are occupied by electrons. That is the reason why QPC conductance shows consecutive steps at multiples of 2e2 /h as the gate voltage is increased (Fig. 1.1).  1.3  Thermopower  Ballistic transport through 1D subbands gives rise to quantization of electrical conductance in QPCs. This quantum ballistic effect should occur as well for the thermal properties of QPCs, leading to discrete steps or peaks as a function of gate voltage. Thermopower of QPCs has been specifically calculated by Streda [23], who found that thermopower shows a peak aligned with electrical conductance  8  risers but vanishes on conductance plateaus. This prediction was experimentally confirmed by Molenkamp and coworkers in 1990 [24]. Two years later, the same group performed further measurements that discovered the quantization of thermal conductance and Peltier coefficient in QPCs, both of which exhibit quantized plateaus as electrical conductance [25]. The rest of this section will be devoted to discuss the thermopower of QPCs as it is one of the subjects of this thesis. In general, current flow can be driven not only by potential difference e∆V but also by temperature difference ∆T across the constriction. In linear response regime, this can be expressed as I = G∆V + GT ∆T,  (1.9)  where G and GT are conductance and thermoelectric coefficient, respectively. Experimentally, one does not typically measure GT directly but instead measures the amount of voltage it takes to cancel the transmission current generated by the temperature difference ∆T . Thermopower, S (also referred to as the Seebeck coefficient), is thus defined as the amount of voltage induced by the temperature difference at the state of vanishing current: I = G∆V + GT ∆T = 0 =⇒ S ≡ −lim  ∆V ∆T  = I=0  GT . G  (1.10)  For QPCs, electrical conductance G, along with thermoelectric coefficient GT and thermopower S, can all be expressed in terms of transmission probability in the Landauer formalism. This is done by expanding the Fermi functions fi (E), (i = L or R corresponding to left or right reservoirs) around the equilibrium chemical potential µ and temperature T as fi (E) ≃ f (E) − ∂E f (E)(µi − µ ) − ∂E f (E)(E − µ )(Ti − T )/T,  (1.11)  valid for a small bias and temperature difference. Inserting Eq. (1.11) into Eq. (1.4),  9  linear transport coefficients are obtained by comparison with Eq. (1.9): ∫  G(µ , T ) =  2e2 Nm ∑ h n=1  GT ( µ , T ) =  2ekB Nm ∑ h n=1  S(µ , T ) =  dE tn (E)[−∂E f (E)],  (1.12)  (  ) E−µ dE tn (E) [−∂E f (E)], kB T ( ) E−µ Nm ∫ dE t (E) ∑ n kB T [−∂E f (E)] kB n=1 . ∫ Nm e dE tn (E)[−∂E f (E)] ∑n=1 ∫  (1.13)  (1.14)  A useful approximation to thermopower is made by expanding transmission coefficient t(E) in (Eq. 1.14) to the lowest order around µ , giving ∫  S(µ , T ) =  Nm 2ekB ∑n=1 dE [tn (µ ) + ∂µ tn (µ )(E − µ )]  (  E−µ kB T  )  [−∂E f (E)]  G(µ , T )  h  , (1.15)  valid when kB T is much smaller than variation of the transmission around the chemical potential. Equation (1.15) can be simplified and in the end becomes the famous Mott formula for thermopower of noninteracting electrons, S=  π 2 kB2 T 1 ∂ G ∂ lnG ∝ . 3e G ∂ µ ∂µ  (1.16)  To make comparison with experiment data, one often assumes that the chemical potential µ is proportional to gate voltage Vg , and then replaces µ with Vg . As a result, Mott formula becomes S∝  ∂ lnG . ∂ Vg  (1.17)  Figure 1.3 shows experimental data of electrical conductance and thermopower, as well as calculated S based on Eq. (1.17). Calculated S shows a peak whenever conductance changes from one plateau to another, which is qualitatively in agreement with the experimental data. However, if an obvious 0.7 structure shows up in linear conductance, the Mott formula is found to break down [26]. This situation will be discussed in Sec. 1.6.2.  10  1.4  Nonlinear conductance B = 0T  B = 0T  (a) 6  (b)  3  3 2.5  Vg (mV)  G (e2/h)  2.5 2  4  1.5 1  2  0.85  2  -300 1.5  -400  1 0.85  0.25  0  -1000 B = 8T  0  0.25 2.5 0.25 -1000  1000 (d) 2  1.25 1.5  2  1  2.5  2.25  -300  1.75 1.5  1.25  0.85 0.5  0  1000  2  1.75  Vg (mV)  G (e2/h)  (c) 4  0  B = 8T  1  1  -400  0.25  0.85 0.5 0.25  -1000 0 1000 Vdc (µV)  -1000 0 1000 Vdc (µV)  Figure 1.4: (a, c) Waterfall plot of nonlinear conductance at different gate voltages Vg at B = 0T (a) and B = 8T (c). Plateaus appear as dense regions where many traces come together. Conductance on the plateau is labeled in units of 2e2 /h. Note that plateaus at finite Vdc and G > 2e2 /h are not flat but bend upwards, which is suggested arising from electronelectron interactions [27]. (b, d) Transconductance (dG/dVg ) versus Vg and Vdc at B = 0T (b) and B = 8T (d). Black regions are plateaus with low dG/dVg , while red and yellow bands mark conductance risers with high dG/dVg .  Application of a d.c. bias, Vdc , across the QPC would shift the value of conduc11  tance plateaus [28]. This can be seen in Fig. 1.4, where the nonlinear conductance data is presented in two ways: waterfall plot (Figs. 1.4a, c) and transconductance (Figs. 1.4b, d). The waterfall plots show differential conductance G as a function of Vdc at different but equally spaced gate voltages Vg . In this representation, conductance plateaus appear as dense regions where many traces accumulate. The waterfall plot in Fig. 1.4a shows nonlinear conductance at zero field and low temperature, with the integer plateaus [G = N × 2e2 /h, N = 1, 2, 3...] at zero bias and the so-called “half-integer plateaus” [G = (N − 1/2) × 2e2 /h] at finite bias.2 As mentioned in Sec. 1.1, half-integer plateaus also occur when a large in-plane magnetic field lifts the spin degeneracy. When both voltage bias and magnetic field are present, conductance quantization is further reduced and quarter-integer plateaus show up at G = (N/2 − 1/4) × 2e2 /h (Fig. 1.4c) [29]. Transconductance, the derivative of G with respect to Vg , is another useful tool for presenting and analyzing nonlinear conductance data. In Figs. 1.4b and d, black areas with low transconductance are plateaus, while red and yellow bands with high transconductance are conductance risers. Usually one assumes that the chemical potential µ of a QPC is a linear function of Vg , so transconductance dG/dVg ∝ dG/d µ , showing a peak whenever a subband aligns with either the source chemical potential (µs ) or the drain chemical potential (µd ). Previous literatures use this method to extract location of subbands as a function of Vg and Vdc . We note that such a method is built on the assumption of µ ∝ Vg , which is an approximation and is not true in some cases, especially for G < 2e2 /h at zero field. Therefore, transconductance is only useful to get a rough estimation of subband location but could be wrong in details. In Ch. 4, we will develop a more reliable technique based on thermopower for imaging subbands and their behaviors with varying Vg and Vdc . For now, we rely on transconductance for extracting subband location and then justify results based on the Landauer-B¨uttiker formalism. A schematic subband energy diagram at various Vg and Vdc is shown in Fig. 1.5. Notice that differential conductance at finite Vdc increases by 0.5 × 2e2 /h when µs or µd passes through a subband. This indicates that each subband lying between µs and µd only con2 Two plateaus at 0.25 and 0.85 × 2e2 /h are observed instead of the predicted G = e2 /h plateau. This phenomenon is believed to arise from electron interactions [8, 9, 27].  12  Vg 3 2.5 ε3 μs  2  ε2 ε1  2  μd  1.5 1 1 0.5 0 Vdc  0  Figure 1.5: Schematic of the subband energy diagram as a function of Vg and Vdc . The black lines and white areas correspond respectively to the conductance risers and the the plateaus, similar to the transconductance plot in Fig. 1.4. The schematic is symmetric with respect to Vdc = 0. For plateaus at Vdc ≤ 0, their conductance is labeled in units of 2e2 /h. For plateaus at Vdc ≥ 0, subbands (ε1 , ε2 ...) are shown with respect to the source and drain chemical potentials (µs and µd ).  tributes e2 /h to the total conductance, different from 2e2 /h for each subband below  µs and µd . A simple calculation based on Eq. (1.4) can help understanding this difference. Assume a classical transmission coefficient: t(E) = 1 for E ≥ εn and t(E) = 0 for E < εn , where εn is the bottom of the nth subband. Also assuming zero temperature and µs > µd , the net current carried by the nth subband can be written as  In =  ∫ 2e µs  h  µd   (εn > µs ),   0 t(E)dE = 2e(µs − εn )/h (µd < εn < µs ),   2 2e V /h (εn < µd ). 13  (1.18)  Here µs = µ0 + β eV and µd = µ0 − (1 − β )eV , with µ0 the chemical potential at zero bias and β the fraction of bias voltage dropped between the source and the bottleneck (the narrowest point of the constriction). Conductance of the nth subband is obtained by taking the derivative of Eq. (1.18):  (εn > µs ),   0 2 Gn = 2e β /h (µd < εn < µs ),   2 2e /h (εn < µd ).  (1.19)  It is worth noting that conductance of each subband below µd is always equal to 2e2 /h, but conductance of each subband lying within the bias window depends critically on how external bias drops across the constriction. QPCs are usually considered to be of smooth boundaries and electrons can pass through them adiabatically. For these constrictions, half of bias voltage would drop between the source and the bottleneck [30], thus β = 0.5 and Gn = e2 /h for µd < εn < µs . Upon application of a magnetic field, spin degeneracy is lifted and the factor 2 in Eq. (1.19) drops off. As a result, each subband between µs and µd contributes e2 /2h, giving rise to quarter plateaus observed in experiments [29].  1.5  Saddle point model  As discussed in previous sections, many characteristics of QPCs agree well with the non-interacting Landauer-B¨uttiker theory. To understand better the transport properties of QPCs, we consider a specific non-interacting model of QPCs which adopts a saddle-point potential to describe the constriction [31]. This model is analytically solvable, so a quantitative comparison with experimental data can be achieved. The starting point is the single-particle Schr¨odinger equation for electrons in a QPC,  [ ] h¯ 2 − ∗ (∂x2 + ∂y2 ) +U(x, y) φ (x, y) = E φ (x, y), 2m  (1.20)  where m∗ is the effective electron mass; x is along the QPC and y is the transverse direction; φ (x, y) is the 2D eigenstate; and U(x, y) is the confining potential from the surface depletion gates. Glazman et al. [32] have shown that the 2D differen14  (c)  (a)  6  Saddle point  4 G (e2/h)  y  x  2 S  G (e2/h)  8 (b) 4  0  -1  µ0  0 1 Vdc (mV)  0  Figure 1.6: (a) An illustration of the saddle point potential expressed in Eq. (1.21). The constriction is along the x-direction. (b) The conductance (solid) and thermopower (dashed) as a function of the chemical potential µ0 calculated by the saddle-point model. (c) Nonlinear conductance calculated using Eq. (1.25). The parameters used in simulations are: h¯ ωx = 0.6meV, h¯ ωy = 2meV and β = 0.5.  tial equation of Eq. (1.20) can be separated into two 1D equations if the variation of confining potential in the x-direction is much slower than in the transverse ydirection (adiabatic constriction). Eigenvalues of the y-direction equation are discrete, each corresponding to a transverse mode (subband). To proceed further, we adopt the saddle-point model as an approximation for confining potential. In this model, one neglects all but the linear and quadratic terms in the Taylor expansion of the confining potential near the narrowest point (saddle point) of the constriction, therefore U(x, y) becomes 1 1 U(x, y) = U0 − m∗ ωx2 x2 + m∗ ωy2 y2 , 2 2  (1.21)  where U0 is the potential of the saddle point; ωx and ωy define the curvatures of 15  QPC potential along the longitudinal and transverse directions, respectively. Substituting Eq. (1.21) into Eq. (1.20), the resulting y-direction equation denotes a harmonic oscillator, with the transverse quantization energy εn = (n − 1/2)¯hωy +U0 . For the nth subband, one finds that the transmission coefficient takes a conveniently simple form [33], tn (E) =  1 2π  1 + e− h¯ ωx (E−εn )  .  (1.22)  Expressions for differential conductance and thermopower can be obtained by inserting Eq. (1.22) into Eqs. (1.12) and (1.14). The integrations can be done numerically and the results are shown in Fig. 1.6b. Note that G and S are simulated as a function of the zero-bias chemical potential µ0 while they are measured as a function of the gate voltage Vg as shown in Fig. 1.3. For a rough analysis, it usually assumes a linear relation between µ0 and Vg , so calculation results and experimental data can be compared directly. However, the actual relation between µ0 and Vg is very complicated, and a more accurate comparison requires self-consistently solving the electrostatic problem. At zero temperature, differential conductance can be expressed in an analytical form. For small Vdc (linear response regime), we have G(µ ,Vdc → 0) =  2e2 Nm 1 . ∑ − h n=1 1 + e h¯2ωπx (µ −εn )  (1.23)  For large Vdc , differential conductance is a weighted average of two zero-bias conductances, one for a chemical potential of µs = µ0 + β eVdc , and the other for a chemical potential of µd = µ0 − (1 − β )eVdc , i.e., G(µ ,Vdc ) = β G(µs , 0) + (1 − β )G(µd , 0), = [G(µs , 0) + G(µd , 0)]/2,  (β = 1/2).  (1.24) (1.25)  When a subband lies inside the bias window (µd < εn < µs ), G(µs , 0) differs from G(µd , 0) by 2e2 /h, giving rise to a half-integer plateau at G(µ ,Vdc ) = G(µd , 0) + e2 /h. The simulated results using Eq. (1.25) are shown in the form of waterfall plot in Fig. 1.6c, which in general agrees with the experimental data (Fig. 1.4a) for G > 2e2 /h. Below the first plateau, experimental data shows features that are sig16  nificantly different from results of the saddle-point model. These features are generally believed to arise from electron-electron interactions in low-density QPCs.  1.6  Interaction effects  Many phenomena of QPCs, such as conductance quantization and thermopower peaks, can be described by the Landauer-B¨uttiker formalism without taking into account electron-electron interactions. This is quite surprising because electron interactions in 1D are increasingly important compared to those in higher dimensions. Unlike in higher-dimensional systems, interacting electrons in 1D are not described by Fermi liquid theory, but by Tomanaga-Luttinger liquid theory [34]. This theory predicts that, for an infinitely long 1D wire, the conductance quantization would be renormalized to G = g(2e2 /h), with g = 1 for non-interacting electrons, g < 1 for repulsive interactions and g > 1 for attractive interactions [35]. This apparent disagreement with experiments is resolved by taking into account the effects of noninteracting leads. Experimentally, QPCs are all of finite length and are connected to large leads with negligible interactions. To model a realistic 1D wire, Maslov and Stone[36] used a position dependent g parameter that is chosen to model the strong interacting quantum wire and noninteracting leads. The calculation results show that the low-frequency conductance of such a system is dominated by the noninteracting leads, and the universal value 2e2 /h is restored. This result provides grounds for noninteracting explanations such as the saddle-point model, which indeed has had great success in describing QPC transport properties when conductance is higher than 2e2 /h. Below 2e2 /h, however, deviations from the noninteracting pictures have been found. In this region, the electron density is low and the interaction effect is believed to be more important, thus it is not surprising that the noninteracting model fails to capture some features observed in experiments. Since Thomas et al. investigated the famous 0.7 structure in 1996 [6], researchers have found more and more transport anomalies that are believed to arise from electron interactions. The rest of this section is devoted to introducing several interaction effects in lowdensity QPCs. Theoretical explanations of these phenomenon will be discussed in the next section.  17  3  G (e2/h)  4 (a) B = 0 T ||  2  (b) B||= 0T  2  0 -250  50mK 300mK 620mK 1000mK 1500mK  -200  -150  1  0 -50 Vg (mV)  B||=6T  0  50  Figure 1.7: Temperature and magnetic dependence of the 0.7 structure. (a) Conductance G versus gate voltage Vg measured at B|| = 0 for different temperatures. With increasing temperature, the 0.7 structure becomes more pronounced. (b) Conductance G versus Vg for a range of parallel magnetic fields from 0 to 6 T in 1 T steps. Successive traces are offset horizontally for clarity. The 0.7 structure at zero field evolves to the spin-split e2 /h plateau at high field.  1.6.1 0.7 structure Conductance of QPCs is quantized in units of 2e2 /h at zero magnetic field and the quantization becomes e2 /h when the spin degeneracy is lifted at high magnetic field, as predicted by the single-particle model. It was therefore surprising when Thomas et al. [6] showed that a shoulder-like feature was consistently observed around 0.7 × 2e2 /h. This feature, now known as the 0.7 structure or 0.7 anomaly, is actually visible even in the earliest experiment on QPCs, but it was overlooked for many years and thought to be a resonant structure. Thomas et al. [6] performed measurements in some very clean QPCs which showed nearly 30 quantized integer plateaus, in addition to a well-defined 0.7 structure. They concluded that the 0.7 structure is not related to impurities, but rather an intrinsic phenomenon of QPCs. Since then, numerous measurements have been performed to try to understand the 0.7 structure. This feature has been studied not only in GaAs electron gas, but also in GaAs hole systems [37–40], cleaved-edge-overgrowth 1D wires [41, 42], Si 18  [43], GaN [44], and InGaAs heterostructures [45]. Universal behaviors of the 0.7 structure are summarized below. 1. Temperature dependence: As temperature is increased, the 0.7 structure becomes more pronounced and its conductance evolves downwards. At 4.2K, the 0.7 structure remains visible at about 0.6 × 2e2 /h [46], while all integer conductance plateaus are smeared off by the thermal broadening. This fact suggests that the 0.7 structure is not a ground-state property but a thermally activated phenomena. Kristensen et al. further showed that temperature dependence of the conductance at fixed gate voltage follows an Arrhenius form, G(T )/G(0) = 1 −Cexp(−TA /T ), with TA the activation temperature [47, 48]. 2. Magnetic field dependence: With the application of an in-plane magnetic field, the 0.7 structure shifts down in conductance, and eventually evolves to the spin-polarized e2 /h plateau at high field [6]. Thomas et al. concluded from this behavior that the 0.7 structure is related to spontaneous spin polarization at zero field. 3. Bias dependence: On increasing source-drain d.c. voltage Vdc , the 0.7 structure increases smoothly from 0.7 × 2e2 /h at zero bias to ∼ 0.85 × 2e2 /h at high bias (Fig. 1.4a) [49]. The 0.85 plateau, together with another feature at 0.25 × 2e2 /h, are not in agreement with the single-particle models given in Secs. 1.4 and 1.5 [8, 9, 27]. Further experiments has suggested that the 0.7 structure is actually not alone but has analogs at high conductance. Additional plateaus are occasionally observed at 1.7 and 2.7 × 2e2 /h with much fainter appearance [49, 50]. Upon application of a magnetic field, the 1.7 and 2.7 structures evolve smoothly to 1.5 and 2.5 × 2e2 /h plateaus, respectively.  1.6.2 Enhanced thermopower According to the Mott formula (Eq. 1.17), a plateau in the conductance should be accompanied by a zero in the thermopower. Such a prediction holds for plateaus at N × 2e2 /h for N = 1, 2, 3..., however, it breaks down in the region around the 19  4  6  Conductance Thermopower  2  4  Vth (μ V)  G (e2/h)  0.7 Structure  2  0  -150  -100 Vg (mV)  -50  0  Figure 1.8: The electrical conductance G (solid) and the thermovoltage Vth (dashed) versus gate voltage for a QPC with prominent 0.7 structure (grey region). Thermopower can be read from Vth as S = −Vth /∆T . According to the Mott formula, a plateau in G should be accompanied by a zero in S. However, experimentally S remains a finite value in the region around the 0.7 structure.  0.7 structure (Fig. 1.8). Appleyard et al. have found that experimentally the thermopower around the 0.7 structure is not zero, but rather a finite value roughly independent of gate voltage [26]. Application of a magnetic field restores the prediction of the Mott formula by converting the 0.7 structure into a spin-polarized e2 /h plateau, accompanied by a decline in the value of thermopower to zero. Generally speaking, thermopower is related to entropy: S = −S /e with S the entropy per carrier. The Mott formula includes entropy of electron motion (current flow) but neglects entropy of spin configuration, so it is only valid in the single-particle picture where spin-flip process is absent. In case of strong electronelectron interaction, the spin degrees produce a significant contribution, therefore enhance the value of thermopower from the prediction of Mott formula [51]. Based on this consideration, Appleyard et al. conclude that the 0.7 structure is related to some kind of many-particle interactions [26]. Similar effects are also observed in the Kondo regime of quantum-dot systems, where thermopower is a finite positive 20  1.6  (b)  (a)  G (e2/h)  1.4 1.4  150mK 300mK 650mK  40mK 70mK  0T  3.0T  1.5T  4.5T  1.2  1.4 1.2 -200  0  200  -200  0  200  Vdc (µV)  Vdc (µV)  Figure 1.9: Zero-bias anomaly in nonlinear conductance of QPCs. (a) Temperature dependence of zero-bias peaks for different gate voltages, at temperatures from 40 to 650mK. (b) Evolution of the ZBP with in-plane magnetic field from B|| = 0 to 4.5T.  value instead of zero as predicted by the Mott formula [52]. Another interpretation is that the 0.7 structure is actually not a true plateau in conductance but originates from the pinning of subband energy with respect to the Fermi energy [26]. Recall that we replace ∂ G/∂ µ with ∂ G/∂ Vg in Eq. (1.17) based on the assumption of linear relationship between µ and Vg . However, this relation is not always true especially when the subband starts being populated. Experiments reveal that, as the Fermi energy is raised above subband edge, the subband is reluctantly populated and rises together with the Fermi energy, making µ unchanged with varying Vg [9, 53]. In this case, ∂ µ /∂ Vg = 0, so it is not appropriate to substitute ∂ G/∂ µ by ∂ G/∂ Vg . More details about the subband pinning will be discussed in Sec. 1.7.3.  1.6.3 Zero-bias anomaly Upon application of a voltage across the constriction, a sharp and narrow conductance peak centered at zero bias is observed at low temperature. This peak is termed zero-bias anomaly since it does not agree with the smooth and broad peaks predicted by the single-particle model (see Fig. 1.6c). Due to their proximity to the 0.7 structure, zero-bias peaks (ZBPs) within 1e2 /h  21  G < 2e2 /h are of particular  interest and have received extensive attention. Experimental findings of ZBPs in this region at various temperatures and in-plane magnetic fields (B|| ) are summarized in Fig. 1.9. Basically, ZBPs are suppressed and eventually disappear at high temperature or in large magnetic field. A characteristic feature of the magnetic field dependence is that the ZBP splits into two smaller side peaks with spacing of (3 − 5)Ez , where Ez = gb µB B|| is the Zeeman energy in the bulk GaAs (|gb | = 0.44) [7]. These behaviors of ZBPs closely resemble those of Kondo peaks observed in quantum dots [54–56]. Motivated by the similarities, a Kondo model is proposed to account for the occurrence of ZBPs around the 0.7 structure (1e2 /h  G < 2e2 /h) [7]. Although  this model answers some questions, it also raises others. For example, how could Kondo physics possibly occur in an open system like QPC? Could Kondo physics be extended to the conductance regimes away from the 0.7 structure, such as the low-conductance limit? The first question will be answered in Sec. 1.7.2 when the Kondo model is introduced. The second question is one of the major subjects of this thesis, and will be discussed in Ch. 2.  1.7  Theoretical models  Although QPCs are the simplest nanoscale devices, the interaction effects described in Sec. 1.6 have not been fully understood after more than one decade of investigation. Many theoretical models are proposed to account for these phenomena, but none of them is able to explain all the experimental features consistently. Nevertheless, there is an overall consensus that many-particle interactions are the origin of the effects. In this section we will go through a few of the available explanations to the transport anomalies in low-density QPCs.  1.7.1 Spontaneous spin polarization Already in their early paper in 1996, Thomas et al. [6] proposed such a mechanism to account for the presence of the 0.7 structure. Based on the fact that the 0.7 structure evolves smoothly to the 1e2 /h plateau with increasing magnetic field, they speculate that the 0.7 structure may originate from a spin polarization of the 1D electron gas in zero magnetic field. The polarization continues to grow as 22  1.3  Ε↑  Εσ/E1  1.2 1.1  Ε↓  1 0.9 0  0.5  1 µ/E1  1.5  2  Figure 1.10: Energy of spin-up E↑ (black) and spin-down E↓ (grey) subbands as a function of chemical potential µ , calculated based on the nonequilibrium Green-function technique where the interaction is incorporated at the Hartree-Fock level. The dashed line shows the chemical potential µ . Constant E1 = h¯ 2 π 2 /(2mW 2 ) is the energy of the first transverse mode in the narrow channel of width W . (Figure modified from Ref. [57])  magnetic field is increased, and gives rise to the fully spin-polarized plateau at high field. The speculation of spontaneous spin polarization has been cast into a phenomenological model by Bruus et al. [58], who made two assumptions: 1) the energies of spin-up and spin-down subbands depend on the electron density; 2) when populated, the spin-up subband (higher energy) rises with the Fermi level (pinning) while the spin-down subband (lower energy) stays put, therefore it opens a spin gap at zero magnetic field. Reilly improved this model by assuming that the spin gap continues to open even when the chemical potential is above the spin-up subband edge [11, 59, 60]. The phenomenological model has been fairly successful in reproducing temperature, magnetic field and bias dependencies of the 0.7  23  structure, as well as the enhanced thermopower. Some doubts are cast on the model of spontaneous spin polarization [61, 62], as it seems in contradiction with the Lieb-Mattis theorem [63], which predicts an antiferromagnetic singlet ground state for 1D systems. However, the Lieb-Mattis theorem only applies for strictly 1D systems, in a mathematical sense [64]. Here the electrons in QPCs are actually three-dimensional, albeit confined to a channel of small width. This deviation from true one dimensionality can, in principle, give rise to a spin-polarized ground state in a interacting electron system. For example, there exists a ferromagnetic state in a quasi-1D zigzag Wigner crystal [5, 14]. The microscopic driving force of spontaneous spin polarization is the exchange interaction between electrons. The parallel alignment of electron spins decreases the exchange energy of the system, but increases the kinetic energy due to Pauli exclusion principle. On lowering the electron density, the exchange interaction becomes more important and eventually the gain in exchange energy wins. As a result, electrons favor parallel alignment of their spins even though the magnetic field is absent, and the spontaneous spin polarization takes place. Follow-up numerical calculations [10, 57, 64] confirm exchange interaction as the driving mechanism for the opening of a spin gap. When a spin-degenerated subband starts to be populated, the upper energy level pins to the chemical potential whereas the lower energy level does not (Fig. 1.10). These spin-dependent behaviors lead to energy splitting between subbands, and consequently to the 0.7 structure and its analogs.  1.7.2 Kondo effect As temperature falls, the electrical resistance of a metal usually drops and in the end saturates for T  10K. When magnetic impurities are added, however, the  electrical resistance increases rather than saturating at low temperature. This unusual phenomenon is known as the Kondo effect, which remained unexplained for several decades until Jun Kondo formulated the first satisfactory theory in 1964 [65]. According to Kondo’s theory, conduction electrons in the metal are trying to screen the magnetic impurities, causing the formation of Kondo clouds that block the flow of current. With decreasing temperature, the Kondo clouds become bigger  24  Energy (meV)  (a)  (b)  0 -1 -2 -200  0  200  -200 Position (nm)  0  200  Figure 1.11: (a) Spin-up and (b) spin-down local densities of states (LDOS) calculated by the spin-density-functional theory, for the QPC at a gate voltage corresponding to the 0.7 structure. The Kohn-Sham potential for electrons in the lowest transverse mode is also shown (white dashed line). The potential for spin-up electrons has a double-barrier formed near the center of the QPC, and supports a quasi-bound state about 0.5 meV below the Fermi energy (green dashed line). (Figure modified from Ref. [16])  and bigger, giving rise to the observed increase in resistance. A more controllable platform for studying the Kondo effect is quantum dots, where the Kondo effect manifests itself as an increase in conductance with decreasing temperature and a conductance peak (Kondo peak) centered at Vdc = 0. Note that in metals the Kondo clouds increase scattering and hence resistance. In quantum dots, it is the scattering between conduction electrons and localized spin in the dot that makes electrons transport through the otherwise blockaded dot. That is why Kondo effect increases conductance of quantum dots. As already mentioned in Sec. 1.6.3, there are a number of remarkable similarities between Kondo peaks in quantum dots and ZBPs within 1e2 /h  G < 2e2 /h  [7]. These similarities include: 1. Both Kondo peaks and ZBPs in QPCs are suppressed and finally disappears as temperature is increased. For Kondo peaks, temperature dependence at different gate voltages follows a universal form derived by numerical renor25  malization group methods [66]. According to Ref. [67], this form can be well approximated by an empirical formula, G(T ) = G0 f (T /TK ),  (1.26)  where G0 is the saturation conductance at zero temperature; TK stands for the Kondo temperature and f (T /TK ) = [1 + (21/S − 1)(T /TK )2 ]−S ,  (1.27)  with S = 0.22 for spin 1/2. For ZBPs in QPCs, temperature dependence of the peak maximum at various gate voltages is found to follow a similar scaling formula, G(T ) = (e2 /h) f (T /TK ) + e2 /h.  (1.28)  The reason for adding the second term e2 /h in Eq. (1.28) is to represent a fully transmitting channel, and the first term stands for a channel with reduced transmission. 2. In quantum dots, the width of Kondo peaks (defined as the full width at half maximum, FWHM) is set by Kondo temperature TK [56]. For ZBPs in QPCs, FWHM is also related to TK extracted from fitting to Eq. (1.28) as FWHM ∼ 2kB TK /e. 3. As the applied in-plane magnetic field B|| is increased, Kondo peak splits into two small side peaks with spacing scaled with B|| [55]. ZBPs in QPCs also split with B|| with spacing of about (3 − 5)Ez . The comparison inspires a Kondo explanation to ZBPs and consequently the 0.7 structure [7]. This model assumes that, at low electron density, there exists a localized state with nonzero spin in the middle of the constriction. Kondo screening of this state by conduction electrons in electrodes leads to enhanced conductance when the temperature and d.c. bias are less than TK , the Kondo temperature. Within this model, ZBPs within 1e2 /h  G < 2e2 /h and the 0.7 structure can be attributed  to the suppression of the Kondo-enhanced conductance by finite bias and by finite temperature, respectively. 26  The proposal of the Kondo explanation was soon followed by spin density functional calculations [15, 68] which support the existence of an exchange-induced localized state at the center of the constriction. Further calculations [16, 69] suggest that the exchange-induced localized state may survive at high conductance and high magnetic fields, giving rise to 1.7 structure and other 0.7 analogs. However, these results are highly debated because other spin density functional calculations did not find localized states [10, 70–73] and Hartree-Fock calculations [74] could not confirm the Kondo scenario either. Recently, more rigorous calculations that go beyond the mean-field approximations have been performed based on the quantum Monte Carlo technique for an inhomogeneous quasi-1D system [75]. Calculation results show that formation of localized states depends on the gradient between the high and low-density regions. For a short constriction with very smooth potential landscape, there exists a single localized electron with dynamic spin polarization in the low-density region, which is in agreement with the Kondo model. Experimentally, it is also controversial whether a Kondo-like localized state exists in QPCs and whether the Kondo effect is responsible for the 0.7 structure and the ZBPs. The formation of a localized state is supported by some experimental results [40, 76]. However, measurements on quantum wires with an intentionally made localized state suggest that the Kondo effect and the 0.7 structure are separate and distinct phenomena [77]. Recent investigations on ZBPs also reveal characteristics that cannot be accounted for by the Kondo model [78, 79]. To conclude, there remains no consensus on whether the ZBPs and the 0.7 structure have a Kondo origin and this subject is still under debate. To shed more light on this subject, in Ch. 2 we will present a systematic study on ZBPs and compare the measurement results with the Kondo theory and other available models.  1.7.3 Subband motion Generally speaking, subband motion refers to dynamics of 1D subbands as a function of gate voltage Vg and applied bias Vdc , including subband splitting and subband pinning discussed in Sec. 1.7.1. In this subsection we will restrict discussions to subband motion with varying Vdc , and focus on its effects on nonlinear conductance.  27  G (e2/h)  2 (a)  (b)  1  0  -1  0 1 Vdc (mV)  -1  0 1 Vdc (mV)  Figure 1.12: (a) Calculated differential conductance as a function of Vdc for G ≤ 2e2 /h, assuming that subband energy stays fixed with respect to the center of bias window. (b) Differential conductance versus Vdc , calculated in the same way as (a), but including a linear rise of subband energy with Vdc at a rate of γ = 0.2meV/mV. Other calculation parameters are h¯ ωx = 0.6meV, h¯ ωy = 2meV, η = 0, and β = 0.5.  In Sec. 1.5, the energy of subbands is assumed fixed with respect to the center of the bias window when Vdc is varied. Here we assume that subbands are no longer fixed but move up with increasing Vdc , so the saddle-point potential in Eq. (1.21) is rewritten as 1 1 2 U(x, y) = U0 + γ |Vdc | + η Vdc − m∗ ωx2 x2 + m∗ ωy2 y2 , 2 2  (1.29)  where γ and η are coefficients for the linear and quadratic terms in the Taylor expansion, respectively. Differential conductance is calculated numerically by combining Eq. (1.29) with Eq. (1.4) and Eq. (1.22). A nonzero γ term gives rise to a nonlinear conductance plateau at (0.5− γ )×2e2 /h, instead of a half-integer plateau predicted by the single-particle model. This is one of the explanations available for the anomalous 0.25 plateau [28, 80]. The effect of the quadratic term η is to change the slope of nonlinear conductance plateaus. As shown in Fig. 1.4a, the half-integer plateaus at 1.5 × 2e2 /h and 2.5 × 2e2 /h are not flat but bend upwards with increasing Vdc , which can be reproduced by using a negative η in calculations 28  [81]. Recently, Chen et al. [78] pointed out that a positive γ term also gives rise to a sharp and narrow conductance peak around Vdc = 0. Figure 1.12a shows simulation results for γ = 0—as expected, there is no ZBP. In contrast, Fig. 1.12b shows differential conductance characteristics calculated for γ = 0.2meV/mV—a sharp ZBP is now present. It is worth emphasizing that the ZBPs occur in simulations whenever γ > 0, and γ only changes the size of the peak. Further simulations show that the occurrence of the ZBPs stems from the rise of subband energy with Vdc , irrespective of the precise functional form of subband rising [78]. There is a fundamental reason for the rise of the subband energy with increasing d.c. bias. For 1D systems, the density of states shows a singularity at the bottom of the subband. At low conductance where the zero-bias chemical potential is below the subband µ < εn , increasing Vdc injects a large amount of electrons into the subband, thereby greatly enhances the Coulomb energy. To minimize the Coulomb energy, the subband moves up with increasing Vdc . This situation has been captured in numerical calculations employing a nonequilibrium Green’s function technique within the Hartree approximation of spinless electrons [27]. Although the rise of the subband with increasing Vdc can account for the emergence of the ZBP, there have been no predictions for a spin dependence of this effect, conflicting with the universal disappearance of the ZBP for large in-plane field. Indeed, the spin degree of freedom is also absent from the calculations in Ref. [27]. A more sophisticated theoretical study with spin effects taken into account is thus needed to study the subband motion and consequently the ZBPs at various magnetic field.  1.8  Organization of this thesis  In the rest of this thesis, we present results of three projects that are independent but coherent in understanding spin effects in QPCs. In Ch. 2, we begin the exploration by questioning the Kondo explanation to the formation of ZBPs. This leads us to examine characteristics of ZBPs not only within the region of 1e2 /h 2e2 /h,  but focused especially on the low-conductance limit (G <  1e2 /h)  G< and the  high-conductance regime (G > 2e2 /h). Comparison of the experimental results  29  with the Kondo model and the subband-motion model shows that both explanations display shortcomings. The Kondo model fails to account for the magnitude of ZBP splitting with magnetic field and the consistent observation of ZBPs in the highconductance regime. For the subband-motion model, the spin degree of freedom is irrelevant, so it conflicts with the spin dependence of ZBPs. In Ch. 3, we provide experimental evidence for the spin dependence of ZBPs. The first observation of nuclear spin polarization in QPCs in an in-plane magnetic field is presented and discussed. The dependence of ZBPs height on nuclear polarization confirms the spin-related nature of this feature. This result also indicates that the subband-motion model must also be spin-related to account for the behaviors of ZBPs. In Ch. 4, the spin-dependent motion of subbands is confirmed and imaged by the differential thermopower spectroscopy. This is a technique that has several advantages over the commonly used transconductance spectroscopy. Using this technique, subbands of different spin species are clearly shown exhibiting different behaviors. Specifically, the spin-down subbands pass through the chemical potential without pinning, whereas the spin-up subbands pin above the chemical potential before passing through. The magnetic field difference of subband motion provides an explanation for behaviors of ZBPs. Below G = 2e2 /h, thermopower spectroscopy reveals features that may arise from localized states and spin-related many-particle interactions.  30  Chapter 2  Zero-bias anomaly in low and high-conductance regimes 2.1  Introduction  Zero-bias anomaly in nonlinear conductance of QPCs has stimulated considerable interests since it was suggested as the signature of Kondo physics in low-density 1D systems [7]. In Kondo picture, the 0.7 structure, as well as its dependence on temperature and magnetic field, can all be attributed to the suppression of Kondo enhancement in QPC conductance. The Kondo model is successful in describing most effects associated with the 0.7 structure, however, some recent measurements reveal characteristics of zero-bias peaks (ZBPs) that do not agree with this picture [78, 79]. This casts doubts on the relation between ZBPs and the Kondo model, and consequently the Kondo explanation to the 0.7 structure. To discern the nature of ZBPs, it is necessary to perform a systematic measurement on this feature and compare results with available theoretical models. The reason why narrow conductance peaks around zero bias are called “anomalies” lies in their deviation from the single-particle model. In the single-particle picture, QPCs are commonly modeled as saddle-point constrictions with adiabatic transmission of electrons (see Sec. 1.5). Figure 2.1b shows the calculated differential conductance G for various electrostatic potentials using this model, and Fig. 2.1a shows experimental data for ZBPs. A comparison of Figs. 2.1ab for 31  (a)  (b)  EXP  CALC  G (e2/h)  6  4  2  0  -1000  0  1000  -1000  0  1000  Vdc (µV)  Figure 2.1: (a) Measured differential conductance versus d.c. bias, Vdc , from the third plateau down to pinch-off at T = 40mK and B|| = 0T. The arrow marks a narrow ZBP above 2e2 /h. Bold and dashed lines are selected for comparison to saddle-point calculations. (b) Calculated conductance traces, using the saddle-point model discussed in Sec. 1.7.3, for a range of electrostatic potentials at B|| = 0T. Subband energy is assumed staying fixed with respect to the center of bias window, so γ = 0 and η = 0 are used in (Eq. 1.29) Except where noted, parameters of the QPC adopted throughout this chapter are h¯ ωx = 0.6meV, h¯ ωy = 1.3meV and β = 0.5.  G < 2e2 /h reveals a striking difference between the two: the calculated differential conductance shows a smooth and broad zero-bias peak that gradually turns into a dip as the conductance is lowered, whereas the experimental data shows sharp and narrow ZBPs from just below 2e2 /h all the way down to pinch-off. Most previous measurements of ZBPs in point contacts have focused on the medium-conductance regime (1e2 /h  G < 2e2 /h), where the 0.7 structure is ob-  served. Within this conductance regime, occurrence of the ZBPs can be successfully attributed to Kondo screening of an impurity that forms self-consistently in the QPC [7, 16]. Only a few papers discuss ZBPs that persist down into the low-conductance regime (G ≪ e2 /h) [7, 79], and it is tempting to attribute the same mechanism to the formation of ZBPs at low conductance. As pointed out  32  in Refs. [7] and [79], however, there are several quantitative differences between medium-conductance ZBPs and those below G ∼ 1e2 /h. For example, ZBPs in both low- and medium-conductance regimes split with in-plane magnetic field, but the magnitude of the splitting is very different well above and well below G ∼ 1e2 /h [79]. Less attention has been paid to details of differential conductance of a QPC in the high-conductance regime (G > 2e2 /h). There have been occasional reports in the experimental literature of “1.7 structure” [49, 50, 77, 82], and attempts to fit such a feature within a consistent theoretical framework [11, 16]. But this 1.7 structure is not clearly visible in many QPCs, and there is no consensus as to whether it is an intrinsic phenomenon. Nonlinear conductance features in this regime have received even less attention, apparently in the belief that they are accounted for by the single-particle model [81]. But going back to the first reports of zero bias anomalies in QPCs [7], sharp features are often observed at zero bias near the midpoint of the second and the third conductance riser, i.e. near 1.5 × 2e2 /h and 2.5 × 2e2 /h (solid bold curve in Fig. 2.1a), which are inconsistent with the singleparticle model (solid bold curve in Fig. 2.1b). In this chapter, we present the magnetic field and temperature dependencies of ZBPs from defect-free quantum point contacts in both low- and high-conductance regimes. Different from ZBPs in the medium-conductance regime, where a singleparticle mechanism also contributes to ZBPs and it must be subtracted before attempting to discern many-particle physics from these features, ZBPs at low conductance and near the midpoints between plateaus are absent in the single-particle picture (Fig. 2.1b), making them ideal for studying the many-body effects in QPCs. The chapter is organized as follows: devices and measurement setup are described briefly in sec. 2.2. Magnetic field and temperature dependencies of ZBPs in the low- and high-conductance regimes are presented in Secs. 2.3 and 2.4, respectively, and compared with those of medium-conductance ZBPs. In Sec. 2.5, implications of the measurement results on theoretical models are discussed.  33  2.2  Devices and measurement setup  Three 1µ m-long and six 0.5µ m-long QPCs were defined by electrostatic gates on a GaAs/AlGaAs heterostructure with a two-dimensional electron gas (2DEG) 110 nm below the surface. The lithographic width of the QPCs was 225 nm. At T = 1.5K, the electron density and mobility of the 2DEG were ns = 1.11 × 1011 cm−2 and µ = 4.44 × 106 cm2 /Vs, respectively. Differential conductance measurements were performed in a dilution refrigerator with base electron temperature ∼ 40mK, using a lock-in with a d.c. source-drain bias superimposed on top of a 10µ V a.c. excitation, i.e., Vbias = Vdc + Vac . A magnetic field up to 12T was applied within 0.5◦ of the plane of the sample, so the primary effect from the field was through the induced spin-splitting. Data presented in this chapter were measured over a range of gate voltages Vg , source-drain d.c. bias Vdc , temperature T and in-plane magnetic field B|| . For some cooldowns the in-plane field was aligned along the QPC axis, and other times perpendicular to it, but no consistent effect of field orientation was observed. The data for the figures in this chapter came from three different devices; consistent behaviors were observed in all nine devices, independent of length.  2.3  ZBPs in low-conductance regime  Differential conductance signatures of the QPCs in this experiment were similar to those reported across the literature (Fig. 2.2a) [7, 79]. ZBPs were observed from just below the first plateau (G = 2e2 /h) all the way down to pinch-off. A logarithmic plot of G (Fig. 2.2b) shows that the ZBPs could be resolved down to 10−4 e2 /h, the lowest conductance measured in this experiment. At this low conductance, the detailed shape of differential conductance curve was found to differ between QPCs and even cooldowns. In many cases the ZBP was very sharp, with peak conductance as large as five times higher than the conductance off-peak even below G ∼ 10−3 e2 /h (Fig. 2.2c). The strength and visibility of the ZBP below 0.1e2 /h depended on device details, but all measured QPCs showed ZBPs at least down to 10−1 e2 /h. We conclude, therefore, that the presence of a ZBP in the lowconductance limit is a universal characteristic. ZBPs can be characterized by a full width at half maximum (FWHM) and a peak height, δ G. Previous reports have consistently shown that the FWHM de34  100  G (e2/h)  (a)  10-3 10-4 (b)  -1000  1.0  0.5  1.0  -1000  0  1000  Vdc (µV)  0  Vdc (µV)  1000  (c)  0.5  0.0  0.0  -100  0  100  Vdc (µV)  300 (d) 250  0.7×2e2/h  0.4 FWHM δG/Gmax  200  0.2  150  0.0 10  -4  -3  10  -2  10  -1  10  δG/Gmax  FWHM (µV)  10-2  G (×10-3e2/h)  G (e2/h)  1.5  10-1  0  10  Gmax(e2/h) Figure 2.2: (a) Differential conductance versus source-drain d.c. bias, Vdc . Each trace represents a different gate voltage, Vg , evenly spaced with intervals of 1mV, at T = 40mK and B|| = 0T. (b) Logarithmic plot for data in (a). The ZBP was clearly resolved down to 10−4 e2 /h. (c) An example of sharp ZBP observed below 10−3 e2 /h. (d) FWHM (left axis) and the relative peak height δ G/Gmax (right axis) of ZBPs versus conductance maximum, Gmax , extracted from data shown in panels (a) and (b). Both FWHM and δ G/Gmax show a minimum around 0.7 × 2e2 /h and remain basically flat in the low-conductance regime.  35  creases monotonically as G drops from 2e2 /h to ∼ 0.7 × 2e2 /h [7]. Below this conductance, there is a sharp rise in FWHM, which then remains constant down to pinch-off [7, 44, 79]. This behavior is clearly seen in Figs. 2.2b and 2.2d. Low conductance ZBPs from the devices in this experiment had FWHMs within the range 80 − 200µ V. The peak height, δ G , can be defined as the difference between the conductance on top of the peak, Gmax , and the average of local minima on either side. Existing literature describes a similar non-monotonic dependence of δ G on Gmax , with a local minimum at G ∼ 0.7 × 2e2 /h [79]. One significant feature of δ G that can be easily seen in log-scale plots such as Fig. 2.2b, but to our knowledge has not been previously pointed out, is that the relative peak height δ G/Gmax saturates at a value that does not change over orders of magnitude in conductance (see also Fig. 2.2d). This remarkable consistency of relative peak height over a wide range of conductance was observed in many QPCs and cooldowns. The saturation value varied within the range 0.3 − 0.8 from device to device. A similar saturation of  δ G/Gmax can be noted in Fig. 2b of Ref. [79] but was not discussed in that work.  2.3.1 Magnetic field dependence Zero-bias conductance peaks in QPCs are suppressed by in-plane magnetic fields on the scale of several Tesla—a phenomenon observed in this experiment and consistent with reports from across the literature. For ZBPs above 1e2 /h, a splitting was often observed before the peak was fully suppressed. The magnitude of the splitting, ∆pp , was typically between (3 − 5)Ez [7, 79], where Ez = |gµB B|| | is the Zeeman energy using the bulk GaAs g-factor, g = −0.44. We compare ∆pp to Zeeman rather than orbital energy scales because the magnetic fields were applied in the sample plane, causing relatively minor orbital effects. As the gate voltage was tuned to bring G below 1e2 /h the splitting in all devices dropped to less than 2Ez , consistent with the gate voltage dependence of the splitting reported in Ref. [79]. For even lower conductances, however, ∆pp saturated to a value that did not change down to pinchoff (see, e.g., Fig. 2.3b) [83]. The detailed magnetic field dependence of the ZBPs below G ∼ 1e2 /h varied widely from device to device, even for lithographically-identical QPCs free of dis-  36  (a)  0T (b)  5T (c)  9T  100  G (e2/h)  10-1  10-2  10-3  -200  200  -200 200  -200  200  Vdc (µV) 0T 4T  G (e2/h)  10-1  8 6 4 2  1.5T 7T  10  -1  0T 2.5T  1T 5T  0  200  8 6  (d) -200  0  200  Vdc (µV)  (e) -200  Vdc (µV)  Figure 2.3: Evolution of the low-conductance ZBPs in an in-plane field. (a) B|| = 0T, (b) B|| = 5T, and (c) B|| = 9T. Individual traces in (a-c) represent evenly-spaced gate voltages as in Figs. 2.2a and 2.2b, with every other trace removed above G = 1e2 /h for clarity. (d) An example of a ZBP with splitting much less than 2Ez . (e) An example of a ZBP that collapses before clear splitting is observed. In panels (d) and (e), gate voltages are different for different fields, chosen to maintain the conductance at Vdc = −400µ V. Traces for B|| > 0T are offset vertically for clarity.  37  order (resonances). These diverse behaviors may help explain the range of reports that have appeared in the literature [7, 78, 79, 83]. Figure 2.3 summarizes the magnetic field dependencies that were observed in this experiment, all for ZBPs with similar zero-field widths and heights. In Fig. 2.3b, a clear splitting is observed at intermediate field, and the magnitude saturates to 1.8Ez in the low-conductance regime. In Fig. 2.3d, a splitting is again easily seen at 7T , but the magnitude is only 0.5Ez . In Fig. 2.3e, the peak collapses much more rapidly with field, reducing in height by 66% at 1T compared to 15% at 1.5T in Fig. 2.3d. Small bumps consistent with remnants of a split ZBP are visible at 5T, but the visibility of these features is qualitatively worse than in the other two devices. As seen in these examples, the magnitude of splitting and resilience of the ZBP in a finite magnetic field are not clearly correlated. Some of the factors that influence ZBP splitting were explored in Ref. [78], where a transition from splitting to non-splitting behavior was reported by laterally shifting the QPC. In the present experiments, it was observed that ZBP splitting in some QPCs changed from hour to hour with other parameters held unchanged, apparently due to minor rearrangement of dopant potentials that were too small to affect the zero-field conductance. This observation indicates that ZBP splitting is exquisitely sensitive to the energy profile of the QPC, in contrast to the ZBP itself, which was observed in every QPC measured and whose shape was significantly less sensitive to fine details in the QPC potential.  2.3.2 Temperature dependence As temperature is increased, ZBPs become lower and eventually disappear, independent of conductance (Figs. 2.4a, c) [7]. Using a Landauer description of ballistic transport at bias voltage, V , and temperature, T [3], ∫ ∞  I=  −∞  dE t(E,V, T )[ f (E, T ) − f (E + eV, T )],  (2.1)  it is seen that the suppression of ZBPs at high temperature can result from broadening of the Fermi functions, f (E, T ), or from temperature-dependent changes in the transmission coefficient t(E,V, T ), or both. To distinguish these effects, experimental data up to 650mK are compared to calculations that include thermal 38  4  (a)  G (e2/h)  2  40mK 70mK 150mK  CALC 300mK 650mK  10-1  2  (b)  10-1 6  6 4  4  2  2  10-2  10-2 -200  0  -200  200  1.7 (c) 1.6  0  200  Vdc (µV)  Vdc (µV)  G (e2/h)  EXP  CALC  1.6  1.5  1.4  1.5 1.4 1.3  1.4  (d)  EXP  1.2 -200  0 Vdc (µV)  200  -200  0  200  Vdc (µV)  Figure 2.4: Evolution of the ZBPs with temperature from T = 40mK to 650mK. (a, c) Experimental data for ZBPs in the low-conductance (a) and medium-conductance regimes (c). (b, d) Calculation results for ZBPs in the low-conductance (b) and medium-conductance regimes (d). For each group of curves in (b) and (d), ZBPs at T ≥ 70mK were approximated by substituting the measured G(V, T =40mK) for G(V, T = 0) in Eq. (2.2).  39  broadening but exclude any temperature dependence of t (Fig. 2.4). Assuming that the voltage bias drops equally on both sides of the QPC [81, 84] and that the barrier itself is not directly affected by the applied bias, differential conductance at finite temperature G(V, T ) can be expressed as the convolution of its zero-temperature value G(V, T = 0) with the derivative of the Fermi function: ∫ ∞  G(V, T ) =  −∞  dV ′ G(V ′ , 0)  ( ) ′) ∂ f µ + (V −V , T 2  ∂V ′  ,  (2.2)  where µ is the chemical potential. In the low-conductance regime, the simulation results (Fig. 2.4b) closely resemble the experimental data (Fig. 2.4a), indicating that thermal broadening due to f (E, T ) is the major contributor to the suppression of ZBPs. In the mediumconductance regime, however, conductance around Vdc = 0 in the measurement (Fig. 2.4c) is substantially lower than in the calculation (Fig. 2.4d), and cannot be accounted for simply by thermal broadening. This suggests that the functional form of t(E) is directly affected by temperature at moderate conductance, but not at low conductance. This picture will be proven true in Ch. 4 where the temperature dependence is found to be non-uniform across low- and medium-conductance regimes (see Fig. 4.6).  2.4  ZBPs in high-conductance regime  Figure 2.5 examines the anomalous ZBPs in the high-conductance regime in more detail, showing the striking difference between calculation based on the saddlepoint model and experimental data near 3e2 /h (Figs. 2.5a,b) and 5e2 /h (Figs. 2.5d,e). The saddle-point model predicts a transition from peak to dip at 3e2 /h, but experimentally ZBPs are clearly observed down to ∼ 2.6e2 /h. The narrow features near zero bias that are visible in Fig. 2.5 were robust to changes in the relative gate voltage applied to the split gates, and were observed in all devices. Analogous narrow features were frequently observed at the next higher transition between plateaus, between 4e2 /h and 6e2 /h. The distinction between saddle-point model and experiment for this transition was primarily in the cusp at zero bias above 5e2 /h. As in Fig. 2.5d, ZBPs were seldom visible below the midpoint (G = 5e2 /h). In transi40  3.5 (a)  EXP  (b)  CALC (c)  CALC  G (e2/h)  3.0  2.5  γ= 0, η= 0  (d)  EXP  (e)  γ= 0.05, η= -0.12  CALC (f)  CALC  5.5  5.0  -500 0 500  γ= 0, η= 0  γ= 0.02, η= -0.12  -500 0 500 Vdc (µV)  -500 0 500  Figure 2.5: Experimental data and calculation results for nonlinear conductance around G = 2e2 /h (a-c) and G = 4e2 /h (d-f). Panels (a) and (d) are close-ups of experimental data shown in Fig. 2.1b. Calculations presented in (b) and (e) assume fixed subband energy with respect to the center of bias window, while in (c) and (f) include effects of subband motion with increasing Vdc (see Sec. 1.7.3). Calculation parameters are indicated in the figure (the unit of η is meV−1 ).  tions between even higher plateaus, narrow ZBPs were difficult to distinguish from measurement noise. The magnetic field dependence of high-conductance ZBPs is summarized in Fig. 2.6. Similar to their analogs below e2 /h, narrow ZBPs between 2.6e2 /h and 3e2 /h were rapidly suppressed with increased magnetic field, and eventually became zero-bias dips for large in-plane field [7]. One thing different is that the narrow ZBPs here did not display any splitting before collapse in magnetic field. However, splitting at finite field was a very solid phenomenon for broad ZBPs above 3e2 /h. This splitting increased with magnetic field, and eventually evolved into a conductance plateau at 3e2 /h (Fig. 2.6d). Measured by Ez , the Zeeman energy in bulk GaAs, the magnitude of peak splitting in Figs. 2.6cd ranges from 5 41  (a)  EXP (b)  3.5 (e)  CALC (f)  EXP (c)  EXP (d)  EXP  3.5 3.0  G (e2/h)  2.5 0T  2T  4T  CALC (g)  6T  CALC (h)  CALC  3.0  2.5  Ez = 0meV  -500 500  Ez = 0.2meV  -500 500  Ez = 0.6meV  Ez = 0.4meV  -500 500 Vdc (µV)  -500 500  Figure 2.6: Evolution of ZBPs within (2 − 4)e2 /h in an in-plane magnetic field. (a-c) Experimental data measured at (a) 0T, (b) 2T, (c) 4T and (d) 6T. Bars in panels (c) and (d) indicate the scale of 2Ez at corresponding B|| . (e-h) Calculation results using the saddle-point model with Zeeman energy (e) Ez = 0meV, (f) 0.2meV, (g) 0.4meV and (h) 0.6meV. Parameters γ = 0 and η = 0 are used in the calculations.  to 6Ez , larger than that of 3 ∼ 5Ez observed within (0.7 ∼ 1) × 2e2 /h and that of 0.5 ∼ 1.8Ez observed in the low-conductance limit [7, 79, 85].  2.5  Comparison with theoretical models  The ZBP is frequently discussed in connection with a possible Kondo effect in QPCs [7, 16]. Existence of a symmetrically-bound localized state in the middle of constriction has been justified by spin-density-functional theory (SDFT) for QPCs in the medium-conductance regime (Fig. 2.7a) [16]. Kondo screening of this state would lead to enhanced conductance when the temperature and applied bias are less than the Kondo temperature, TK , giving a ZBP with width ∼ 2kB TK /e. In the low-conductance regime, the fact that the ZBP width remains constant, from 42  (a)  (c) 0  10  10-1  G (e2/h)  (b) 10-2 10-3 10-4 10-5 -2  -1  0  1  2  Vdc (mV)  Figure 2.7: (a) Energy profile of a symmetric Kondo-type localized state. (b) Top: geometry of a symmetric QPC with two asymmetric Kondo-type localized states on either end. Bottom: energy profile of this QPC. Arrows indicate the position of localized states. (c) Calculated differential conductance versus d.c. bias using the saddle-point model. The subband energy is assumed to rise linearly with Vdc at a rate of 0.2 meV/mV (γ = 0.2, η = 0).  G < 0.7 × 2e2 /h all the way down to 10−4 e2 /h, then implies that TK is not affected by the overall conductance. But TK scales exponentially with the coupling of the localized state to the leads, so it is difficult to explain the insensitivity of TK to QPC conductance over three orders of magnitude [67, 86]. A way around this seeming inconsistency, and still within the framework of Kondo physics, could be that localized states in QPCs with low conductance are coupled to leads through strongly asymmetric barriers [87]. This scenario is supported by SDFT, which predict asymmetrically-bound localized states on either end of a QPC near pinchoff (Fig. 2.7b) [16]. The localized states are separated by an opaque barrier that would limit the overall conductance, but each connected to the reservoir through a transparent barrier that sets TK [88]. Since G and TK are determined by different barriers, their dependence on Vg could, in principle, be different. One piece of experimental evidence supporting the asymmetric Kondo model is that ZBPs at low conductance are often observed to be somewhat asymmetric, 43  and not to be centered at exactly zero bias; similar features have been observed in quantum dots with asymmetric contacts [89, 90]. In the high-conductance regime, SDFT predicts that the formation of a localized state in the constriction is also possible but very sensitive to QPC parameters [16]. This is the reason why not all QPCs exhibit the 1.7 structure. However, experimentally the narrow ZBPs near 3e2 /h and 5e2 /h are a fairly solid observation across all measured QPCs, which does not agree with the predicted instability of the localized state at high conductance. A classic signature of Kondo-related zero-bias peaks is that they split by 2Ez in magnetic field: this has been referred to as the “smoking gun” of Kondo effect [91]. Despite the frequent observation of splitting in a magnetic field, in the present experiment and others [7, 78, 79], this smoking gun is less than convincing because the expected splitting magnitude, 2Ez , is observed in none of the low, medium or high-conductance regimes. Effective g-factor, g∗ , could be extracted by comparing the actual peak splitting to the expected 2Ez . Results of |g∗ | show oscillation as the subband is filled one by one: |g∗ | is lower than 0.44 for G ≪ 1e2 /h, and increases to greater than 0.44 for 1e2 /h  G  2e2 /h; as the second subband is filled, |g∗ |  drops again because no splitting was observed for narrow ZBPs within the range 2.6e2 /h < G < 3e2 /h, and then increases to greater than 0.44 for 3e2 /h < G < 4e2 /h, etc [92]. The enhanced g-factor observed in the medium-conductance regime (1e2 /h G  2e2 /h) has often been interpreted as splitting of a Kondo-related ZBP with  the exchange-enhanced g-factor that defines subband splittings in QPCs and lowdensity 2DEGs [6, 7]. This reasoning cannot help to explain the reduced g-factor in low and high-conductance regime. One explanation for peak splitting less than 2Ez would be if the 2DEG wavefunction penetrated significantly into the AlGaAs layer, but this effect is expected to be significant only when electron density is high or the 2DEG is close to the surface [93], neither of which are the case in this experiment. Alternatively, theoretical calculations that consider details of the peak shape at finite bias predict a somewhat reduced peak splitting ∆pp ∼ 4/3Ez when Ez is on the order of kB TK and ∆pp ∼ 1.7Ez even at Ez ∼ 100kB TK [94], but this is still much larger than the peak splitting observed in some ZBPs (e.g., Fig. 2.3d). A comparison between experimental data and calculation results for broad 44  ZBPs within the range 3e2 /h  G < 4e2 /h suggests that ZBP splitting in magnetic  field does not necessarily involve the many-body Kondo effect. In Fig. 2.6e-h, differential conductance calculated based on the saddle-point model shows a broad ZBP in this region that splits into two side peaks with increasing magnetic field. This behavior qualitatively agrees with those of broad ZBPs observed in experiments (Fig. 2.6a-d), indicating a single-particle mechanism for the peak splitting within 3e2 /h  G < 4e2 /h. Such a mechanism should also contribute to ZBP split-  ting in the medium-conductance regime. Here we do not claim that ZBP splitting can be completely explained within the single-particle picture, as the simple model is only of qualitative significance. Many-body interaction certainly plays a role in the phenomenon, especially in the low-conductance regime where the singleparticle model predicts no splitting. Further investigation is needed to determine whether this many-body interaction is Kondo type or not, since ZBP splitting can no longer be viewed as a signature of this effect. A recent paper by Chen et al. [78] suggested that the occurrence of ZBPs in transport properties of QPCs does not need to have the Kondo mechanism either, because they can be reproduced with the saddle-point model if subband energy is assumed to rise with Vdc for the sake of minimizing the Coulomb energies (see Sec. 1.7.3). The Coulomb interaction is especially strong in the low-conductance regime due to the low electron density and the weak screening effect. Figure 2.7c shows calculation results of the saddle-point model, assuming subband rising linearly at a rate of γ = 0.2meV/mV. Calculated nonlinear conductance shows a clear ZBP persisting down to the low-conductance limit with constant FWHM and δ G/Gmax (Fig. 2.7c), agreeing remarkably with the experimental results in Fig. 2.2b. In the high-conductance regime, the electron density is much higher and the screening is greatly enhanced, hence the effect of interactions is much weaker. By setting a tiny rate of γ = 0.05meV/mV in the saddle-point model, sharp and narrow ZBPs are reproduced from 3e2 /h to 2.6e2 /h (Fig. 2.5c). These ZBPs become exponentially low when conductance approaches 2e2 /h, therefore cannot be resolved below about 2.6e2 /h and only conductance dips are observed. When the third subband is occupied, screening is even stronger and γ is much smaller, thus ZBPs are hardly observed below 5e2 /h. As even more subbands are occupied, γ approaches 45  0 and subband do not rise with Vdc any more. The single-particle picture recovers in this limit, agreeing with calculations for an adiabatic constriction [30]. The above scenario provides an explanation for the existence of ZBPs at low, medium and high-conductance regimes, however, the spin degree of freedom is totally absent in the model, suggesting that subband energies should rise even for spin-polarized carriers. In this sense, ZBPs would not disappear with increasing magnetic field, which conflicts with the experimental observations. In the next chapter, we will present measurements of nuclear polarization in QPCs, which provide further evidence demonstrating the spin-related origin of ZBPs. In Ch. 4, experimental data of thermopower spectroscopy will clearly show the spin dependence of subband motion.  46  Chapter 3  Nuclear polarization in an in-plane magnetic field 3.1  Introduction  Despite extensive experimental and theoretical work to understand the electron spin physics of QPCs below the 2e2 /h plateau, the effects of nuclear spin on QPC conductance have only been studied deep in the quantum-Hall regime [95], where many of the interaction effects discussed in Sec. 1.6 disappear. Over the last decade, however, it has become increasingly clear that understanding the electron spin physics of semiconductor nanostructures requires a careful consideration of the influence of nuclear spin via the hyperfine interaction [96, 97]. This is especially true for nanostructures defined in GaAs and other III-V materials, where large atomic masses lead to a large electron-nuclear coupling constant through the Fermi contact interaction [98]. The hyperfine interaction gives rise to an effective magnetic field acting on electron spin that is proportional to the local nuclear spin polarization. Significant nuclear polarizations can be built up, also via the hyperfine interaction, when a nonequilibrium population of electron spins relaxes, flipping nuclear spins to conserve angular momentum: a process known as dynamic nuclear polarization (DNP) [99]. For example, a large d.c. bias applied between spin-polarized edge states in the quantum Hall regime often leads to DNP, which can then change transport 47  characteristics significantly. This mechanism has been used to generate and detect nuclear polarization in QPCs, giving rise to hysteresis in the conductance as a function of d.c. bias [95]. In this chapter, we show that DNP in QPCs is not limited to the quantum Hall regime: it is a generic feature of nonequilibrium transport through a QPC in an external magnetic field. The magnetic fields applied in our measurements were primarily in the plane of the electron gas; the out-of-plane component was much too small to give rise to Landau quantization. Nuclear polarization is manifested by hysteresis in the differential conductance as a function of d.c. bias across the QPC, with polarization and relaxation timescales that are consistent with other nanostructures in GaAs [95, 100, 101]. DNP leads to changes in conductance features such as ZBPs, which provides evidence for their their spin correlation. The rest of this chapter is organized as follows: Secs. 3.2 and 3.3 introduce hyperfine interaction and DNP in the quantum-hall regime. In Sec. 3.4, evidence is shown for nuclear spin polarization in QPCs in an in-plane magnetic field. Analysis on this effect is performed in Sec. 3.5.  3.2  Hyperfine interaction  Electrons in solids experience interactions not only with each other but also with nuclei. The interaction between electrons and nuclei is termed “hyperfine interaction”, and the corresponding Hamiltonian is written as N  HHF = ∑ Ak Ik · S,  (3.1)  k  where Ik and S are the spin operator for nucleus k and the electron spin, and Ak is the hyperfine coupling constant. For the conduction band in III-V semiconductors, electron state is primarily s-type and HHF is dominated by the so-called Fermi contact interaction. Given a nucleus k located at rk and the electron wavefunction  ψ (r), the contact interaction has the form [98], Hck =  8π µ0 k γe γ h¯ |ψ (rk )|2 Ik · S, 3 4π n  48  (3.2)  G (e2/h)  0.8  0.7  -0.5  0  Vdc (mV)  0.5  Figure 3.1: Inset: schematic of the device used in Ref. [95]. Measurements were taken in the quantum-Hall regime with filling factor ν = 2. The dark regions 1, 2, 3 are ohmic contacts, and A, B, C are gates. The spinresolved edge channels are represented by curved lines. Main panel: differential conductance versus Vdc for the QPC formed by gates B and C. There is hysteresis in two sweep directions (indicated by arrows) of Vdc . (Figure modified from Ref. [95])  where γe = −2µB and γnk are the gyromagnetic ratios for a free electron and the nucleus k, respectively. In a material containing several different nuclear isotopic species, it is common to define an average hyperfine coupling constant A so that Ak = ν A|ψ (rk )|2 with ν the volume of a crystal unit cell containing one nuclear spin. In GaAs, the three naturally occurring isotopes,  69 Ga, 71 Ga  and  75 As,  have  the same nuclear spin I = 3/2 and the relative abundances are respectively 0.3, 0.2 and 0.5, giving rise to A ∼ 90µ eV. If the nuclear spin in GaAs is fully polarized, it leads to an fairly strong effective magnetic field of |IA/g∗ µB | ≈ 5T.  3.3  Dynamic nuclear polarization in quantum-Hall regime  Because the hyperfine interaction couples electron and nuclear spins, a flip of an electron spin gives rise to a simultaneous flop of a nuclear spin in the opposite direction so as to conserve the net spin of the entire system. By using transport [95– 97], optical [102], or resonance methods, nonequilibrium population of electron spins can be induced in the GaAs 2DEG. Once the pumping is turned off polarized electron spins relax to their equilibrium states, leading to “flip-flop” scattering that  49  transfers angular momentum from electron to nuclear spins and hence builds up significant nuclear polarization. This process is known as the dynamic nuclear polarization (DNP) [99]. In the quantum Hall regime, nonequilibrium electron spin population is created in spin-polarized edge channels. When a large d.c. bias, Vdc , is applied between edge channels, electrons flow from one edge channel to another and flip their spins, which triggers flops of nuclear spins in the vicinity and thus generates a net nuclear polarization. This nuclear polarization affects the electron transport through QPCs, giving rise to a characteristic hysteresis in the two sweep directions of Vdc (Fig. 3.1) [95].  3.4  Evidence for nuclear spin polarization in an in-plane magnetic field  In this section, we present evidence that DNP can also occur in QPCs in an in-plane magnetic field. Devices and measurement set-up in this experiment are the same as those described in Sec. 2.2. Nonlinear conductance characteristics of measured QPCs were typical of reports in the literature for low-disorder QPCs, both at zero field and high field (see, e.g., Fig. 2.1a). A clear ZBP was observed throughout the range 0.01e2 /h  G  2e2 /h at zero field. The ZBP collapsed at high field, and a  spin-resolved plateau appeared at 1e2 /h. Looking more closely at the high field data, however, the differential conductance was slightly dependent on sweep direction (Fig. 3.2a), giving rise to hysteresis in traces of G as Vdc was swept from negative voltage to positive and back to negative. The sweep rate used to gather these hysteresis curves was 8µ V/s, chosen to be fast enough that relaxation was minimized during the sweep, but slow enough that artificial hysteresis due to a lag in instrumentation was eliminated. Subtracting off the average of the two sweep directions, Gavg , the hysteresis is visible primarily within the range |Vdc |  200µ V, corresponding to the bias window where the ZBP  is observed (Fig. 3.2b). The magnetic field and time dependence of the hysteresis provide insight into its origin (Figs. 3.2c and 3.3). Hysteresis was absent at zero magnetic field, then grew with field on a scale of hundreds of mT. The fact that hysteresis appears only 50  1.0  4 5  0.5 0.0  1 2 3 4 5  6 7  -200  (c) 0.4 G (e2/h)  (b) 1 2 3  G - Gavg (e2/h)  B|| =4.9T  0  200  -200  0  200  Vdc (µV) scan direction  0.12T  -200µV to 200µV 200µV to -200µV  0.3  6 7  0.02 e 2/h  0.48T  Ph (e2/h)  G (e2/h)  (a) 1.5  0.02 0.01 0.00 0.0  0.5 1.0 B|| (T)  1.00T 1.46T  -200  0 Vdc (µV)  200  Figure 3.2: (a) Differential conductance curves showing hysteresis for Vdc swept from negative to positive (solid) and from positive to negative (dashed). Labels 1-7 denote curves at different Vg . (b) Hysteresis measured as deviation from average conductance (described in text) for corresponding curves 1-7 in (a). Arrow indicates vertical scale. (c) Evolution of ZBP and hysteresis in an in-plane magnetic field B|| from 0.12T to 1.46T (different QPC to that used for panels a and b). Inset: B|| dependence of peak-height difference ∆Ph extracted from curves in main panel of (c)  in the presence of a field suggests that it is related to spin instead of, for example, to thermal effects or bias-induced switching of charged dopant sites. Hysteresis timescales were measured by applying |Vdc |  50µ V for a time TP ,  then switching Vdc rapidly to zero and monitoring the conductance over several minutes. Fig. 3.3a shows relaxation traces associated with TP from 20s to 1800s. 51  TP = 180s  TP = 600s TP = 1800s  0.4 0.3 -200  TP = 20s  200 Vdc(µV)  0.02 0  1000 (b)  0..02  t (sec)  B|| = 4.9T Vdc = +300µV  0..01  2000 G (e2/h)  G - Geq (e 2/h)  TP = 60s  0.04  0.00 G - Geq (e 2/h)  B|| = 4.4T  G (e2/h)  (a)  0.06  3000  1.4 1.2 1.0 -200 V (µV)200 dc  0..00 Vdc = -300µV  0  200  t (sec)  400  600  Figure 3.3: (a) Relaxation of zero-bias conductance away from its equilibrium value, Geq , for various TP at B|| = 4.4T (dot). Solid lines are fits of data to Eq. (3.3). Curves are offset horizontally by 500s for clarity. Inset: zero-bias peak measured at the same gate voltage and B|| . (b) Fits of Eq. (3.3) (solid) to conductance relaxation curves (dot) of another QPC defined in the same wafer. Black and grey curves represent polarizing bias Vdc = −300µ V and +300µ V, respectively. Inset: hysteresis measured at the same gate voltage and B|| .  The conductance relaxes over time in all traces, with a typical time constant on the order of 100s. This time constant is much longer than relaxation times associated with electron spin in GaAs, but consistent with previous reports of nuclear spin relaxation in GaAs 2DEG nanostructures [95, 100, 101]. These relaxation measurements suggest that the hysteresis cannot be due to electron spin polarization, leaving nuclear spin as a likely explanation.  52  3.5  Theoretical analysis and discussion  The observation of similar hysteresis and conductance relaxation effects in all QPCs measured demonstrates that nuclear polarization is a generic feature for QPC nonlinear transport in an in-plane magnetic field. When the nuclear polarization is small, the change in zero-bias conductance away from its equilibrium value, Geq , can be assumed to be proportional to nuclear polarization. A careful analysis of the conductance change during the nuclear spin relaxation process can therefore shed light on the relaxation mechanisms involved. Nuclear spin relaxation in GaAs nanostructures occurs through spin-lattice relaxation, coupling with conduction electrons (Korringa relaxation) and by spin diffusion. Qualitative evidence for the importance of the diffusion mechanism can be found in the dependence of relaxation time on the high-bias polarization time, TP , as shown in Fig. 3.3a. The signal measured immediately after returning to zero bias (t = 0) grows with TP up to TP = 180s but saturates and even begins to decrease for longer polarization times. This implies that longer polarization times lead initially to larger nuclear spin polarization but, for TP  180s, the polarization rate is  matched by spin-lattice relaxation and out-diffusion. However, relaxation curves associated with longer TP consistently relax more slowly, indicating that the area over which nuclei are polarized continues to increase with polarization time and providing evidence for nuclear spin diffusion. Spin-lattice relaxation and spin diffusion can be included in the dynamics of nuclear spin polarization I(r,t) in a 3-dimensional system by solving:  ∂ I(r,t)/∂ t = D∇2r I(r,t) − I(r,t)/τR + S(r,t),  (3.3)  where the first term on the right-hand side represents nuclear spin diffusion with uniform rate D and the second term represents spin-lattice relaxation with characteristic time τR . Korringa relaxation effectively adds to the second term in Eq. (3.3), but the Korringa rate is expected to be an order of magnitude slower than the spinlattice rate at 40mK [103]. The last term, S(r,t) = I0 ζ (r)[θ (t) − θ (t − TP )], describes the QPC as a spatially localized source of nuclear spin polarization during 0 < t < TP , with geometry ζ (r) and polarization rate I0 . Approximating the QPC as a spherical source of nuclear polarization with ra53  dius a, ζ (r) = e−r  2 /a2  , Eq. (3.3) can be solved analytically. The parameters I0 ,  τR , and a were fit to the data for 5 different polarization times (Fig. 3.3a) using published values for the nuclear spin diffusion constant in GaAs, D ∼ 10−13 cm2 /s [98]. a and τR were constrained to be the same for all polarization times because they are related to GaAs material properties and the QPC itself. I0 was fit independently for each curve, reflecting the possibility that the polarization rate may depend on TP [101, 104]. The spin-lattice relaxation rate extracted from these fits, τR = 2600 ± 100s, is consistent with NMR measurements of spin-lattice relaxation times in GaAs [105]. It is an order of magnitude longer than decay times seen in Fig. 3.3a, however, indicating that the contribution of spin-lattice relaxation is overwhelmed by that of spin diffusion. The value of a extracted from the fits is related to the volume over which polarization occurs. For the QPC in Fig. 3.3a, a was found to be 65 ± 5nm, similar to the Fermi wavelength λF = 75nm in the bulk 2DEG. Data from a second QPC defined on the same wafer yielded a = 42 ± 5nm (Fig. 3.3b), using the same τR and D. In practice, however, Eq. (3.3) ignores spatial dependence in the spin diffusion rate, which would be expected due to different electron densities [106]. Furthermore, the geometry of the polarizing region, ζ (r), is not known. It must not be thicker than the electron wavefunction transverse to the 2DEG, i.e., a disk rather than a sphere, and it may be closer to a rod than to a disk due to the 1D geometry of the QPC itself. More complicated ζ (r) require numerical solutions to Eq. (3.3). Taking into account numerical solutions of Eq. (3.3) for a range of reasonable geometries, the dimensions of polarization source must all be less than ∼ 5λF , indicating that polarization happens in the QPC rather than in the leads. The effect of nuclear polarization in Fig. 3.2c is similar to that of the external field: it creates an additional nuclear field, BN = Btot − B|| , in the total effective magnetic field, Btot , as one might expect from a uniformly polarized 2DEG. The direction of BN depended on the polarity of the polarizing bias (see also Fig. 3.3b), perhaps due to asymmetry in the QPC potential. Comparing the magnitude of the hysteresis with the effect of B|| , an estimate BN ∼ 0.1T at B|| = 0.48T can be extracted from Fig. 3.2c. Values up to BN ∼ 0.3T were found for other gate settings and other B|| . Another similarity between the effects of nuclear polarization and in-plane field in this measurement is the insensitivity of the conductance to 54  G (e2/h)  1.0 0.9 0.8 0.7  (a)  -200 1.29  0  200  (b)  -200  Vdc (µV)  1.08  (c)  1.28  B|| = 5.9T  0  200 1.6  (d) (g)  1.07  1.4 1.2  1.27 1.06  0.94  (e)  1.4  (f)  0.36  1.2 1.0  0.93  G (e2/h)  G (e2/h)  0.25 0.20 0.15 0.10  B|| = 3.9T  0.4 0.35  0  400  0 t (sec)  400  -200 200 Vdc (µV)  0.2  Figure 3.4: (a, b) Examples of hysteresis effect that are more complicated than those shown in Fig. 3.2c. (c)-(f) A transition between symmetric and antisymmetric conductance relaxation curves at B|| = 4.9T as readout conductance decreases from G = 1.27e2 /h to 0.35e2 /h while keeping the conductance during the polarization step unchanged at 1.27e2 /h. Black and grey curves correspond to Vdc = −300µ V and +300µ V, respectively. (g) Zero-bias peaks measured at the same readout gate voltage as in (c)-(f).  DNP at high bias. In the conductance regime explored in this paper (below the 2e2 /h plateau) the external field was observed to have a strong effect on the ZBP (|Vdc |  200µ V) but only a weak effect at higher bias. The absence of hysteresis  outside |Vdc |  200µ V can then be explained by an insensitivity of the differential  conductance to spin effects at low conductance and high d.c. bias. The hysteresis behavior in Fig. 3.2c, in which DNP mimics external field, is consistent with the assumption of Eq. (3.3) that nuclear polarization is generated at one site in the QPC, then diffuses to the surrounding area creating a broad re55  gion of polarization aligned uniformly parallel or anti-parallel to the external field. More complicated hysteresis curves require more sophisticated explanations, however. In Figs. 3.4a and 3.4b, the conductance peak shifts to slightly positive or negative bias depending on the sweep direction, but does not change significantly in height. A shift in the total effective magnetic field due to nuclear polarization cannot, by itself, explain this behavior. Taking all measured QPCs into account, no consistent correlation was observed between the type of hysteresis (as in Fig. 3.2c vs Figs. 3.4a and 3.4b) and the external parameters such as field, gate voltage, etc. Conductance relaxation measurements showed a non-trivial bias dependence that is similarly difficult to explain with a single field of nuclear polarization. For a given sign of bias (positive or negative), nuclear polarization depended only slightly on |Vdc | for |Vdc |  50µ V. Changing the sign of the applied bias, on the  other hand, sometimes led to significant changes. Relaxation curves for QPCs polarized under opposite biases were always symmetric for 1e2 /h  G  2e2 /h. In  Fig. 3.4c, for example, positive and negative bias polarizations both decrease in conductance as they relax. At lower conductance, the curves were often antisymmetric: as in Figs. 3.3b and 3.4e, positive and negative bias polarizations often relaxed in opposite directions. The origin of symmetric and antisymmetric behaviors was investigated by separating polarization and readout processes: polarizing at one conductance, rapidly changing the gate voltage to give a different conductance when bias was removed, then measuring relaxation at the new conductance. A transition from symmetric, to antisymmetric, and back to symmetric relaxation curves can be seen in Fig. 3.4c-f, covering a range of readout conductances but maintaining the same polarization conductance. Analogous transitions between symmetric and antisymmetric relaxation curves occurred for all QPCs, and at many different fields, but the readout conductance where the transitions occurred varied widely. In some QPCs the antisymmetry remained for polarization and/or readout conductance down to ∼ 0.1e2 /h. These data indicate that the same polarization state can lead to relaxation curves with arbitrary sign depending on the readout conductance, despite the fact that an external B|| always decreased the conductance. This observation is impossible to explain with a single field of nuclear polarization. It is necessary, therefore, to consider a more sophisticated model of nuclear spin configuration in 56  G (e2/h)  0.6  (a)  0.4  (b)  B|| = 0T B|| = 4.9T calculated ZBP at B|| = 4.9T  0.34  Vdc = -300µV  0.33  Vdc = +300µV  0.2 -200  0 200 Vdc (µV)  0  200 400 t (sec)  600  Figure 3.5: (a) Comparison between measured ZBP at B|| = 4.9T, and the expected curve based on a simple Kondo-like splitting. The dashed line was calculated by taking the ZBP at B|| = 0T and splitting it by 2gµB B|| , using g = 0.44. (b) Zero-bias conductance relaxation curves measured at settings for panel (a) (B|| = 4.9T), indicating dependence of zerobias-peak height on nuclear polarization in the QPC.  the QPC and the microscopic mechanism of how it is generated under high bias. Unlike measurements in the quantum Hall regime, there are no edge channels in an in-plane magnetic field, so the flip-flop scattering that leads to DNP cannot be due to scattering between spin-polarized edge channels as described in Ref. [95]. Instead, a mechanism known as spin injected dynamic nuclear polarization (SIDNP) may be responsible for the present data [107, 108]. SIDNP was studied in ferromagnetic (FM)/ nonmagnetic (NM) heterostructures [109], where spinpolarized current injected from the FM layer creates nonequilibrium spin magnetization in the NM layer and the nonequilibrium magnetization polarizes nuclei via the hyperfine interaction. Nonequilibrium magnetization can also be generated by injecting electrons through a spin-polarized QPC, then transferred to nuclei in the vicinity of the QPC by DNP. Considering both source and drain contacts, injection through a spin-polarized QPC more closely resembles a NM/FM/NM system than the NM/FM system from Ref. [109]—essentially two SIDNP junctions back to back. Applying a voltage to drive electrons from source to drain across a spin-up polarized QPC creates an excess of spin-up electrons in the drain, while leaving an excess of spin-down electrons in the source. Flip-flop relaxation of spin-up and spin-down electrons creates 57  opposite nuclear polarizations. One might therefore expect opposite nuclear polarizations on either end of the QPC, leading to a dipole field acting on conduction electrons. A full explanation for bias-dependent nuclear polarization would have to take into account this dipole field, as well as device-dependent asymmetries in the QPC itself. The in-plane field dependence of QPC conductance features has historically been used to discern whether or not they originate from spin-related effects. When the in-plane field dependence is simple (e.g. features split with a voltage corresponding to the Zeeman energy), this is a useful tool. But large in-plane fields also affect orbital electron characteristics, and the connection to spin is ambiguous when experimental data cannot be easily correlated with Zeeman energy. Nuclear polarization, on the other hand, affects only the spin degree of freedom for electrons moving through a QPC. For this reason, the question of whether a particular conductance feature is spin-related can be answered simply by observing whether it is affected by DNP. As an example, the splitting of the ZBP in Fig. 3.5 does not follow the predictions of a Kondo model [7], and in fact the field dependence is very weak up to ∼ 5T, and the dominant feature in the 4.9T data is a ZBP that is not split by magnetic field. One might, therefore, attribute the ZBP entirely to non-spin-related phenomena. But from the dependence of the ZBP height on nuclear polarization (Fig. 3.5b), a degree of spin-dependence in the feature can be confirmed. This observation does not rule out an additional contribution to the ZBP that does not depend on spin. A recent theoretical proposal in Ref. [110] suggests that more detailed measurements of the nuclear relaxation time may further narrow the range of possible explanations for conductance anomalies in QPCs. Chapter 2 has shown that many characteristics of ZBPs could be reproduced by a phenomenological model of subband motion. The driving mechanism for subband motion is suggested to be Coulomb interactions [78], without involving the spin degree of freedom. However, the spin dependence of ZBPs is confirmed by DNP. Therefore, if subband motion is responsible for ZBPs, it must also be spin-dependent. In the next chapter, we will show the spin dependence of subband motion imaged by thermopower spectroscopy.  58  Chapter 4  Imaging many-body effects via differential thermopower spectroscopy 4.1  Introduction  Among numerous explanations for the 0.7 structure, the phenomenological model proposed initially by Bruus et al. [58] and then improved by Reilly et al. [11, 59, 60] has been fairly successful in accounting for most experimental results associated with this feature. This model suggests that, when a spin-degenerate 1D subband is filled by electrons, it splits into two subbands with the higher-energy one rising together with the chemical potential (subband pinning) and the lower-energy one dropping quickly. Because the subband can be populated by varying either the gate voltage, Vg , or the d.c. bias, Vdc , effects of subband splitting and rising are manifested not only in linear conductance as the 0.7 structure, but also in nonlinear conductance at Vdc > 0. Specifically, subband splitting has been suggested as the origin of the anomalous 0.25 and 0.85 plateaus [8, 9], and subband rising may be! responsible for the occurrence of the zero-bias anomaly [78, 85]. In general, subband motion as a function of Vg and Vdc constitutes a coherent explanation for many transport anomalies.  59  However, the subband motion is only a phenomenological explanation that lacks a microscopic origin, so it is unable to answer some fundamental questions. For example, how does the spin degree of freedom affect the subband rising? Moreover, the subband motion is a hypothesis made from some measurement results, but has never been directly observed in experiment. To justify this explanation and to shed more light on electron transport in 1D nanostructures, a spectroscopy technique is needed to image subband motion as a function of Vg and Vdc . Experimentally, differential conductance spectroscopy is the most commonly used method for extracting information of subbands in QPCs. As has been discussed in Sec. 1.4, a peak in transconductance (dG/dVg ) shows up whenever a subband edge aligns with either the source or drain chemical potentials (µs or µd ). Based on this rule, band structure and subband motion can in principle be mapped out by tracking where the transconductance peaks are and how they move with increasing Vg and Vdc . Nonetheless, several factors make results of conductance measurements difficult to interpret and sometimes even misleading. First, the magnitude of the differential conductance, G ≡ dI/dVdc , is not a reliable measure for the number of subbands lying inside the bias window, because G depends sensitively on the functional form of potential drop across the QPC (see Sec. 1.4). Second, the magnitude of transconductance depends on subband edge with respect to both µs and µd , which complicates the interpretation of spectroscopic data. Third, transconductance is defined as dG/dVg = (dG/d µ )(d µ /dVg ), but the chemical potential needs not be a simple function of Vg . Therefore, peaks in transconductance do not necessarily mean subbands passing through the chemical potential. In view of these drawbacks in differential conductance spectroscopy, a new and better spectroscopy technique is highly desirable. But few other probes are available to study open systems consisting of just a few electrons. The lack of an effective imaging tool has become a severe obstacle for further understanding electron spin transport in QPCs. In this chapter, we describe the technique of differential thermopower spectroscopy that can directly image the motion of QPC subbands as a function of Vg and Vdc . Unlike transconductance, differential thermopower probes the energy dependence of transmission directly, without involving complicated relation between  µ and Vg . Moreover, spectroscopic data of differential thermopower is straightfor60  ward for interpretation, because effects at the source and drain chemical potentials are separated and the magnitude of thermopower peaks only depends on the reservoir whose temperature is varying. Results of this experiment clearly show pinning, splitting and other motions of subbands when they are populated. Subbands of different species are observed exhibiting very different behaviors, confirming the role of spin degree of freedom in subband motion.  4.2  Differential thermopower  If V and T are defined as potential and temperature difference between the source and the drain reservoirs, the linear thermopower described in Sec. 1.3 can be written as S ≡ −V /T , measured in the limit of zero current flow (I = 0). When temperature varies as a function of time, we shall introduce differential thermopower, S ≡ ∂ V /∂ T . The rest of this chapter will be all about differential thermopower, so for simplicity it will also be denoted as S. Experimentally, measurements of differential thermopower have several advantages over those of linear thermopower. First, a high signal-to-noise ratio can be achieved in differential thermopower measurements, because only the thermopower signal at a specific frequency is picked up and all other frequency components are ignored. Second, a d.c. current is allowed to pass through the QPC and the resulting d.c. voltage does not interfere with the thermopower measurement. The d.c. voltage across the QPC is crucial in this experiment, as it provides an energy scale for the thermopower features. To derive an analytical expression for differential thermopower, choose the source as the reference, which is kept at the lattice temperature Tl , and assume that the drain temperature is raised up by ∆Td . This temperature difference generates a heat current (drain −→ source) added to the voltage induced current (source −→ drain), thus reducing the total current transmitting through the QPC. Since the total current needs to be maintained as a constant, voltage across the QPC would increase by ∆V from the initial value V0 to compensate the heat current, i.e., I(0,V0 ) = I(∆Td ,V0 + ∆V ).  61  (4.1)  In the linear response regime, this condition is expressed as  ∂I ∂I ∆Td = − ∆V = −G∆V, ∂ Td ∂V  (4.2)  where differential conductance G ≡ ∂ I/∂ V . The differential thermopower comes out from Eq. (4.1) as ∆V 1 ∂I =− ∆Td G ∂ Td { } ∫ 2e Nm 1 ∂ = − ∑ dE tn (E)[ fs (E) − fd (E)] . G ∂ Td h n=1  S =  (4.3)  The second equality is based on the expression for 1D transmitting current in the ballistic transport regime, Eq. (1.4). Because fs (E) is independent of Td , it drops out after taking derivative with respect to Td . As a result, the product of S and G becomes a quantity that only relies on µd and Td : S×G =  2e Nm ∑ h n=1  =  2e Nm ∑ h n=1  ∫ ∫  dE tn (E) ×  ∂ fd (E) ∂ Td  dE tn (E) ×  (E − µs )e(E−µs )/kB Td . (kB Td )2 [1 + e(E−µd )/kB Td ]2  (4.4)  In general, S × G only depends on the reservoir whose temperature is oscillating, and it is irrelevant to the reservoir whose temperature is constant. The transmission coefficient could be approximated by the classical expression, tn (E) = θ (E − εn ), where θ (x) = 1 for x > 0 and θ (x) = 0 for x < 0. This approximation makes Eq. (4.4) become 2e Nm S×G = ∑ h n=1  ∫ ∞ εn  dE  (E − µd )e(E−µd )/kB Td . (kB Td )2 [1 + e(E−µd )/kB Td ]2  (4.5)  The expression under the integral sign in Eq. (4.5) is an odd function with respect to (E − µd ), so the integration achieves maximum when εn = µd . In other words, the quantity S × G shows a peak whenever the chemical potential of the heated reservoir aligns with a subband edge. This property is very useful in understanding  62  6 4 0 2  G (e2/h)  Vdc (mV)  -1  1 0 Vg Figure 4.1: Simulated S × G as a function of Vg and Vdc . The simulation is based on the saddle-point model, with h¯ ωx = 0.6meV and h¯ ωy = 2meV. Red bands mark traces of peak and blue regions indicate low value of S × G. Black trace is the simulated linear conductance of the QPC (right axis).  the thermopower spectroscopy of QPCs as a function of Vdc and Vg . It is worth emphasizing that S × G predicted by Eq. (4.5) is never lower than zero due to the step function of transmission coefficient. The physical reason is that, for a subband, energy levels are available from the subband edge up to infinity, so electron is the major contributor to transport properties. Also note that electrons (holes) contribute positive (negative) thermopower since S = ⟨E⟩/eT with ⟨E⟩ the average energy of carriers with respect to the chemical potential. As a result, thermopower of a subband is a non-negative number. If we consider a localized state with discrete energy levels εn , transmission coefficient tn (E) = δ (E − εn ), and Eq. (4.4) becomes S×G =  2e Nm (εn − µd )e(εn −µd )/kB Td ∑ (kB Td )2 [1 + e(εn −µd )/kB Td ]2 . h n=1  (4.6)  Therefore, thermopower of a localized state oscillates between positive and negative values [52, 111]. Let us now come back to the thermopower of a QPC and consider how quantity S × G will look like in the plane of Vdc and Vg . In the simplest approximation, Vg induces a capacitive shift to the subband energy, giving εn = CVg + β Vdc + εn0 63  where C is the capacitive coupling coefficient; β = 0.5 assuming an equal bias drop on both sides of the QPC; and εn0 is the subband energy at Vg = 0. Here and in the rest of this chapter, we assume that drain temperature is wiggled and potential of drain is fixed at 0. According to Eq. (4.5), quantity S × G shows a peak when the subband energy is equal to the drain potential, i.e., 0 = CVg + β Vdc + εn0 . This equation represents a series of straight lines extending from upper left to lower right in the plane of Vdc and Vg . Figure 4.1 shows results of numerical calculations based on Eq. (4.4) and a saddle-point expression of transmission coefficient, Eq. (1.22). A peak is indeed seen whenever µd is aligned with the bottom of the subband, tracing straight diagonal lines as expected from the linear equation for εn .  4.3  Devices and measurement setup  This experiment was performed on narrow channels of a GaAs 2DEG which were used before to generate pure spin currents [112, 113] and to study ballistic spin resonance [114, 115]. The channels were defined using electrostatic gates on the same GaAs/AlGaAs heterostructures reported in Chs. 2 and 3 (electron density ns = 1.11 × 1011 cm−2 and mobility µ = 4.44 × 106 cm2 /Vs at T = 1.5K). On each wall of the channel, there were two 500nm-long QPCs with spacing of 5µ m, giving a total of four QPCs in one channel. The data presented in this chapter came from three 2µ m-wide channels aligned along [110] and [110] crystal axes, but no consistent effect of channel orientation was observed. Thermopower measurements required an electron temperature difference across the QPC. This was achieved by passing an a.c. heating current IH = (30 − 50)nA at frequency fH through the channel. Energy dissipation in the channel resulted in an increase in the average kinetic energy of electrons. Due to frequent electronelectron scattering events, the electron temperature in the channel quickly equilibrated at Te . On the other hand, the regions outside the channel remained at the lattice temperature Tl . So a temperature difference ∆T = Te − Tl was created across all QPCs. Since the heating power of IH varied as IH2 R, ∆T and the resulting thermovoltage (S∆T ) alternated at frequency 2 fH , allowing thermal effects to be distinguished from resistive voltages. Heating current raised the electron temperature  64  Ref QPC  (a) (b)  Reference  IH  Vth  Iac+Idc  QPC  Vac  Figure 4.2: (a) Optical image of a typical device used in this chapter. Thermovoltage is measured across a pair of QPCs on opposite sides of the channel, with one as “the QPC” and the other one as “the reference”. The reference QPC is set on conductance plateau. (b) Schematic of measurement setup. An a.c. current IH at frequency fH heats electron gas inside the channel, giving rise to thermovoltage Vth at 2 fH . A small a.c. current Iac at fac and a d.c. current Idc are also injected into the channel through the QPC. The voltage Vac at fac gives the differential conductance of the QPC, G = ∂ Iac /∂ Vac . Inset: scanning electron micrograph (SEM) of the QPC (gates are light gray in this image).  in the channel to 200 − 240mK for Tl = 150mK.1 Thermopower measured within this range of electron temperature can be scaled onto a single curve, demonstrating that the measurements are linear. As illustrated in Fig. 4.2, the thermovoltage was in fact measured across two QPCs: the conductance of one QPC (referred to as “the reference”) was maintained at the center of a conductance plateau, while the conductance of the other (referred to as “the QPC”) was swept. Because the thermopower of the reference was zero (Sref = 0), thermovoltage directly reflected the thermopower of the QPC, Vth = (S − Sref )∆T = S∆T . Besides the heating current IH , a d.c. current Idc and a small a.c. current Iac = 1nA were injected from the QPC into the channel. The goal of injecting Iac was 1 Method  for determining electron temperature in the channel is described in Appendix B.  65  to make a simultaneous measurement of QPC conductance and to consequently obtain the quantity of S × G. Note that Iac was at a different frequency fac , so the conductance measurement would not interfere with the thermopower measurement. The d.c. current Idc was used to generate a static potential difference across the QPC and hence enable measurements of thermopower at various Vdc . The magnitude of Vdc was not measured directly in the experiment, but could be calculated from G and Idc as Vdc =  ∫ Idc di 0  G(i)  .  (4.7)  To induce spin splitting, a magnetic field, B|| , up to 8T was applied within 0.5◦ of the plane of the sample. Another magnetic field, |B⊥ | < 100mT, was applied perpendicular to the sample to adjust the out-of-plane component from B|| . The magnitude of thermopower was observed fluctuating significantly as a function of B⊥ , a phenomena named “universal thermopower fluctuation” [116]. However, this fluctuation did not affect major features of thermopower spectroscopy. Details about B⊥ dependence of thermopower is described in Appendix A.  4.4  Differential thermopower at zero field  The waterfall plot and the transconductance of a measured QPC are shown in Figs. 4.3ab, where all the signatures of a resonance-free QPC are present, including 0.7 structure, 0.25 and 0.85 plateaus, and ZBPs from just below the first plateau all the way down to the low-conductance limit. Results of thermopower spectroscopy for this QPC is shown in Fig. 4.3c, where the colorscale indicates magnitude of the quantity S × G. Compared with simulation results in Fig. 4.1, experimental data in Fig. 4.3c shows some identical characteristics: in both cases, multiple peaks are observed moving from upper left to lower right, corresponding to subbands with different indices. Looking more closely, however, two images exhibit qualitative differences that arise from motion of subbands. One of the most obvious differences is that thermopower peaks do not move linearly after they cross Vdc = 0, but stay close to Vdc = 0 line for a range of Vg , indicating subband pinning to the chemical potential. The rest of this chapter will be devoted to discussing subband motion imaged by thermopower spectroscopy and their behaviors at various in-plane mag-  66  (a) 4  4  (b)  Vdc (mV)  -1  1  2  (c)  2  G (e2/h)  G (e2/h)  3  Vdc (mV)  -1  1  2 4  1  0  0 -1  0 1 Vdc (mV)  -450  -400 Vg (mV)  ×10-6  2  0  -350  Figure 4.3: Results of differential conductance measurements and differential thermopower spectroscopy for the same QPC at B|| = 0T. (a) Waterfall plot of nonlinear conductance. (b) Transconductance, dG/dVg , versus Vg and Vdc . Blue regions are plateaus; red and white bands indicate conductance risers. (c) S × G versus Vg and Vdc . Dashed lines mark traces of thermopower peaks. Black traces in panels (b) and (c) are linear conductance of the QPC (right axis).  netic fields and temperatures. We begin by considering the motion of the second subband, which crosses the chemical potential at around Vg = −370mV. Dashed lines are traces of thermopower peaks picked up by a peak-tracking program. For Vg < −370mV, the subband lies above µd at Vdc = 0, and stays unfilled until a negative Vdc lowers the subband energy to µd . In this regime, the subband moves linearly with Vg , in agreement with the prediction by a capacitive coupling model. However, after the subband crosses µd at around Vg = −370mV, it is obvious that the thermopower peak deviates from its original direction and moves nearly horizontally below Vdc = 0. 67  This means that the subband pins just underneath the chemical potential, and almost stops populating even though Vg is increased. As Vg is further increased above -350mV, the pinning is broken and the subband moves away from µd . Similar pinning effects also occur for the third and higher subbands, though in a much weaker form. For the first subband, thermopower spectroscopy reveals more subtle features than that of higher-index subbands. In the medium-conductance regime (1e2 /h G < 2e2 /h), a thermopower peak is observed moving almost horizontally with increasing Vg , corresponding to a subband that pins just below the chemical potential. When the conductance is decreased below e2 /h, the subband goes above µd at Vdc = 0, so the thermopower peak is expected to move continuously to negative Vdc . However, the peak pinning below Vdc = 0 fades out with decreasing conductance, and another peak at negative bias fades in at the same time. In the subband picture, this discontinuity in the trace of thermopower peak indicates a sudden jump in subband energy. But it is more likely that the subband picture is invalid at low conductance and the explanation of this phenomenon may involve tunneling via localized states. As will be shown in the next section, signature of localized state is observed in magnetic field dependence of thermopower spectroscopy. ized.  4.5  Differential thermopower at finite field  To gain more insight into the microscopic origin of subband motion, we examine how the thermopower peaks are modified by an in-plane magnetic field, B|| , which explicitly breaks spin degeneracy. Fig. 4.4a shows thermopower spectroscopy for the same device and settings as in Fig. 4.3c, but with B|| = 8T. Each thermopower peak observed at zero field is expected to split into two peaks with spacing Ez = |gµB B|| |, where g = −0.44 is the g-factor in bulk GaAs. Nonetheless, the data in Fig. 4.4a shows several notable features that are different from this expectation. First, subband splitting depends on subband index, and is significantly larger than Ez . Here the subband splitting is measured as the spacing between two thermopower peaks at the same gate voltage. On 1e2 /h, 3e2 /h and 5e2 /h spin-polarized plateaus, the subband splittings are 4.5Ez , 3.4Ez and 3.0Ez , respectively. The en-  68  -1  (a) B = 8T  6  ×10-6  4 2 0 -2  4  G (e2/h)  Vdc (mV)  Ez  2  0  1  -400 (b) B = 0T  (c) B = 2T  -1  -200  -300 (c) B = 4T  (d) B = 6T 4  4  Vdc (mV) 1  3  G (e2/h)  ×10-6  4 2 0  2 -380 (e) B = 0T  -340  -380 (f) B = 2T  -340  -380 -340 (g) B = 4T  -360 -320 (h) B = 8T 2  -0.5  Vdc (mV)  1  G (e2/h)  ×10-6  2 1 0  0.5 0 -80  -60  -80  -60  -60  -40  -60  -40  -20  Vg (mV)  Figure 4.4: (a) Thermopower spectroscopy, S × G, for a QPC at 8T. White bar in the top left corner indicates the scale of Ez . (b-d) Magnetic field dependence of thermopower peaks for the second subband, measured in the same QPC as in (a). (e-h) Magnetic field dependence of thermopower peaks for the first subband, measured in a different QPC from (a-d). Thermopower data in panels (a), (b-d) and (e-h) is shown respectively in the same colorscale. Black trace in each panel is linear conductance of the QPC (right axis).  69  hanced subband splitting in QPCs is consistent with previous literatures [6, 7, 92], and this phenomenon is usually attributed to exchange interactions. But how exactly the exchange interactions affect the subband splitting remains poorly understood [10, 57, 117]. Second, subbands of different spin species exhibit different behaviors: the spindown (lower energy) subbands straightforwardly go through the chemical potential, whereas the spin-up (higher energy) subbands hover over the chemical potential for a range of Vg and then quickly pass through. These behaviors help understand why spin-degenerate subbands at zero field pin below the chemical potential—that is simply the compromise between two spin subbands. As magnetic field is increased, the spin-up subband gradually moves above the chemical potential, and the spindown subband falls down in energy, giving rise to a significant subband splitting (Figs. 4.4b-d). This finding confirms the spin-dependent behaviors of subbands proposed in Refs. [53, 92, 118] and provides foundation for phenomenological models [11, 48, 58]. Spin-dependent behaviors of subbands indicate that, during the population process, the spin-up subbands rise up whereas the spin-down subbands do not. These results manifest the importance of exchange and correlation interactions in subband motion, since Coulomb interaction alone leads to a universal rise in subband energy [27], as mentioned in Sec. 1.7.3. There have been attempts to study the effects of exchange and correlation within the framework of spin-density functional theory [64]. In future, we look forward to further theoretical studies on this topic that go beyond the mean-field method. Field dependence of thermopower spectroscopy provides an explanation for behaviors of ZBPs. At zero magnetic field, thermopower peaks pass the Vdc = 0 line and then stay just below it for a range of Vg , indicating that subbands rise up in energy and pin to the chemical potential. The rising of subbands gives rise to ZBPs, as discussed in Sec. 1.7.3. At high field, although the thermopower peaks for spin-up subbands move nearly horizontally when they are still away from the Vdc = 0 line, all spin-polarized thermopower peaks pass Vdc = 0 without changing direction. In other words, all subbands at high field do not rise up near Vdc = 0, therefore ZBPs disappear. At the end of this section, we examine magnetic field dependence of ther70  2  ×10-6  2 1 0  0.5  G (e2/h)  1  2 (b) Vth×G (×10-6Ve2/h)  (a) B = 8T  Vdc (mV)  -0.5  0T 2T 4T  1  0 6T 8T  0 -60  -40 Vg (mV)  -20  -500  0 500 Vdc (µV)  Figure 4.5: (a) Thermopower spectroscopy, S × G, for G < 2e2 /h at 8T. This figure is the same as Fig. 4.4h but re-plotted using a different colorscale to emphasize negative thermovoltage (shown in blue). (b) Evolution of S × G at G = 0.3e2 /h with magnetic field from B|| = 0T to 8T.  mopower peaks for the first subband (Figs. 4.4e-h). For thermopower spectroscopy at B|| = 8T, there are clearly two trajectories of peaks, the upper one and the lower one, corresponding to spin-up and spin-down subbands, respectively. In the area between the two peaks, S × G becomes negative (Fig. 4.5a), which is incompatible with the subband picture discussed in Sec. 4.2. Instead, it points to the formation of a localized state where negative thermopower is possible (Eq. 4.6). When the magnetic field is decreased, the negative thermopower at low conductance increases and eventually turns into a positive peak at zero field (Fig. 4.5b). This behavior indicates some kind of spin-flip process that enhances the thermopower (see discussions in Sec. 1.6.2). Such spin-flip processes are suppressed by magnetic field [51], therefore the negative thermopower is restored at high field. Combination of a localized state and spin-flip processes points to the Kondo effect (Sec. 1.7.2). Nonetheless, in next section, inconsistency will be shown between the Kondo effect and temperature dependence of thermopower spectroscopy.  4.6  Temperature dependence  Differential conductance of QPCs shows nontrivial temperature dependence since the anomalous 0.7 structure becomes more and more prominent with increasing 71  -0.5 (a) T = 100mK  (b) T = 400mK  (c) T = 1100mK  4  2 1 ×10-6  2  ×10-6  ×10-6  0.0  6 4 2 0  0  0  1  -0.5  0 1  (d) 100mK  (e) 400mK  (f) 1100mK  G (e2/h)  Vdc (mV)  0.5  0.0 0  1  0.5  -120  -100  -120  -100  -120  -100  0  Vg (mV)  Figure 4.6: Thermopower spectroscopy for G < 2e2 /h at various lattice temperatures of (a) 100mK, (b) 400mK and (c) 1100mK. Heating current IH is maintained at 50nA, so magnitude of thermopower decreases with increasing temperature. To avoid the effect of colorscale, data in each column is normalized and plotted again in panels (d-f). Black trace in each panel shows linear conductance of the QPC (right axis).  temperature. Thermopower spectroscopy of a QPC at various temperatures also displays features that could not be simply explained in terms of thermal broadening. These features are summarized in Fig. 4.6, where thermopower spectroscopy for G < 2e2 /h is measured from T = 100mK up to 1100mK. Thermal broadening is about 3.5kB T and should apply uniformly to features at all gate voltages. However, broadening of thermopower patterns in Fig. 4.6 is apparently very non-uniform: features at Vg < −125mV seem to get broadened just by temperature, but the thermopower peak around Vg = −115mV, where the 0.7 structure occurs, broadens significantly more than 3.5kB T . This is consistent with the result that transmission coefficient is directly affected by temperature in the medium-conductance regime, but not in the low-conductance regime  72  (see Sec. 2.3.2). The broad pattern of thermopower around Vg = −115mV, especially at high temperature, seems not to be associated with subbands because the thermopower peak for a subband should appear as a narrow trace. This result casts doubt on the subband picture and its explanations to the 0.7 structure and the ZBPs. At such low resistance, many-particle interactions is increasingly strong and the concept of “subband” breaks down. The thermopower pattern around Vg = −115mV becomes more and more prominent with increasing temperature, indicating thermal activated processes involved. Electrons flip their spin during these processes, enhancing the otherwise low thermopower and thus leading to an unexpected broad thermopower pattern. These processes do not seem like Kondo effect which gets suppressed with increasing temperature. The detailed mechanism of the processes is unclear. What also remains unclear is why these processes take place most strongly in the gate voltage corresponding to the 0.7 structure.  4.7  Discussion  Our initial intention was to employ thermopower spectroscopy to image subband motions in QPCs, but experimental results reveal a more complicated picture about electron spin transport in these devices. At G > 2e2 /h, discrete subbands form and they are populated one by one with increasing gate voltage, Vg . Soon after a subband is populated, its energy pins just below the Fermi level for a range of Vg , which could arise from a sudden change of gate capacitance due to the divergence of 1D density of states at subband edge. The pinning effect also occurs at high magnetic field, but appears on odd conductance plateaus (G = N ×e2 /h, N = 1, 3...) where the spin-down (lower energy) subband is well below the Fermi level but the spin-up subband is still above it. As a result, the spin-down subbands pass the Fermi level faster than the spin-up ones, confirming the spin-dependent subband motion proposed by Refs. [53, 92, 118]. Since the pinning effect in both spindown and spin-up subbands takes place in the same range of Vg (see Fig. 4.4), it can also be attributed to a sudden change of gate capacitance. However, on the odd conductance plateaus, the Fermi level is away from any subband edges, so the divergence of 1D density of states is not involved here. This result questions  73  the 1D-density-of-states explanation to the zero-field pinning effect, as the pinning effect should have the same origin at various fields. At G < 2e2 /h, the many-body interactions are increasingly significant and results of thermopower spectroscopy show that the subband picture is inaccurate to describe electron transport in this regime. Instead, the negative thermopower in Fig. 4.5 and the broad thermopower pattern in Fig. 4.6 indicate that the anomalous transport properties of QPCs is related to spin-flip scattering and may involve localized or resonant states. This is consistent with the spin-density functional theory (SDFT) which suggests that localized states can self-consistently form in low-conductance QPCs as a result of many-particle interactions [16]. SDFT also suggests a Kondo coupling between the localized states in the QPC and the conduction electrons in reservoirs, which would be suppressed by either magnetic field or temperature. However, as discussed in Secs. 4.5 and 4.6, the spin-flip process is only suppressed by increasing magnetic field but becomes more and more prominent with increasing temperature. So a new kind of many-particle interaction may be responsible for electron spin transport in low-conductance QPCs, and the exact mechanism requires future investigations.  74  Chapter 5  Conclusions In this thesis, we have studied spin-related electron transport in QPCs, aiming at understanding anomalous conductance features such as the 0.7 structure and the zero-bias conductance peak (ZBP). The first two experiments focused on ZBPs, while the final experiment brings together the full range of QPC conductance features. We began by investigating behaviors of ZBPs in the low (G < 1e2 /h) and highconductance (G > 2e2 /h) regimes. Previous experiments focused on the ZBPs at moderate conductance, in the range of 1e2 /h  G < 2e2 /h. In our measure-  ments, sharp and narrow ZBPs were observed not only at moderate conductance, but also at low conductance down to 10−4 e2 /h and at high conductance around the midpoint of conductance risers. In the low-conductance regime, the width of the ZBP remains constant over orders of magnitude in conductance. We considered how our experimental observations compare to predictions from two theories: the Kondo model and the subband-motion model. Our observations are inconsistent with the symmetric Kondo model that has been proposed to explain ZBPs in the medium-conductance regime [7], and points to an asymmetric Kondo model. The asymmetric Kondo model describes localized states on either end of a QPC near pinch-off, each strongly coupled to one of the reservoirs but weakly coupled to each other. This model is consistent with spin-density functional calculations [16]. Nevertheless, within the Kondo picture it is very difficult to explain the magnitude or absence of ZBP splitting with magnetic field, and the consistent observation of 75  narrow ZBPs in the high-conductance regime. The subband-motion model does reproduce the occurrence of ZBPs in all three conductance regimes, as well as the fact that the width remains constant over orders of magnitude in conductance. Nontrivial subband motion–such as might give rise to ZBPs–has been predicted due to Coulomb interactions in 1D systems [78]. But the spin degree of freedom is irrelevant to these models, in contrast to the spin dependence of ZBPs. To test for spin dependence of ZBPs directly, we next presented measurements of nuclear spin polarization in QPCs. Nuclear spin polarization is often generated in GaAs QPCs when an out-of-plane magnetic field gives rise to spin-polarized quantum-Hall edge states and a voltage bias drives transitions between the edge states via electron-nuclear flip-flop scattering [95]. In this thesis, we reported a similar effect for QPCs in an in-plane magnetic field, where currents are spin polarized but edge states are not formed. The nuclear polarization gives rise to hysteresis in the d.c. transport characteristics with relaxation time scales around 100s. Since the nuclear polarization only affects spin degree of freedom for electrons moving through the QPC, it provides a useful test for spin sensitivity of transport phenomena. The shape of ZBPs depends on nuclear polarization, confirming the spin dependence of this feature. From this measurement, we concluded that the subband-motion model would have to be significantly modified by spin-related effects if it were to be responsible for ZBPs. To test for spin dependence of subband motion, we had to go beyond commonlyused conductance measurements and develop a new technique of spectroscopy. We built on the method demonstrated by Appleyard et al. [26], employing differential thermopower to probe electron transport in QPCs. In this measurement, thermovoltage was generated by heating up one of the reservoirs using an a.c. current. Unlike conductance spectroscopy, this technique directly probes the energy dependence of transmission, and reflects features at a single chemical potential even when a finite source-drain bias is applied. At G > 2e2 /h and at high magnetic field, the thermopower spectroscopic data clearly shows different behaviors for subbands with different spin species: the spin-down (lower energy) subbands quickly go through the chemical potential, while the spin-up (higher energy) subbands pin above the chemical potential for a range of Vg before passing through. This observation confirms the spin dependence of subband motion and provides an 76  explanation for the disappearance of ZBPs at high magnetic field. At G < 2e2 /h, thermopower spectroscopic data is more subtle, and the subband picture seems to break down as a result of strong interactions. 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The fluctuation patterns are exactly repeatable over time, indicating that the phenomena does not arise from noise. Although the magnitude of thermopower is B⊥ dependent, major features of thermopower spectroscopy for QPCs are not affected by a small B⊥ . Subband pinning, subband splitting and other characteristics described in Ch. 4 remain qualitatively the same for B⊥ from 0 to 100mT. Some minor features, e.g , a ridge on the first conductance plateau, occur only at certain B⊥ . These features may arise from some kind of resonance instead of subband motion in QPCs.  86  (a)  Vth (µV)  1  0  0 -1  20  40  (b)  G (e2/h)  Vdc (mV)  4  ×10-6  0  80  (c)  2 1  1  60  B┴ (mT)  -1  ge Rid  -80  -60  -40  -80  -60  -40  2  Vg (mV)  Figure A.1: (a) Typical universal thermopower fluctuations as a function of B⊥ on the conductance riser (red) and the conductance plateau (blue). (b,c) Thermopower spectroscopy for the second subband at (b) B⊥ = 0mT and (c) B⊥ = 52mT. Data is plotted in the same colorscale. Thick dashed line indicates the thermopower ridge on the first conductance plateau. Black trace shows linear conductance of the QPC (right axis).  87  B Measurements of electron temperature Electron temperature in nanostructures is usually determined from the QPC thermopower as described in Ref. [120, 121]. Within the saddle-point model, thermovoltage peaks of QPCs have a height [122] Vthi =  π 3 kB2 (Te2 − Tl2 ) , 24e¯hωx i + 1/2  (6.1)  where Te and Tl are electron and lattice temperatures, respectively; i is the subband index; h¯ ωx is the characteristic tunnel broadening of the QPC (see Sec. 1.5). Once h¯ ωx is determined by bias spectroscopy for differential conductance, the electron temperature can be obtained from Vthi using Eq. (6.1). However, this technique could not give a reliable electron temperature in our experiment, because a huge universal thermopower fluctuations were observed as a function of perpendicular magnetic field, B⊥ (Fig. A.1a). Such fluctuations arose from quantum interference between different electron trajectories, which became weaker at higher temperature. So we employed universal conductance fluctuations (UCFs) to estimate electron temperature. Figure B.1a shows the setup for measuring UCFs in the channel. A small a.c. current Iac = 1nA generated a potential drop Vac along the channel measured by a voltmeter, and a d.c. current Idc = (0 − 200)nA heated up electron gas in the channel to a temperature Te . Differential conductance G = dIac /dVac showed fluctuations as a function of perpendicular magnetic field, B⊥ , as shown in Fig. B.1b. The lattice temperature Tl could be controlled precisely by a heater installed in the fridge and measured by a RuO2 resistance thermometer, whereas Te depended on both Tl and Idc . To determine Te , the first step was to measure UCFs at Idc = 0 when Te = Tl . UCFs at higher Tl had a smaller amplitude, which set a scale for the electron temperature. The second setp was to measure UCFs at a fixed Tl but at various Idc . Te for different Idc was then determined by mapping the magnitude of UCFs to those obtained in the first step. Note that UCFs were superimposed on a slow variation of G (Fig. B.1b), which came from electron orbital effects induced by B⊥ . To avoid ambiguity in extracting UCFs, we performed Fourier transform to G, followed by fitting to lines with a  88  (a)  Iac+Idc  Vac (b)  Vth (µV)  1.5  1.0  0.5 0  10  B┴ (mT)  Power  50mK 150mK 300mK 600mK  20  30 0nA 30nA 60nA 90nA  (c)  (d) 1/ΔB┴  Figure B.1: (a) Experimental setup for measuring UCFs in the channel. (b) Typical UCFs as a function of B⊥ . (c) Fourier transform of UCF data measured with Idc = 0nA at lattice temperature Tl = 50, 150, 300 and 600mK. (d) Fourier transform of UCF data measured with Tl = 150mK at Idc = 0, 30, 60, and 90nA. Solid lines in panels (c) and (d) are linear fits to Fourier spectrums.  89  constant slope. The y-intercept of these lines represented the magnitude of UCFs. From the results in Figs. B.1cd, we obtained Te = 200mK and 240mK for Idc = 30nA and 50nA at Tl = 150mK.  90  

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