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Waveguides for spin-polarized currents in diluted magnetic semiconductor - nanomagnet hybrids Cheung, Kelly 2006

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Waveguides for spin-polarized currents in diluted magnetic semiconductor — nanomagnet hybrids by Kelly Cheung' •' B.ASc, University of British Columbia, 2004 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF T H E REQUIREMENTS FOR T H E DEGREE OF Master of Science The Faculty of Graduate Studies (Physics) The University of British Columbia September, 2006 © Kelly Cheung, 2006 11 Abstract Diluted magnetic semiconductors in their paramagnetic phase exhibit a giant Zeeman response. This effect can be used in conjunction with external inho-mogeneous magnetic fields to engineer spin-polarized charge-carrier eigenstates with certain desirable features. In this thesis, we solve Schrodinger's equation numerically and examine the charge-carrier wavefunctions in D M S waveguides in the presence of a highly inhomogeneous external magnetic field from an in-finitely long rectangular nanomagnet. The low-energy eigenstates are found and their dependence on various parameters, such as size of the nanomagnet, thick-ness of the D M S waveguide, strength of coupling between the charge-carriers and the magnetic spins in the semiconductor, and addition of other external magnetic fields, is characterized. This geometry is shown to be ideal for creat-ing spin-polarized currents under the nanomagnet's edges. \ iii Contents Abstract ii Contents iii List of Figures iv Acknowledgements . v Chapter 1 Introduction 1 Chapter 2 Formulation of the problem 4 2.1 Solving Schrodinger's equation using a complete basis 4 2.2 Uniform cubic B-splines 6 2.3 Magnetic field of the rectangular magnet 9 2.4 The DMS layers 13 2.5 Simplifying Schrodinger's equation 15 Chapter 3 Results 18 Chapter 4 Optimal Configuration 28 Chapter 5 Conclusion 29 Bibliography 30 Appendix A B-splines 32 Appendix B Checking the Program 33 iv List of Figures 2.1 Uniform cubic B-splines 8 2.2 Geometry of interest 9 2.3 The magnetic field lines around an infinitely long rectangular nanomagnet . . 11 2.4 The magnetic field parallel to the plane at different distances below the nanomagnet 12 2.5 The magnetic field perpendicular to the plane at different dis-tances below the nanomagnet 12 2.6 Magnetic fields at a distance of 10 nm from a nanomagnet of size b = c = 150 nm .- 13 2.7 Magnetic fields at a distance of 10 nm from a nanomagnet of size b = 75 nm and c = 300 nm 14 2.8 Program takes 0 ( n 3 ) time 16 2.9 Convergence of the ground state eigenvalue with the number of partitions in x and z 17 3.1 Charge density for magnetization perpendicular to the plane . . 18 3.2 Spin +x for magnetization perpendicular to the plane 19 3.3 Spin +z for magnetization parallel to the plane 20 3.4 Spin +z wavefunction for magnetization perpendicular to the plane, and addition of a uniform field parallel parallel to z . . . . 21 3.5 Wavefunction for magnetization parallel to the plane, and addi-tion of a uniform field parallel parallel to —x 22 3.6 The ground state dependence on z2 22 3.7 The ground state energy dependence on c 24 3.8 The ground state energy versus geg 24 3.9 The ground state energy versus additional external magnetic field 25 3.10 Charge density for 4 layers of D M S 26 3.11 Ground state energy plotted against 1 nm layers separated by 1 nm : 27 B . l The smallest eigenvector and eigenvalue for a harmonic oscillator in x and a potential well in z 32 B.2 The smallest eigenvector and eigenvalue for a harmonic oscillator variant 1 in x and a potential well in z 32 B.3 The smallest eigenvector and eigenvalue for a harmonic oscillator variant 2 in x and a potential well in z 33 V Acknowledgements I would like to thank my supervisor, Dr. Mona Berciu, for the countless number of hours she has invested in teaching me, giving me ideas, and providing me resources to complete my work. In addition, I would like to thank Mandy Wong, James Charbonneau, and Jinshan Wu for making the past two years very enjoyable. Last, but not least, my family for not complaining about me always working in the middle of the night. This research has been enabled by the use of WestGrid computing resources, which are funded in part by the Canada Foundation for Innovation, Alberta Innovation and Science, B C Advanced Education, and the participating research institutions. WestGrid equipment is provided by I B M , Hewlett Packard and SGI. 1 Chapter 1 Introduction Diluted magnetic semiconductors (DMS) doped most usually with M n , of gen-eral type I I I i _ x M n x V and I I i _ x M n x V I , have been studied primarily due to their ferromagnetic properties below a Curie temperature T c . So far, G a i - z M n ^ A s [1] has been the most studied III-V D M S because it has the highest reliable critical temperatures recorded: 160 K for bulk samples (where M n is uniformly doped in the entire sample) [2] and 172 K in digitally doped heterostructures (where M n is doped in very narrow two-dimensional layers, ideally of atomic width, which are separated by several undoped atomic layers) [3]. In I I I i - ^ M n ^ V DMSs, sub-stitution of a fraction x of the type-Ill element with M n introduces both local M n spins (5 = 5/2) and holes into the system, since M n has valence II and a half-filled 3d shell, M n : 3d54s2. It is widely accepted that magnetization is due to charge-carrier mediated, effectively ferromagnetic, interactions between the M n spins [1,4]. In I I i - ^ M n ^ V I DMSs, substitution of a fraction x of the type-II element with M n introduces local M n spins, but a second type of charge dopant is needed to introduce charge-carriers. Because the charge-carriers and M n spins are usually located on different sub-lattices, their interactions are much weaker and the resulting Curie temperatures are much smaller, below a few K (for a review of II-VI D M S , see Ref. [5]). In the ferromagnetically ordered phase, the partial spin-polarization of the charge-carriers allows for creation of partially spin-pplarized currents. The use and need of such spin-polarized currents in spintronic devices is the main reason why the D M S have been such an active research area recently. However, there are obvious drawbacks in this approach. To achieve a high degree of spin-polarization, one needs to be well below Tc. Since the critical temperatures are still rather low, this implies that devices based on this scheme could only operate at extremely low temperatures. , Another approach to control the spatial location and the degree of spin-polarization of charge-carriers in D M S at rather high temperatures (above Tc) has been proposed recently [6, 7, 8]. This scheme is based on using the giant Zeeman response to an external magnetic field, observed in D M S in their para-magnetic phase, for T > T c . For example, for a 0.5 T field, the Zeeman splitting in a non-magnetic semiconductor is «0 .06 meV, while large Zeeman responses, in the range of 15-30 meV for 0.1-0.5 T, have been seen experimentally in II-VI D M S using photoluminescence spectroscopy [9, 10]. The origin of this giant Zeeman effect is the exchange interaction between a charge-carrier (say, electron) of spin s with the M n impurity spins Si located Chapter 1. Introduction 2 at positions R4: Hex = J2j(f-Rz)Si-s. (1.1) i where J(f) is the exchange interaction. Within a mean-field approximation (justified since each carrier interacts with many impurity spins) 5, • s —> (5j)s + St{s): and the average exchange energy felt by the electron becomes: Hex = N0xa(S)s • (1.2) Here, x is the molar fraction of M n dopants, A^ o is the number of unit cells per unit volume, (5) is the average expectation value of the M n spins, and a = f dfu* ^_Q(r)J(r)uc j;=0(r) is an integral over one unit cell, with uc ^0(r) being the periodic part of the conduction band Bloch wavefunctions. The usual virtual crystal approximation, which averages over all possible location of M n impurities, has been used. Exchange fields for holes can be found similarly; they are somewhat different due to the different valence-band wavefunctions. . In a ferromagnetic D M S , these terms explain the appearance of a finite mag-netization below TC: as the M n spins begin to polarize, (5) ^ 0, this exchange induces a polarization of the charge-carrier spins (s) ^ 0. In turn, exchange terms like Si(s) further polarize the M n , until self-consistency is reached. In paramagnetic D M S , however, (5) = 0 and charge carrier states are spin-degenerate. The spin-degeneracy can be lifted if an external magnetic field B is applied. Of course, the usual Zeeman interaction —g^BS- B is present though this is very small since typically g « 2. Much more important is the indirect coupling of the charge-carrier spin to the external magnetic field, mediated by the M n spins. The origin of this is the exchange energy of Eq . (1.2), and the fact that in an external magnetic field, the impurity spins acquire a finite polarization {§) j) B, |(5)| = SBs{gHBSB/(kBT)), where 5 is the value of the impurity spins (5/2 for Mn) and Bs is the corresponding Brillouin function (for simplicity of notation, we assumed the same bare ^-factor for both charge-carriers and M n spins; also, here we neglect the supplementary contribution to the M n spin-polarization coming from the 5j(s) terms, since for the geometries we study an impurity spin interacts with at most a spin-polarized charge-carrier). The total spin-dependent interaction of the charge-carriers at T > TC and in the presence of the external magnetic field is, then: Tiex = -g^BS- B +.N0xa(S)s = gespBS- B (1.3) where , N ° X A C R (9HBSB\ for electrons, with an equivalent expression for holes. Since Bs{x) a x if x <^ 1, it follows that for low magnetic fields geg becomes independent of the value of B, although it is a function of T and x. Its large effective value is primarily due to the strong coupling a (large J(r)) between charge-carriers and M n spins. Experimentally, values as large as geg « 500 — 2000 have been measured. Chapter 1. Introduction 3 We combine this giant Zeeman response with a small-to-moderate exter-nal magnetic field B(r) with a strong spatial variation on a nanometer scale, and use the resulting large, spatially modulated effective Zeeman interaction geflUBS- B(r) to create spin-polarized charge-carrier wavefunctions with various desired symmetries. In this thesis, we investigate the resulting wavefunctions in the presence of external magnetic fields created by long lines of permalloy Ni8oFe2o, Co, or Fe nanomagnets. The ability to create nanomagnets of various shapes and sizes, such as disks [11, 12, 13, 14] and lines [15], has already been demonstrated. We find that spin-polarized wavefunctions are strongly trapped under the nanomagnet's edges, in a geometry ideally suited for generation of spin-polarized currents at relatively high temperatures. This should be of obvi-ous interest for spintronic device design. This numerical work is a continuation of Berciu and Janko's study of charge carrier wavefunctions in the presence of a cylindrical nanomagnet [6], which ex-hibit localization in 3D. Recently, Wong [16] has considered the wavefunctions in the presence of an infinitely long rectangular nanomaget, but using a sim-plified 2D geometry (i.e., ignoring the effect of the width of the D M S layer). Here we extend her work by considering a D M S layer with a finite thickness, as well as the case of multiple layers of D M S , relevant for digital doping. A somewhat related study has been published recently [17], however, there are important differences between our and their work, on which we comment in our conclusions. In the next chapter, we describe the geometries of interest to us, and the approach we use to solve the resulting Schrodinger's equations. Results are presented in Chapter 3, and the characteristics of the ideal configuration are discussed in Chapter 4. Chapter 5 contains our conclusions and final discussion. Some further technical details are presented in two appendices. 4 C h a p t e r 2 Formulation of the problem In this chapter, we first describe the general approach to solving Schrodinger's equation based on expanding the wavefunction in a complete but not necessarily orthogonal basis set. The set we use in this work, cubic B-splines, is then briefly introduced. We then consider the particular case of interest to us. We first derive and characterize the magnetic field created by an infinitely long rectangular nanomagnet inside a finite-width DMS layer, and then use its characteristics to simplify the resulting Schrodinger equation and also to characterize the various symmetries of the eigenfunctions. 2 . 1 So lv ing Schrodinger 's equation using a complete basis Assuming we know the magnetic field B(r), the task we face is to find the low-energy charge-carrier eigenstates inside the paramagnetic DMS layer. The general Schrodinger's equation to be solved is: ±{p-qA(f))2-geS^.B(f) q>(f) = EV(r) (2.1) Here, m and q are the effective mass and charge of the charge carrier. Several approximations have already been made to arrive at this form. For example, we have ignored electron-electron interactions. This is justified if the carrier concentration is either large enough such that screening is very effective, or so low that only a small number of electrons might be trapped inside the DMS (the case of interest to us). Also, as explained in the introduction, we assume that the Mn spins create a smooth polarizing field for the charge-carriers, ignor-ing possible local fluctuations due to the fact that the Mn spins are randomly distributed in the DMS. This last simplification is called the virtual crystal approximation and is used extensively to describe DMS physics. Finally, note that geff could be either positive or negative, depending on whether the coupling between charge-carriers and the Mn spins is.ferro- or antiferromagnetic. Both cases are possible, depending on material details. Here, we assume that <?eff > 0 for the Hamiltonian of Eq. (2.1). The solutions for <?eff < 0 for the case of interest to us here can be obtained from the ones for geg > 0 via a TT rotation of the eigenspinor about the y-axis (see discussion below). We make one more simplification, which is to ignore the terms containing the vector potential A(r) in the kinetic energy. This is justified since their Chapter 2. Formulation of the problem 5 contribution is negligible compared to that of the Zeeman term, which is hugely enhanced through ges. As a result, we have to solve the following equation: 2m *(r) = £* ( f ) (2.2) We choose three complete sets of basis functions: {Xn(x)} in x direction, {Ym(y)} in y direction, and {Zp(z)} in z direction, where n,m,p are integers denoting a particular function from each of the three sets. We rewrite \&(f) as a linear combination of the basis functions: = Z~2 anmpXn(x)Ym(y)Zp(z) n,m,p where a n m p are spinors in order to take into account the spin degrees of freedom (we assume that charge carries have spin | . Generalization to higher spins, of interest for hole doping, can be achieved easily. However, even for holes, at low doping only states in the heavy band, with spin-projections jz = ± | , are occupied, and the system behaves effectively as a spin | with a geg increased by a factor of 3). Notice that each basis set generally contains an infinite number of functions, so we have to introduce cut-offs: n m a x , m m a x , pmax- The values for the cut-offs must be chosen such that for each set, increasing the number of basis functions taken into account does not change the final solution. Wi th this, we get: 2m yZ anmpXn(x)Ym(y)Zp(z) n,m,p = E anmpXn(x)Ym(y)Zp(z) ,n,m,p (2.3) We multiply the equation with each possible combination of the three sets: Xj(x)Yk(y)Zi(z) (as shown below, we use real functions in our sets), and then integrate over the whole space: dif[Xj(x)Yk(y)Zl(z)} f~2 OLnmpXn(x)Ym(y)Zp(z) = E J cPrlXjWYkiyWz)] _^ anmpXn(x)Ym(y)Zp(z) (2.4) Using integration by parts for the kinetic energy, we arrive at the final form: 2m CX-nmp XjnYkmZip + XjnYlmZip + XjnYkmZ'ip — Chapter 2. Formulation of the problem 6 — 9eSPB ^ ] Bjkl,nmpO!nmp — F ^ ^ &nmpXjnYkmZlp (^-5) n,m,p n,m,p where we used the shorthand notation: /oo d x X ^ x ) ^ ^ ) (2.6) - O O /CO ^n(2/)*m(2/) (2.7) -oo /CO dzZi(z)Zp(z) (2.8) -oo oo oo dXj(x) dXn(x) X ' = T" ' (2.9) and Bjkl,nmp — Id T | -5(f) x ^ ) ^ ) ^ ) * ^ ) ^ ) ^ * ) (2.12) Once the magnetic field 5 ( f ) is known and the particular basis sets have been chosen, these integrals can be calculated. Of course, each basis set should be chosen so as to make this computation as efficient as possible. Because each combination (jkl) gives a different equation, we are left with 2 n m a x x mmax x Pmax matrix equations with a total of 2nmax x mmax x pmax unknown coefficients for spin \ . These can be written in the general vectorial form Ax = EBx, where A and B are symmetric matrices, which we can solve using existing linear algebra subroutines. 2.2 Uniform cubic B-splines As we show in the following, the magnetic field of interest to us is indepen-dent of y; as a result, the wavefunctions are plane-waves in the y. direction, Yk(y) = elky/y/2~Tr. On general physical grounds, we expect the eigenfunctions to be localized in the x and z directions. Usually this would lead to a choice of an orthonormal basis set for these directions, such as, for instance, the eigen-functions of a I D harmonic oscillator. Such a choice would simplify the above general equation since for orthonormal basis sets, the matrix B would be the identity because in this case Xjn = 5j>n, etc. However, because the integrals appearing in Eq . (2.12) are quite complicated, one would expect most of these integrals to be finite. Calculating these many integrals is very time-consuming. Chapter 2. Formulation of the problem 7 As a result, we use a different strategy. We use a non-orthogonal but com-plete basis set for the x and z directions, namely uniform cubic B-splines. Since we are searching for localized eigenfunctions, the interval spanned by these cu-bic B-splines in either direction is chosen to be large enough so that it contains the whole support of the low-energy eigenfunctions. These intervals are par-titioned into Nx, respectively Nz, segments of equal length; the end points of these segments define the so-called knot sequence or vector. The use of this uni-form mesh is not necessary, and one may easily use a non-uniform mesh, with a larger sampling where the wavefunction varies faster. Such generalizations were not needed in the case of interest to us. Once the knot sequence is defined, one generates a- basis of cubic B-splines as described in Appendix A . Here we review the essential properties of these functions. Cubic B-splines provide a complete basis for all piecewise functions defined on this knot vector, that are continuous and have a continuous deriva-tive, provided that on each interval the function is a polynomial of rank 3 (hence the name "cubic"). For a uniform mesh with TV mesh intervals of length Ax = 1, we use the knot sequence [0,0, 0 , 0 , 1 , . . . , JV - 1, JV, JV, JV, JV] to generate JV + 3 cubic B-splines: B-3(x) (1 0 •xf if x e [0,1) otherwise B-2{x) 7^3 - f x 2 + 3x if x e [0,1) if x e [1,2) otherwise £ _ i ( x ) = _ I i ~ 3 7 1 2 3 % 3/ 3 3/ 3 X 2 (3-x) 3 2X if x e [0,1) if x e [1,2) if x e [2,3) otherwise For i = 0,.. . ,JV- -4: Bi(x) = 6 - i x 3 - X 3 -_ 3i+4 2 _ 3i+8 „ 2 2"° 2 (i+4-x)3 6 0 x2 _ 3i2+8i+4 •x + 3z2 + 16i+20, 3 +4t 2 +4t •3 + +8i2+20i 2 22 3 if x e [i if x e [i if x e [i if x e [i otherwise + l , i H 2,i-\ 2) 3) + 3,z + 4) and finally, BNS(X) ( (x-JV+3)3 6 _ 1 T 3 , 77V-12 2 12 T 4 7JV2-24Af+18 X+ + 7AfJ-36Af';+54iV-18 12 11 „ 3 _ 11AT-6-2 12 4 ^ 11N 3 -18N 2 HW / : -12]V, I o 12 if x e [JV — 3, JV — 2) • if x e [ J V - 2 , J V - 1 ) if x e [ JV-1 , JV) otherwise Chapter 2. Formulation of the problem 8 Figure 2.1: Uniform cubic B-splines with knot vector [0,0,0,0,1,2,3,4,5,6,7,8,8,8,8]-(x-N+2)3 4 21N-7 3 7 J V 3 - 1 8 i V 2 + 12JV ^ X 2 21N2-36JV + 12 x | if x e [ J V - 2 , J V - 1 ) if x e [N — 1, JV) otherwise and J V - l ( x - J V + l ) 0 if x e [JV -otherwise 1,JV) Fig. 2.1 shows as an example 11 uniform cubic B-spline functions with knot vector [0,0,0,0,1,2,3,4,5,6,7,8,8,8,8]. Far from the edges, each B-splines spans 4 mesh intervals, whereas near the edge, the last 3 B-splines are modified so as to start (or end) on the end point of the interval. The main advantage in using B-splines is that each of them overlaps with only 3 other B-splines. The overlap integrals Xjn and X j n are therefore 0 if \j — n\ > 4, and analytic expressions can be easily calculated if \j — n\ > 4. Similarly, it also follows that most of the integrals appearing in Eq . (2.12) are identically zero, one only needing to evaluate numerically those with \j — n\ < 3, etc. As a result, matrices A and B can be obtained fairly efficiently. This is the main advantage for using B-splines. Typically, we start a calculation with a guess for the extent of the few lower-energy localized eigenfunctions, and a guess for the size of the mesh interval. We then adjust these numbers until the energy and wavefunctions of the few lowest eigenstates of interest converge by being independent on these values. As mentioned above, this approach could be modified for a non-uniform mesh, Chapter 2. Formulation of the problem 9 Figure 2.2: Geometry of interest: an infinitely long rectangular nanomagnet is placed above one or more finite-width D M S layers that are embedded in an undoped semiconductor occupying the remaining z < 0 space. We choose the z axis to be perpendicular to the layers, and the y axis to be the axis of translational invariance. The nanomagnet has a cross-section of 26 x c, and is located between z £ [0, c]. A l l the D M S layers are located below it, Zi < 0. which could make the calculation even more time efficient. Clearly, a different strategy is needed if one is interested in extended wavefunctions, but this was not the case here. 2.3 Magnetic field of the rectangular magnet We now consider the case where the nanomagnet is infinitely long in y and has a rectangular cross section of width 26 in x and height c in z, as shown in Fig . 2.2. The magnetization M(r) inside the nanomagnet is assumed to be uniform, and to be parallel to z or parallel to x. Usually it is parallel to the direction with the longer side (i.e. M ( r ) = Mz if c > 26) in isotropic materials, but there are also layered materials where the magnetization has an easy axis that depends only on the direction of growth. As a result, it is also relevant to study M(f) — Mz even if c < 26. Similar considerations apply to the case M(f) = Mi. To find the magnetic field created by this nanomagnet, we recall Ampere's Chapter 2. Formulation of the problem 10 law and use the fact that there are no electric currents in our system: V x H(r) = J(f) = 0 (2.13) It follows that we can define a magnetic scalar potential 4>M(T) s o that H(f) = -V4>M(f). ' (2.14) But H{f) == -V<f>M(r) = —B(r) - M(f) (2.15) Mo and using the condition that V - S ( r ) = 0 , (2.16) we find an equation for the magnetic scalar potential: y2Mff) = V-M(r) (2.17) The general solution of Eq . (2.17) is [18]: *-« • -hu:^ 4TT JS \r-r\ 4 i r j v where 5 is the surface of the nanomagnet, and V is its volume. In our case the magnetization is uniform within the magnet; therefore, 4-7T Js \r — r | If we consider the case of magnetization perpendicular to the D M S layers, we only need to integrate over the z = 0 and z = c nanomagnet surfaces: M {x,y,z) = -—J dy' J ^dx'-yj{x - x')2 + (y - y')2 + z2 1 f°° , , fb , , M /OO fO dy' j dx' -oo J—b 47T . / - O O J_b ^{x - X>)2 + (y _ y>)2 + ( z _ c ) 2 (2.21) One integral can be done easily. Before performing the second one, we recall that what we need is H(r) = —V(f>M(r)- Taking these derivatives first allow us to then perform the second integral. The magnetic field B(r) = poH(r) outside the nanomagnet is found to be, in this case:. Bx(x,z) = ^ b x ( x , z ) (2.22) Chapter 2. Formulation of the problem 11 300 250 200 150 ~ 100 -150'— -300 _ - v \ . A \ ' ' ' / J ' ' i l l 1 I I 1 i ^ \ v , s / I . . » * * / / / I I I / / — -I I I " J i i 1 1 / / < 1 ' \ \ \ \ , „ , 1 \ \ \ V _ _ „ . I 1 \ \ \ -200 -100 0 x (nm) 100 200 Figure 2.3: The magnetic field lines outside an infinitely long rectangular nano-magnet with magnetization perpendicular to the plane and b = c = 150 nm. By(x, z) = 0 Bz(x,z) = !^-bz(x,z) where bx(x,z) = ln and (x - b)2 + z2 x-b)2 + (z- c ) 2 x + b) + ln bz(x,z) — 2 tan - I (z-c)\ (x + b)2 + (z- cf (x + b)2 + z2 (x-b) 2 t an" 1 (z-c) +2 t an" 1 •(x-bY - 2 t a n " 1 '{x + b)' z z (2.23) (2.24) (2.25) (2.26) The magnetic field when the magnetization is parallel to the plane, M || x, can be obtained by a 7r /2 rotation from the fields above, namely Bz —> — Bx and Bx —> Bz. Note that because By = 0 in all cases, we can obtain the solutions for the Schrodinger equation with M > 0 from those with M < 0 by 7r-rotations of the eigenspinors about the y-axis. Same holds true for a sign change in geg. Fig. 2.3 shows the magnetic field created outside such a nanomagnet. As expected, it has a cross-section similar to that of the field created by a magnetic dipole, and is invariant to translations in the third direction. For a magneti-zation perpendicular to the plane, Bx is always strongest at the corners of the magnet; its strength drops rapidly with distance away from the magnet, and the Chapter 2. Formulation of the problem 12 Figure 2.5: bz(x,z) as a function of x [see Eq . (2.24),(2.26)] at different dis-tances z below a nanomagnet with b = c = 150 nm. Chapter 2. Formulation of the problem 13 x (nm) z (nm) Figure 2.6: Magnetic fields Bx(x, z), Bz(x, z) and Bt(x, z) in units of /XOM/(4TT), at a distance of 10 nm from the surface of a nanomagnet of size b — c — 150 nm. The left side panels show the fields above and below the magnet, whereas the right side panels show the field to the left and to the right of the magnet. maximum shifts slightly to values \x\ > b, as shown in Fig . 2.4. As expected, Bx(x,z) = —Bx(—x,z) is an odd function in x. For the same magnetization direction, Fig. 2.5 reveals that close to the magnet, Bz is largest close to the edges, but as the distance increases, Bz becomes broadly peaked underneath the magnet and then it decreases fast away from it. As expected, this is an even function of x, Bz(x, z) = Bz(—x, z). From the Zeeman interaction, it is clear that a charge-carrier with the proper spin orientation wil l be attracted most strongly to where the total magnetic field Bt = \JB2+B"l is largest. From the analysis above, one expects these regions to be located roughly underneath the nanomagnet's edges. Indeed, this is shown to be the case for both magnetization orientations in Figs. 2.6, 2.7, for magnets with two different aspect ratios. From these considerations regarding the magnetic field created by the nano-magnet, we expect therefore the appearance of spin-polarized wavefunctions which are localized under each edge, and which are extended along the y di-rection. In the presence of an electric field, spin-polarized currents are thus expected to flow underneath the nanomagnet, as if through two waveguides. Some control over their location can be easily achieved by adding a small uni-form magnetic field, which will shift the overall position of the field maxima. 2.4 The DMS layers As shown in Fig . 2.2, we consider geometries with one or more D M S layers of various widths and locations, defined by the coordinates z\,z<2,.. .. Since the Chapter 2. Formulation of the problem 14 x (nm) z (nm) Figure 2.7: Same as previous figure, but for a magnet of size b = 75 nm and c = 300 nm. low-energy eigenfunctions are localized roughly below the nanomagnet's edges, we only need to consider finite length nanolayers, —a<x<a. In all cases we consider, we take a large enough so that the low-energy eigenfunctions are not influenced by edge effects, although we can also easily consider cases where a is comparable or smaller than b. We model such a system very simply. First, we require that all wavefunc-tions become zero for z > 0, i.e. outside the semiconductor. For z < 0, the Zeeman interaction is taken to be proportional to geg inside any of the D M S layers, and to be proportional to g = 2 geg in the rest of the semiconductor. Any sequence of layers can be considered as long as z i , z2, • •• are part of the knot vector defining the mesh in the z direction. We assume the effective mass and semiconductor band-edge to be the same for all z < 0 regions, using a digitally doped (or, for a single thick layer, a bulk-doped) G a M n A s semiconductor as an example. However, one can also envision the situation where the D M S is differ-ent from the spacer semiconductor surrounding it. In this case different effective masses and band-edges must be considered for the different regions. This can be easily accomplished with our scheme of computation. In particular, if the D M S has a significantly narrower gap than the surrounding material, which is the case considered in Ref. [17], then trapping of charge-carriers within the D M S layers is expected even in the absence of the external magnetic fields, as in any regular quantum well. While such a situation can lead to larger binding energies, it also reduces the degree of spin-polarization of the resulting eigenfunctions, since a large contribution to the trapping potential is independent of spin. Chapter 2. Formulation of the problem 15 2.5 Simplifying Schrodinger's equation As already discussed, because for this geometry we have to choose Yk(y) = elky j\/2rr, it follows that for each value of k we only need to solve an effective t 2 . 2 2D equation, with an eigenenergy Exz = E • 2m • 2 / P2 a2 *{x,z) = Exz<v(x,z) (2-27) We rewrite Eq . (2.27) in terms of the spin-up and spin-down components (defined with respect to the z axis): $(x,z) •4>l(x,z) ipl(x,z) as: + . -abx \ ( ^ \ = E X 1 / ^ where we define the energy unit (2.28) - E ° = 2 ^ - <2-29> and a dimensionless constant characterizing the strength of the Zeeman inter-action a = 8nE0 ( 2 - 3 0 ) (this is the value inside the D M S layers. Outside, we use the appropriately scaled down value). For magnetization perpendicular to the plane, since Bx is odd and Bz is even in x, it follows that if ipi is even in x then tpl is odd in x and vice versa. As a result, we only need to solve Schroodinger's equation for a quarter of the space, x > 0 and z < 0. We do it twice, first for the even (I/JJ) - odd (ipi) case. Here ip\(x = 0, z) = 0, showing that the B-spline B-s(x) should not be included in the set spanning the x axis for the spin-down component, only for the spin-up component. The other case, odd (I/'T) - even (tp[) is treated with the appropriate change. In both cases, we exclude Bs(z) and B-2(z) from the sets spanning both components' z axis, since the wavefunction and its first derivative are required to vanish on the semiconductor's surface, ^>(x, z = 0) = 0, jjj(x, z = 0) = 0. We assume a maximum length x < a and a maximum depth z > — d for the region considered, and choose a uniform mesh with Nx and Nz points in the two directions. The values for a, d, Nx and Nz are taken to be big enough so that the low-energy eigenfunctions are not influenced by further increases in any of them. As described previously, in both cases the problem is reduced to an effective equation of the form Ax = EBx, which we solve using the clapack [19] package Chapter 2. Formulation of the problem 16 Figure 2.8: C P U time scaling as a function of the number of mesh points Nx, for various values of Nz. As expected, the scaling is 0 ( n 3 ) . for the C + + programming language using compiler gcc 3.2.3 with optimization level 3. The program have been tested on simpler equations with quadratic potentials with the same symmetries but known analytic solutions, linked to harmonic oscillators. This is briefly reviewed in Appendix B . To examine the time required to solve the equation and the number of mesh points needed in x and z for B-splines, consider the case b — c = 150 nm, a = 262.5 nm, zl = —10 nm, z2 = —50 nm, d = 91.575 nm, and magnetization parallel to z axis. As explained, due to the symmetry of the problem, only the region x e [0,a] and z e [—d, 0] needs to be considered. The equations are solved on a 2 x 3.06 GHz Intel Xenon processor with 4 Gb of R A M provided by U B C / T R I U M F Westgrid. The program solves the equation in 0 ( n 3 ) time, where n x n i s the dimension of the A and B matrices, practically n = 2NXNZ for spin h. Characteristic run times are shown in Fig . 2.8. For cases where the number of partitions is < 30 in both x and z directions, most of the time is spent evaluating the integrals to create the matrix A. For larger number of partitions, the majority of the computation time is used to solve the matrix equation. F ig 2.9 shows how the ground state energy converges when increasing the number of partitions in x and in z. In the case shown, at least 20 partitions in x and 40 partitions in z are required to arrive at a reasonably accurate solution. It is not suprising that more partitions are required in z because the ground state eigenvector greatly varies with distance in z as shown in the next chapter. One can similarly study the dependence on the spatial cut-offs a and d and choose values large enough that the ground-state energy and wave-function be-Chapter 2. Formulation of the problem 17 -4 .041 1 1 —| -H—10 Partitions in z J&— 20 Partitions in z -A— 30 Partitions in z - • — 40 Partitions in z -©— 50 Partitions in z --B EJ 1 - 4 - 4 A \ o ,1 , , 1 , 1 "10 20 30 40 50 60 Number of Partitions in x Figure 2.9: Ground state energy as a function of the number of partitions in x and z. come independent on them. This is achieved when the wavefunction is localized well inside the x e [0,a], z e [—d,0] region considered. This is always the case for the results we show in the next chapter. As stated already, Wong [16] has previously used a similar approach to study the case of a very thin D M S layer. In this limit, the problem is assumed to be purely two-dimensional, with only x and y dependence. The wavefunction is still a planewave in the y direction, so one can reduce the problem to an equation similar to E q . (2.28). However, for an infinitely thin layer, Eq. (2.28) has no derivatives with respect to z, only to x, and the remaining wavefunction is only dependent of x. The dependence of z comes parametrically through the mag-netic field. Because of this drastic simplification, the problem is considerably easier to solve than the more general case we consider. -4.05 y Chapter 3 Results 18 We focus our attention first on the lowest energy eigenvectors, and discuss their eigenenergies Exz afterwards. We take / z 0 M = 1-06 T (typical saturation mag-netization for permalloy; this value is larger for Fe and Co), a moderate Lande factor ges — 500 and an effective mass m — 0.5m e , unless otherwise specified. Such values are very reasonable for holes in several D M S , such as G a M n A s and CdMnTe. We first consider the case of a nanomagnet of size b = c = 150 nm, and a single bulk-doped D M S layer located between z\ = —10 nm and z2 = —50 nm. For magnetization perpendicular to the plane, the ground state is found to be doubly-degenerate, with one even-odd and one odd-even solution. The total charge density \ipi(x, z)\2 + \ip[(x,z)\2 is identical in both cases, and is shown in Fig . 3.1. Note that the wavefunction becomes large just below z = —10 nm where the D M S layer begins, but does not extend all the way to z = —50 nm where the D M S layer ends, showing that charge-carriers are located in the region inside the D M S layer that has the largest total magnetic field (i.e., it is closest to the nanomagnet's edges). -200 -150 -100 -50 0 50 100 150 200 x (nm) Figure 3.1: Total charge profile for the (doubly-degenerate) ground state cor-responding to magnetization perpendicular to the plane. Here, b = c = 150 nm and there is a single D M S layer extending from -10 to -50 nm. Chapter 3. Results 19 Figure 3.2: Left/Right: Overlaps between (1,1)/(1, - 1 ) spinors corresponding to total spins parallel/antiparallel to the x axis, and the symmetric and anti-symmetric combinations of the even-odd and odd-even degenerate ground state eigenfunctions. Magnetization of the magnet is perpendicular to the plane, and the remaining parameters are the same as in the previous figure. The origin of the degeneracy is obviously due to the large distance between the two localization areas. If we abandon the even-odd and odd-even symmetry requirement, we can create a symmetric and an antisymmetric combination of the two eigenstates, one of which is localized entirely under the left edge and has spin pointing primarily in the +x direction, and the other of which is localized entirely under the right edge and has spin pointing primarily in the — x direction, as expected given the total local field (see Fig . 2.6). This is indeed confirmed in Fig. 3.2, and clearly shows that in this case, one can independently induce highly spin-polarized currents under each edge of the nanomagnet. Interaction between the states localized under the two edges (and lifting of this degeneracy) is only expected, and indeed observed, if the nanomagnet is extremely thin, in other words, b is very small, so that the two states start to overlap. Such thin nanomagnets are unlikely to be realized; we show a different way towards achieving this lifting of the degeneracy below. It is also clear that the D M S layer only needs to be about 15 nm thick in this case, since the wavefunctions do not extend to lower depths. Similar considerations apply if the magnetization is parallel to the plane, but the nanomagnet has the same dimensions. As expected given the ir/2 rotation relating the two cases, the only difference is that now the two degenerate eigenfunctions are polarized in the ±z direction, as shown in Fig . 3.3. Going back to the case of magnetization perpendicular to the plane, we can also lift the degeneracy between the two eigenstates by applying an additional uniform magnetic field along the z axis. This will lower the energy of the even-odd solution (because the total z-component of the magnetic field is largest near x = 0, where this solution has only spin-up contribution) and also bring the wavefunction closer to the center of the nanomagnet, as shown in F ig . 3.4. In this case, the expectation value of the spin is also expected to shift towards Chapter 3. Results 20 -200 -150 -100 -50 Figure 3.3: Left/Right: Overlaps between (l ,0)/(0,1) spinors corresponding to total spins parallel/antiparallel to the z axis. Magnetization of the magnet is parallel to the plane, and the remaining parameters are the same as in the previous figure. being parallel to the z axis, since this is the largest magnetic field component underneath the nanomagnet. If we instead add a uniform external field parallel to ± x , the degeneracy is again lifted, with the eigenfunction localized underneath the left/right edge having a lower energy, since its spin-polarization is favored by the orientation of the additional field. Its wavefunction stays close to the edge, where the total field remains largest and actually becomes even more tightly localized than previously, see Fig . 3.5. Such a scheme could be used to ensure that all charge-carriers are trapped underneath a single edge and therefore there is a single spin-polarized current that can be created in the system. These general considerations show that using rather moderate additional external fields, one can further manipulate the location and nature of the spin-polarization of the resulting eigenfunctions, and thus of the resulting spin-polarized currents. We now analyze the dependence of the ground state energy on various pa-rameters of the problem. Of course, ideally one would like the binding of these eigenstates to be larger than room temperature, so that charge-carriers trapped into these states cannot scatter into the continuum. Note that generally there are several bound states, but scattering from the ground state into the higher-energy bound states is less damaging to the degree of spin-polarization, since all the states bound under the same edge have similar spin-polarizations. We first analyze the dependence on the width of the D M S layer, assuming a single layer which always starts at the same distance z\ below the nanomagnet. Results are shown in Fig. 3.6. As expected, if the thickness is very small, the binding energy is close to zero (a weak two-dimensional trap can always bind a state, but usually with a very small binding energy). As the thickness increases, the binding energy quickly increases, until it saturates for a layer width of around 20-30 nm. As already discussed, this is expected, since the Chapter 3. Results 21 lllllli'MIIIHIIII 'It lilt I 111 II I I I I I I I I Ulil l l l l l l l l l l i l l l l l tllllMIIIIIH ll l l III. I I II in um mum min in I I in H I in in -50" -200 0 x ( n m ) Figure 3.4: Overlap between the spin-up spinor and the ground state wave-function if an additional uniform magnetic field parallel to z axis, of magnitude 0.5 (up), 1 (middle) and 2 T (lower panel) is turned on. A l l other parameters are as in previous figures. Chapter 3. Results 22 o -5 -10 -200 -150 -100 -50 0 50 100 150 200 x (nm) Figure 3.5: Charge density of the ground state wavefunction if an additional uniform magnetic field parallel to — x axis is turned on, for magnetization par-allel to the plane. All other parameters are as in previous figures. thickness (z2 - zl) (nm) Figure 3.6: The dependence of the ground state energy on the DMS layer thickness z\ — z%, when z\ is kept fixed at -10 nm. All other parameters are as in previous figures. Chapter 3. Results 23 ground state wavefunction is always "compressed" against the upper surface, where the magnetic field is largest. We can get a very rough estimate for this characteristic width in the following way. If all the kinetic energy was due to the variation of the wavefunction with the total thickness z, it could be roughly estimated as Ekin ~ Ti2/(2mz2). A t the same time, the potential energy is of the order Vpot ~ - f f e f f ^ l ^ f 0 • This would actually be the value if the wavefunction was fully spin polarized and near the maximum magnetic field possible, so it is an overestimate. In the ground-state of a one-dimensional trap, one expects a relationship such as Ekin ~ —jVpot, with 7 < 1 so that the total ground-state energy is negative. For example, 7=5 for a harmonic oscillator. In our case, this implies: h2 7 2mz2 4* if we take 7 = 1/2. For the parameters used, we find z ~ 5 nm, which is a factor of about 4 smaller than what we observe. Given the drastic approximations made, this is reasonably close. From the data in Fig . 3.6 we can draw a few more conclusions. First, comparing the black and red symbols, we see that a bigger (taller) magnet leads to a stronger binding energy. This is reasonable, since a bigger magnet creates a stronger magnetic field (more on this below). The other conclusion comes from comparing the black curve to the blue and green curves. They all correspond to magnets of the same dimension, magnetized either parallel or perpendicular to the plane. The difference is that in the green and blue curves, the Zeeman interaction is replaced by —gegszBz, so that only the Bz component of the magnetic field is actually effectively contributing to trapping. Studying such a case is relevant because there are D M S where the exchange between charge-carriers and M n spins, and therefore the resulting effective Lande factor, is highly anisotropic. As expected, the difference in the binding energy is not too large if the magnetization is parallel to the plane, since in this case the ground state is trapped in the regions of large Bz field. However, for a magnetization parallel to the plane, the wavefunction now migrates to the regions where Bz (not Bx) is largest, and the binding energy is affected more. Note that it is not an accident that the ground state energy shown in Fig . 3.6 (black and red symbols) is the same regardless of whether magnetization is parallel to z or to x. The profile of the total magnetic field is exactly the same because changing the magnetization direction simply switches the values of \BZ\ and |J3X|. Of course, properly speaking for magnetization parallel to the plane, the magnet should be labelled as having b = 75 nm and c = 300 nm. We next vary the height c of the nanomagnet, while keeping its width 2b constant. The results are shown in Fig . 3.7. As c approaches 0, this is equivalent to having no magnet and therefore no trapping. W i t h increasing c the overall magnetic field becomes larger and the binding energy increases. However, the total magnetic field saturates as well if the width c becomes too large, because the contribution from the upper surface to the scalar magnetic field [see Eq . Chapter 3. Results 24 p = c /b Figure 3.7: The dependence of the ground state energy on the aspect ratio p = c/b of the nanomagnet. There is one D M S layer between z% = —10 nm and z2 = —50 nm. A l l other parameters are as in previous figures. Figure 3.8: The dependence of the ground state energy on geg. A l l other parameters are as in previous figures. Chapter 3. Results 25 Figure 3.9: The ground state energy versus additional external magnetic field for a nanomagnet with b = c = 150 nm and magnetization perpendicular to the plane. A l l other parameters are as in previous figures. (2.21)] becomes negligible. This analysis shows that for an aspect ratio p = c/b ss 2 one is already fairly close to the maximum binding energy achievable, meaning that a square-shaped magnet is fairly close to optimum. The most efficient way to increase the binding energy is, not surprisingly, to increase the effective Lande factor geg, since this leads to a linear increase of the trapping potential. This is shown in Fig . 3.8, which demonstrates that very large binding energies (comparable to room temperature) can be achieved for reasonable values of ges- As discussed, the value of geg can be changed as a function of T or of the M n doping x. This figure demonstrates that highly spin-polarized currents could be created at room temperature using this scheme. A n alternative scheme for increasing the binding energy is to add a large uniform external magnetic field. For the situation with the magnetization per-pendicular to the plane, Fig . 3.9 tracks the ground state energy as a function of this additional field. If the additional field is very large, the field created by the nanomagnet becomes irrelevant and one finds a linear decrease of the ground state energy, as expected because of the uniform Zeeman shift. Note that for such large and uniform external magnetic fields, the vector potential terms in the Schrodinger equation (not included here) become relevant, lead-ing to appearance of quantized Landau levels. However, such cases are not of interest to us since in this case the wavefunctions are no longer spatially local-ized under the nanomagnet, but become uniformly spread throughout the D M S layer. The interesting behavior happens for smaller additional fields, where one observes a deviation from the linear dependence. In this case, the wavefunc-tion is still localized under the nanomagnet, and therefore spatially well-defined Chapter 3. Results 26 -30 -35 • - 4 0 1 - 4 5 1 -50 • • -200 -150 -100 -50 0 50 100 150 200 x (nm) Figure 3.10: Charge density in the presence of 4 layers of D M S , located between [-10, -15], [-20, -25], [-30, -35], [-40, -45] nm. The nanomagnet with b = c = 150 nm has magnetization perpendicular to the plane. A l l other parameters are as in the previous figures. spin-polarized currents can be created. For perpendicular magnetization it is more efficient to apply an additional magnetic field parallel to the x axis, since this will favor an already existing spin-polarization. We can also study the case of digital-like doping, modeled as several very thin D M S layers. Figure 3.10 shows the density of charge for the (degenerate) ground state in the presence of 4 such layers. Not surprisingly, the wavefunction is localized in the upper two layers, where the magnetic field is larger. Thus, the location and number of lower D M S layers becomes irrelevant in this case. The ground state energy is also increased, as expected since part of the wavefunction is localized in the higher energy regions in between the layers. The dependence of the ground state energy on the number of D M S layers is plotted in Fig . 3.11, assuming that layers start at z\ = —10 nm, and are each 1 nm thick and separated by 1 nm spacers. The binding energy increases until the layers occupy all volume over which the wavefunction would normally spread, after which it saturates. Even for such thin spacers, however, the ground state energy for a multiple-layer configuration is still larger than the ground state energy for bulk-doping. If layers with anisotropic coupling are used, the binding is even weaker. This conclusively shows that bulk-doping is more advantageous. Chapter 3. Results 27 4 b = c = 150 nm, Mag Z, B z only, layers A—A b = c = 150 nm, Mag X , B z only, layers T b = c = 150 nm, Mag Z or Mag X , bulk Number of 1 nm Layers Separated by 1 nm Figure 3.11: The ground state energy as a function of number of D M S layer, for a configuration with 1 nm thick D M S layers separated by 1 nm spacers. The first layer starts at zl = —10 nm. For comparison, the ground state energy for bulk-doping (with total width equal to that occupied by the layers) is also shown. 28 C h a p t e r 4 Optimal Configuration To get the lowest ground state energy, it is necessary to fine-tune the placement and dimensions for both the nanomagnet and D M S layer(s). In general, the ground state energy decreases with distance between the nanomagnet and the D M S , but they cannot be too close together, since in this case the charge-carriers are no longer confined in the D M S , but can tunnel into the nanomagnet (which are generally metals). This is why we considered a separation of lOnm between their surfaces. This distance could, in principle, be lowered if a thin layer of a large-gap oxide was sandwiched in between. Considering the nanomagnet, an aspect ratio with p = 2, i.e. a square cross-section, is close to optimal, since there is saturation in the ground-state energy after this point as seen in Fig . 3.7. In such case, the magnetization direction does not affect the energy, only the spin direction of the trapped charge-carriers. However, if the D M S has easy-axis anisotropic coupling, the magnetization of the nanomagnet needs to be perpendicular to this easy-axis direction because the magnetic field is strongest in the directions perpendicular to the magnetization. For a fixed aspect ratio, the bigger magnet will have a bigger binding if the depth z\ is kept fixed. For the D M S , the most important parameter is geg, whose increase leads to significant increases in the binding energy. Clearly, one should choose the D M S so as to maximize this parameter. The thickness of the D M S layers, however, is also very important. As shown, bulk-doped D M S provides better trapping than multiple thin layers since the spaces between the layers, which can have strong magnetic fields, are not used to trap the charge-carriers. For bulk-doping, the width of the layer must also exceed a certain value in order to maximize the binding energy. For the cases we have investigated, this minimal thickness is of order 20-30 nm, see Fig . 3.6. Adding a small external field lowers the energy along with changing the con-fining region of the charge-carriers, as shown in Fig. 3.4. This opens up further possibilities in manipulating the binding energy and the spin-polarization of the eigenstates, as well as their spatial location, in some cases. A n external field parallel to the magnetization enables localization of the charge-carriers closer to the center of the magnet. If instead the external field is perpendicular to the magnetization, symmetry is lost at the edges, and one of the edges binds states more strongly than the other. C h a p t e r 5 Conclusion 29 The low-energy bound eigenvalues and eigenfunctions are calculated for charge-carriers in one or more layers of paramagnetic D M S with a giant Zeeman effect, placed in the external magnetic field of an infinitely long rectangular nanomag-net with uniform magnetization. The work in this thesis differs from other work along somewhat similar lines, published in Ref. [17], in that it takes into.account larger layer widths for the D M S . In the previous work, only very narrow layers of a D M S different from the surrounding semiconductor, and therefore with a different gap, were.considered. The magnetic field was assumed to be constant over the D M S width for ease in computation. Moreover, a large contribution to the binding energy came from the quantum well effect, given the band-edge mismatch, with the consequence that the degree of spin-polarization was decreased. Our work is more general, since we can investigate any sequence of D M S layer widths using the exact magnetic field. We find that the eigenstates are trapped in the regions where the magnetic field is the largest in the D M S , usually under the edges of the nanomagnet. The location can be manipulated some-what by applying supplementary magnetic fields. Interestingly, these states are spin-polarized. Given the reasonably large binding energies, this shows the pos-sibility of creating quasi-one dimensional traps for spin-polarized charge-carriers at rather high temperatures. These could be used to guide spin-polarized cur-rents under the nanomagnet's edges, and therefore be of great interest to those working in the area of " spin electronics". 30 Bibliography [1] H. Ohno, J. Magn. Magn. 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Sorensen, LAPACK Users' Guide, (Society for Industrial and Applied Mathematics, 3rd edition, 1999). 32 Appendix A B-splines B-splines are a complete, but not orthgonal, set of polynomials which are used to approximate a function / (x ) from [a, b]. To define the B-splines, the interval needs to be split into N smaller partitions with end points specified by: a = XQ < x\ < X2 < ••• < XN = b. The non-decreasing sequence of end points of the partitions are known as the knot vector. In addition, all the polynomials in the set have the same rank r and each polynomial spans a different r consecutive partitions except at the boundaries. A function is then written as a linear combination of the B-splines where the coefficents for each B-spline separates one function from another. The mathematical properties of B-splines, BiiT(x), are below where i specifies the partition. For an example of B-splines, refer to Fig. 2.1. 1. BiiT(x) > 0, V i , r , x. 2. Bi,r(x) is non zero only within [xi, X i + r + i } . Therefore, within any par-tition, [xi,Xi+i], only r + 1 consecutive Bj>r(x) have non-zero values where j = % — r, i — r + 1 , i . 3. Bi<r is a polynomial of order r. Bitr, its derivatives, up to and including the r — 1th derivative are continuous. 4. Yli^i,r(x) = l j Va: e [xR,XN-R]. For a given rank, B-splines with i = 0 , N — r.form a complete, but not orthogonal, set on this interval. As a result, if we want completeness from [a, b), we need to add points x_ r , a ;_ r +i , . . : , x _ i and XN+I,XN+2, —,XN+r to the knot sequence and extend the number of B -splines to N + r. The B-splines are defined recursively: = {J o « h e e J S ; I , + l 1 < A -" Bt,T(x) = X ' + r+'~ X i W - i ( l ) + * " ' ' * B . , r - i ( i ) (A.2) 33 Appendix B Checking the Program To check our program, we solved three other Schrodinger's equations and com-pared the results to the analytic solutions. The analytic and numerical eigen-values and eigenvectors agree. Figs. B.1-B.3 show how quickly the smallest eigenvalue converges with the number of partitions from [0,7.5] in both x and 1. Harmonic Oscillator in x and Potential Well in z dx2 dz2 %(x,z) = e^(x,z) (B. l ) We recall that the eigenvalues for a harmonic oscillator are en — 2 n + l where n > 0, and the eigenvalues for a'potential well are ema = ( m ^-) 2 where m > 1 and 2a is the width of the well. Let us call the eigenvectors that solve to the harmonic oscillator ipHo(x)- Figure B . l shows the expected eigenfunction; for the harmonic oscillator, the wavefunction goes to 0 near 4, and for the potential well, the wavefunction goes to 0 at the boundary. Due to the symmetry of the potential, the solution can be obtained numerically by considering only the region where x > 0 and z > 0. 2. Harmonic Oscillator Variant 1 in x and Potential Well in z d 2 \ 2 / n *£>(x,z) = e^f(x,z) (B.2) where a < 1. ' To solve this equation analytically, we remove the z dependence knowing the solution is the potential well.' Then we set the spin +x wavefunction = ± spin —x wavefunction. We then get: dx2 •x2(l±a) 4>(x) = e±tp(x) Next, we make the substitution x = u/X to get: d2 1 ± a' 4>{u) = -jpipiu) (B.3) (B.4) Therefore, we set w 2 ( ^ ^ ) 1 to get the eigenvalues: = (2n + ± a in x for n > 0 with the eigenvectors tpHoi^^ ^ ax)- Notice that this variant of the harmonic oscillator shares the same symmetry as the harmonic oscillator, Appendix B. Appendix B Checking the Program 34 Figure B . l : Left/Right: The smallest eigenvector/eigenvalue calculated nu-merically using B-splines in the interval [0,7.5] in both x and z for a harmonic oscillator in x and a potential well in z with a = 7.5. The exact eigenvalue is 1.04386. Figure B.2: Left/Right: The smallest eigenvector/eigenvalue calculated nu-merically using B-splines in the interval [0,7.5] in both x and z for a harmonic oscillator variant 1 in x and a potential well in z with a = 7.5 and a — 0.5. The exact eigenvalue is 0.75097. Appendix B. Appendix B Checking the Program 35 Number of Partitions-, ir Figure B.3: Left/Right: The smallest eigenvector/eigenvalue calculated nu-merically using B-splines in the interval [0,7.5] in both x and z for a harmonic oscillator variant 2 in x and a potential well in z with a = 7.5 and a = 2.5. The exact eigenvalue is -5.20614. and therefore, only the first quadrant needs to be considered when solving the equation numberically. 3. Harmonic Oscillator Variant 2 in x and Potential Well in z dx2 dz2 + x + 2axffx *{ar, z) = e*(x,z) (B.5) Solving this equation, we follow the same initial steps as for variant 1. We remove the z dependence and set the spin +x wavefunction = ± spin —x wave-function. GS)+*2±2a* Next, we subtract and add a2 and simplify to get: tp(x) = e±4>(x) (B.6) (-) \dx2) (x ± of ip(x) = ( e ± + a 2 ) ^ ( x ) (B.7) Using the substitution x ± a = u, we end up with the harmonic oscillator. The eigenvalues in x are en = (2n + 1) - a2 where n > 0. Therefore, the eigenvectors are TPHO{X ± A)- Only the first quadrant needs to be considered when solving this equation because one value of a produces two solutions which can be combined to create even-odd and odd-even solutions in x; as a result, these solutions have the same symmetry as the previous examples. 

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