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Conductivity and magnetoresistance of TTF-TCNQ Tiedje, J. Thomas 1975

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CONDUCTIVITY AND MAGNETORESISTANCE OF TTF-TCNQ by J. THOMAS TIEDJE B.A.Sc., University of Toronto, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1975 In presenting th is thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain shal l not be allowed without my wri t ten permission. Department of The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date 1, ? 9 7 < r ABSTRACT Four probe d. c. e l e c t r i c a l conductivity measurements have been made as a function of temperature on 19 single crystals of tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ). Although the data is mainly for the crystallographic b axis conductivity, some less complete a axis data i s also presented. The temperature dependence of the electronic energy gap i s calculated from the b axis conductivity data. The magnetoresistance for currents along the B axis of TTF-TCNQ, has been measured as a function of temperature Between 17K and 98K in static fields up to 50k0e. For T 5 54K the magnetoresistance Ap/p = {p(50k0e)-p(0)]/p(0) i s less than 0.1% i n magnitude. There i s a peak of about -1.4% at 52.8K. Below 50K Ap/p is small and negative and i s described reasonably well By the formula Ap/p = -(1/2)(ugH/kT) . At a l l temperatures Ap/p was found to Be approximately independent of the orientation of the applied f i e l d with respect to the current. The high temperature behaviour i s consistent with that expected for a metal in the short scattering time limit.(w cT<<1). We attribute the peak at 52.8K to the suppression of the metal-insulator transition By the magnetic, and we show why such Behaviour would Be expected for a Peierls i i i transition. In the low temperature region the crystal acts as a small gap semiconductor for which the T 2 dependence of A p / p i s easily understood. iv TABLE OF CONTENTS Page Abstract i i Table of Contents iv List of Tables v Li s t of Figures v i Acknowledgements ix CHAPTER I 1.1 Background 1 1.2 Conductivity Measurements 8 CHAPTER II 2.1 Introduction 14 2.2 Experiment 17 2.3 Experimental Results 26 2.4 Interpretation 29 2.5 Discussion 45 Appendix: Systematic Error Checks Bibliography 48 52 V LIST OF TABLES Table Page I Summary of conductivity data. 9 II Conductivity of the samples used in the magnetoresistance experiments. 15 I I I Apparent temperature shift of the electronic energy gap in a magnetic f i e l d of 50k0e for T c = 52.8K. 39 LIST OF FIGURES gure 1 TTF and TCNQ molecules. 2 Crystal structure looking along the a axis. 3 One dimensional, h a l f - f i l l e d , electronic energy bands with (a) no gap at the Fermi level and (b) a non-zero gap at the Fermi level. 4 Density of states for the one dimensional bands shown in Figure 3. 5 One dimensional lattices corresponding to the band structures shown in Figure 3. 6 Contact configurations used for (a) b axis and (b) a axis conductivity measurements. 7 Photograph of mounted TTF-TCNQ sample. 8 Top: Conductivity along the crystallographic b axis. Bottom: Conductivity along the b axis in the semi--conducting phase. 9 Top: Conductivity along the crystallographic a axis. Bottom: Conductivity along the a axis in the semi--conducting phase. v i i Figure Page 10 Electronic energy gap determined by f i t t i n g the b axis conductivity. 13 11 Low temperature probe used for conductivity and magnetoresistance measurements. 18 12 Photograph of low temperature probe. 19 13 Chart recorder trace from a magnetoresistance measure--ment at 24K. The temperature has drifted steadily up a total of about 0.03K in the 35 minute trace shown. 23 14 Magnetoresistance measurements for three samples; The dashed lines are the function, - YJT2TJ W ^ T N H = 50k0e. The symbols 0, A, x, #, 1, and 9 indicate experimental runs #1-5; 6; 7,8; 9,10; 11; 12,13 respectively. 25 15 Resistivity derivative curve. The peaks near 38K and 52K were obtained by monitoring the crystal resistance on a chart recorder as the sample temperature drifted slowly. The remainder of the curve was obtained by measuring the resistance change for a 20mK temperature change at 2K intervals. 27 v i i i Figure Page 16 Electronic energy gap near calculated from the model described in the text. The dashed line i s the gap in the presence of a magnetic f i e l d of 50k0e. 36 17 Theoretical magnetoresistance for the model described in the text. The dashed curve is an expansion of the solid curve near T c. 43 18 Magnetoresistance theory (solid line) and experimental data for sample 1. 44 19 Circuit diagram of proportional/integral temperature controller used with the capacitance sensor. 51 20 Capacitance-inductance bridge used for measuring capacitance. 51 ACKNOWLEDGEMENTS It i s a pleasure to acknowledge the active support of Dr. J. F. Carolan under whose supervision this project was carried out. Also Dr. A. J. Berlinsky's many useful suggestions had a strong influence on the outcome of the project, especially with regard to the interpretation of the experimental results. W. I. Friesen's generous assistance in obtaining a numerical solution to the energy gap equation is gratefully acknowledged. Finally, I benefited a great deal from many valuable discussions with Dr. B. Bergersen, Dr. W. N. Hardy, and Dr. G. G. Lonzarich. The TTF-TCNQ samples used in these experiments were synthesized and grown by Dr. L. Weiler and his co-workers i n the Chemistry Department. Their help was v i t a l to the success of the entire project. I am grateful to the National Research Council for financial support in the form of a Science Scholarship. 1 CHAPTER I 1.1 Background The quasi-one dimensional organic conductor tetra-•r-thi'afulvalene-^tetracyanoquinodimethane (TTF-TCNQ) has been the subject of a considerable amount of study in recent years. The material i s interesting for two major reasons. F i r s t i t i s one of a small group of examples of quasi-one dimensional J conductors (Zeller 1973, Elbaum 1974) in which one might expect to observe effects which are peculiar to one dimensional electronic systems (Allender et a l 1974, Lee et a l 1973, Luther and Peschel 1974, Patton and Sham 1973). The term "quasi" is usually interpreted to mean that the material can be viewed as a collection of identical linear chains which are (1) sufficiently weakly coupled that a one dimensional band structure accurately describes the individual chains, and (2) sufficiently strongly coupled that the behaviour of the system is not dominated by the effects of thermodynamic fluctuations as are truly one dimensional systems (Landau and L i f s h i t z 1969). A second reason for interest i n the material is that i t is a member of a new class of organic conductors, whose conductivity (Ferraris et a l 1973, Coleman et a l 1973, Cohen et a l 1974, Groff TTF T C N Q Figure 2 Crystal structure looking along the a axi 3 et a l 1974) is higher than any organic solids previosly synthesized. In principle a good organic conductor opens up the poss i b i l i t y of designing new materials with desireable electronic properties, because of the nearly limitless f l e x i b i l i t y of organic chemistry. In fact L i t t l e (1964) has suggested a possible new mechanism for high temperature superconductivity, which might be realized i n a system of conducting organic chains with polarizeable side chains. Finally i t should be mentioned that anomalously high conductivities have been measured in a small percentage of TTF-TCNQ samples at one laboratory (Cohen et a l 1974). However these measurements have not been duplicated elsewhere. The anisotropy of crystalline TTF-TCNQ results from the stacking behaviour of the relatively large and f l a t TTF and CNQ molecules (Figure 1). In the solid the molecules form segregated TTF and TCNQ stacks (Kisterrmacher et a l 1974) with relatively strong coupling between molecules on the same stack and weak coupling Between molecules on different stacks (Figure 2). As a result 1, the crystallographic B axis, which is the direction with strong intermolecular coupling, is the highly conducting direction. (a) (b) One dimensional, h a l f - f i l l e d , electronic energy bands with (a) no gap at the Fermi level and (b) a non-zero gap at the Fermi level. (a) (b) Density of states for the one dimensional bands shown in Figure 3. U — - * - | 2b |*-• • • • • • e o • e • « (a) (b) One dimensional lattices corresponding to the band structures shown in Figure 3. 5 It i s also usually the long axis of the needle-like crystals. The b axis conductivity i s metallic from room temperature down to about 58K where the conductivity is 10 to 20 times greater than the room temperature value of 400-1000 (ftcm)"' . Below 58K the material undergoes a metal^insulator transition to a small gap (0.02-0.04eV) semiconductor. To set the scale the room temperature conductivity of copper i s about 570 000 (ficm) ' and the maximum conductivity of the best organic conductor before TTF-TCNQ (NMP-TCNQ, Schegolev 1972, Epstein et a l 1972) was ^200 (ftcm)-1. Peierls (1955) pointed out that a one dimensional metal is unstable with respect to a particular type of la t t i c e distortion. This i n s t a b i l i t y can be explained by considering a one dimensional, half-rfilled, tight-binding, electronic energy band as shown i n Figure 3(a). The corresponding density of states i s shown in Figure 4(a). The average electronic energy may be lowered by creating an energy gap at the Fermi level. Such an energy gap w i l l be formed i f the one dimensional l a t t i c e undergoes a distortion with a wavevector 2kp that spans the fermi surface. For the h a l f - f i l l e d band considered here, the distortion required is simply a dimerization of the one dimensional chain as shown in Figure 5(b). Clearly, for an energy Figure 6 Contact configurations used for (a) b axis and (b) a axis conductivity measurements. 6fe F i g u r e 7 P h o t o g r a p h of mounted TTF-TCNQ sample. S c a l e : 7X a c t u a l s i z e . 8 band which i s not exactly h a l f - f i l l e d the distortion which lowers the energy of the electronic system w i l l have a periodicity determined by the position of the Fermi level. In principle this Instability can also exist in three dimensional systems. However the lat t i c e distortion which i s required in order to create an energy gap over the entire Fermi surface, i s in general much more complicated. A one dimensional i n s t a b i l i t y of this type i s commonly referred to as a Peierls distortion, and i t i s a possible mechanism for the metal-^insulator transition in TTF-TCNQ. 1.2 Conductivity Measurements Detailed d.e. b axis conductivity measurements were made on nineteen single crystal samples of TTF-TCNQ from room temperature to about 10K. The measurements were made with a four probe technique using silver paint to make ele c t r i c a l contact to the crystals. A photograph of a sample mounted for conductivity measurements i s shown in Figure 7. In addition the a axis conductivity of two crystals was measured using a technique due to van der Pauw (1961) and Montgomery (1971). In this method leads are connected to each corner of the top face of a crystal as shown in Figure 6(b). First current i s injected at 1,1',and the voltage measured at 2,2', then Summary of Conductivity Data. Sample a R T(ncm) 1 °MAX °RT 18 Apparent a negative below 60K, 22 Behaves as a semiconductor at room temperature. 23 480±70 13.5 24 520±100 12.7 25 720±140 13.2 28 790±90 14.2 30 1150±340 14.9 31 800±240 13.6 36 99±8 8.8 37 430±100 13.6 39 490±150 13.1 40 370±50 10.5 41 390±100 8.8 44 540±90 15.2 45 7401200 12.4 46 390±60 15.2 48 * 50 •k 660±110 177-190±4O 15.9 23.8-9.9 51 430-480±130 4.7-8.3 These samples were measured with current leads on the corners of the ends of the crystal. The apparent conductivi depends on which current connections are used. 9b F i g u r e 8 C o n d u c t i v i t y a l o n g t h e c r y s t a l l o g r a p h i c b a x i s . Bottom: C o n d u c t i v i t y a l o n g t h e b a x i s i n t h e s e m i c o n d u c t i n g phase. TTF-TCNQ b AXIS CONDUCTIVITY dRr- 66O(0.CM] -I JV1J SAMPLE 4-8 IOO 150 2 0 0 2 5 0 T(K) b AXIS CONDUCTIVITY SAMPLE 1 8 4 6 8 I O 0 / T 00 10b Figure 9 Top: Conductivity along the crystallographic a axis. Bottom: Conductivity along the a axis i n the semiconducting phase. 11 • « e o a AXIS CONDUCTIVTY <L = i A (-acM)"' S A M P L E 53 0 100 T(K) 200 300 0 -2 RT -4 -6 a AX/S C O N D U C T I V I T y SAMPLE 53 4 /oo/T ( K " ) the current i s injected at 1,2 and the voltage measured at l',2'. From these measurements a mathematical transformation due to Montgomery (1971) can be used to separate the a and b components of the conductivity tensor. A summary of the conductivity measurements i s shown in Table I. The b axis conductivity of a typical crystal is shown in Figure 8, and the a axis conductivity of another crystal i s shown in Figure 9. Notice that there i s a rapid change in both the a and'b axis conductivities i n the v i c i n i t y of 53K and 38K. The structure at 53K i s attributed to the sudden opening up of an electronic energy gap associated with the metal-insulator transition. From the b axis conductivity data, the feature at 38K also looks l i k e a sudden increase in the electronic energy gap, and i s probably associated with another phase transition at that temperature. The energy gap as a function of temperature may be calculated from the b axis conductivity data in the following way. Assume a h a l f - f i l l e d tight-binding one dimensional electronic energy of width 0.2eV at 54K, and neglect effects due to the temperature dependence of the charge carrier relaxation time. If one treats the energy gap A(T) a s a n adjustable parameter, the conductivity below T (K) Figure 10 Electronic energy gap determined by f i t t i n g the b axis conductivity. 54K can be fitte d using the expression (Ziman 1972), o(T) = with the band structure given by, / 2 E K = ± / A ( T ) + e -where i s the energy band structure above the transition temperature. This expression for w i l l be derived later. The constants appearing in the conductivity expression are the relaxation time T , the. lat t i c e constant b, the volume of a unit c e l l ft, and the charge carrier velocity v^. A graph of A(T) which gives a f i t to the conductivity i s shown i n Figure 10. CHAPTER II 2.1 Introduction This thesis project was originally undertaken i n an attempt to duplicate the anomalously high conductivity measurements mentioned earlier, and then measure the effect of a magnetic f i e l d on this conductivity. Although no anomalously high conductivities were found, and no magnetic f i e l d effects were observed i n the vi c i n i t y of the peak conductivity at 58K, the magnetic f i e l d measurements at lower temperatures do give some interesting information about the nature of the low temperature insulating phase. tion and interpretation of some measurements of the effect of a magnetic f i e l d on the b axis conductivity. The data is presented as magnetoresistance defined by [p(H)-p(0)]/p(0) where p(H) i s the sample r e s i s t i v i t y in the presence of a magnetic f i e l d H. The magnetoresistance of three crystals of TTF-TCNQ has been measured as a function of temperature between 17K and 98K in fields up to 50k0e. The crystals were mounted in three mutually orthogonal orientations relative to the magnetic f i e l d . The magnetoresistance data i s interpreted by treating the material as a one dimensional metal which undergoes a Peierls transition at 52.8K to a semi-conducting state. Both the model calculation and the experimental results indicate that the magnetic f i e l d lowers the metal-semi-conductor transition temperature by an amount, The magnetoresistance that i s calculated for the model is found to be in close agreement with the experimental data. The remainder of this work is concerned with the descrip-Table II Conductivity of the Samples Used in the Magnetoresistance Experiments. Sample aRT(ftcm) 1 a^^Cficm) 1 la 740±200 910012500 2 a 540190 82001300 3 b 390+100 34001900 a decreased ^6% on repeated (>15) thermal cyclings. RT k 0 M A V decreased ^25% on repeated cycling. MAX Note: Samples 1, 2, and 3 li s t e d here correspond to sampL 45, 44, and 41 respectively, lis t e d in Table I. 17 a 2.2 Experiment The single crystal samples of TTF-TCNQ used i n the experiment were prepared in the Chemistry Department of this university. Approximately 2.5 mm by 0.1 mm by 0.02 mm needle - l i k e crystals were mounted with Dupont #4929 s i l v e r paint on 0.001 i n . gold wires i n the standard four probe configuration used to measure the e l e c t r i c a l conductivity along the highly conducting direction of the crystal (needle axis). The s i l v e r paint contact resistances were a l l < 5ft . Indium solder was used to connect the gold wires to the copper current source and voltmeter leads. The room temperature and peak (near 58K) b axis conductivities of the three crystals on which the most extensive measurements were taken, are shown in Table I I. Less detailed magneto-resistance measurements, that were made on two additional samples, were consistent with the results from the f i r s t three samples. Although the crystal conductivities dropped s l i g h t l y after repeated thermal cycling, no discontinuous jumps in the conductivity indicative of cracking i n the sample or contacts were observed. The sample holder mounting configuration i n the cryostat permitted three crystals to be measured at the same time, with three mutually orthogonal orientations of the long axis of the c r y s t a l . The three sample holders were mounted on a probe made of a high thermal conductivity copper. A calibrated s i l i c o n diode temperature sensor (Lake Shore Cryotronics, Type DT 500) was mounted close to the samples, and another s i l i c o n diode and capacitance thermometer Figure 11 Low temperature probe used for conductivity and magnetoresistance measurements. 18a PUMPING LINE He EXCHANGE GAS HEATER OUTER DEWAR TRACE OF He GAS CAPACITOR SENSOR Si DIODE SENSOR COPPER PROBE HEAT SINK POST SUPERCONDUCTING SOLENOID INNER DEWAR VACUUM Si DIODE SENSOR T T F - T C N Q S A M P L E BAKELITE S A M P L E HOLDER He EXCHANGE GAS 1 8 b F i g u r e 12 P h o t o g r a p h of low t e m p e r a t u r e p r o b e . 20 were mounted 8.6 cm above the top crystal on the same piece of copper (Figure 1). Resistance wire was wrapped around the top of the probe as a heater, and the bottom of the probe was enclosed in a small stainless steel dewar can. This small can was placed inside a larger stainless dewar, which in turn was immersed i n a li q u i d helium bath. The end of the outer dewar f i t t e d into the centre of a 50 kOe superconducting magnet. (See Figure 11).. In operation the vacuum jacket on the inner dewar is evacuated, and a fraction of an atmosphere of helium exchange gas is introduced into the space between the small inner can and the outer can, and into the space around the crystals. A much smaller quantity of helium is introduced into the jacket of the outer dewar to allow the system to cool. Typically 250 mW of heater power i s required to hold the probe temperature constant. The small vacuum jacket around the probe was designed to reduce thermal gradients between the sensor and the sample and to minimize thermal d r i f t s . In practice, since the exchange gas can circulate from the inside to the outside of the small dewar can through the holes for the sensor leads, there i s a small thermal gradient ( <.1K) along the length of the probe. Although the small dewar did not eliminate the thermal gradient entirely, i t did reduce the thermal d r i f t to acceptable levels. The calibrated s i l i c o n diode was used to measure the-temperature in zero f i e l d . When the magnetic f i e l d was turned on, a glass 21 ceramic capacitance thermometer (Lake Shore Cryotronics, Type CS-400) was used to maintain a fixed temperature. The calibration of the s i l i c o n diode was checked with a platinum resistance thermometer (Thermal Systems, Type 5001-A), which was in turn calibrated at 4.2K, the hydrogen and nitrogen t r i p l e points and at the ice point. As a result we believe the absolute accuracy of the temperature measurements to be better than 0.2K. The capacitance sensor was measured with a precision variable capacitor (General Radio #1422CB) and a capacitance-inductance bridge of the type described by Thompson (1958). The bridge was sensitive to capacitance changes of .1 pf with a 50 mV rms modulation voltage at 5kHz on the sensor. At 64K and about 25kOe we were able to 4 measure a small "but significant (1 part in 10 ) f i e l d dependence of the capacitor. This observation is in contrast to zero magneto-capacitance (< 1 part in 10^) reported at 4.2K i n up to 140 kOe (Rubin et al 1971). However both results are consistent with the comments of Sample et a l (1974) which suggest that there is a f i e l d dependence in the v i c i n i t y of 60K which disappears as the temperature is lowered to 4.2K. The f i e l d dependence of the sensor was measured with the temperature controllers turned off, and the probe temperature d r i f t i n g very slowly. It is conceiveable that the f i e l d dependence may relax to zero with a r e l a t i v e l y long time constant (as suggested to us by P.M. Chaikin). At any rate by sweeping the f i e l d up and down and progressively pulling the sensor farther out of the magnet we were able to determine that for fields less than about 8k0e at 64K, the sensor exhibits 22 a negligible magneto-capacitance ( < 1.0 mK equivalent temperature change). The capacitor has another inconvenient feature. After cooling from room temperature to the experimental operating point (between 18K and 53K) the capacitance exhibited an exponential relaxation of about 200pf with a time constant of about 45 minutes (Lawless 1972). TPte crystal resistance was measured with a d.e. technique. A combination of resistors and a mercury battery provided constant currents of luA, lOuA, and lOOuA to the sample and the resu l t i n g crystal voltage was measured with a Keithley 140 nanovoltmeter. For magneto-resistance measurements, the nanovoltmeter was offset to zero, and then the output was amplified 30 times. The nanovoltmeter, the capacitance bridge, and the heater current were a l l monitored on a s t r i p chart recorder during the experiment. After cooling from room temperature, and before turning on the magnet, the system was allowed to s t a b i l i z e at the temperature of interest for at least an hour u n t i l the crystal temperature was stable and the capacitor d r i f t manageable. At this point the temperature controller was switched from the calibrated diode to the capacitor and a new baseline established. , Then the magnetic f i e l d was swept from zero to 50 kOe i n four to ten minutes, held steady at 50 kOe for up to twenty minutes, and then swept back to zero, A typical chart recorder trace of the crystal 22b Figure 13 Chart recorder trace from a magnetoresistance measurement at 24K. The temperature has drifted steadily up a total of about 0.03K i n the 35 minute trace shown. 23 C R Y S T A L VOLTAGE — -voltage from a magnetoresistance measurement i s shown i n Figure 13. The principal source of uncertainty was i r r e g u l a r i t y i n the baseline d r i f t . In addition one of our primary concerns was that a spurious temperature s h i f t might have somehow been introduced by the magnetic f i e l d . Accordingly we undertook a series of checks for systematic temperature errors of this type These checks are described i n the Appendix. Also included i n th Appendix are c i r c u i t diagrams of the capacitance bridge and of the temperature controller that was b u i l t to operate with the capacitance sensor. 24b Figure 14 Magnetoresistance measurements for three samples-The dashed lines are the function,. - 2 " | j ^ ^ with H = 50k0e. The symbols 0, A, x , « , k, and S indicate experimental runs #1-5; 6; 7,8; 9,10; 11; 12,13 respectively. T T - l . 6 h (a) -1.2--0.8--0.4-0 -0.2-2 0 hH4 40 T (°K) 60 . SAMPLE I - I ,b axis H • 50 kOe I 1 •T kl O-£> 1 98 °K -I 80 25 -1.6--1.2-g -0.8--o.4[-\ (b) \ \ \ \ i \ f T I \ U II ^ 1 HflJ 0 -0.2-2 0 -'-1 SAMPLE 2 H • 50 kOe • I,b axis 40 T (°K) 60 98 °K J I 1 » 80 -l.6h T I <~ V\ (c) -1.2-3 - 0 . 8 -- 0 . 4 -0 -0.2-1 \ 1 I 2 0 40 60 T (°K) SAMPLE 3 "1 H • 5 0 kOe • I,b axis! 98 °K 80 261 2.3 Experimental Results . The measured magnetoresistance for three different crystals i n three mutually orthogonal directions i s shown i n Figure 3. Note that the general features of the results are independent o f sample orientation. This isotropy suggests that the effect results from an interaction with spins rather than with o r b i t a l motion. Furthermore, the magneto-resistance measurements f a l l naturally into three temperature regions. The f i r s t i s the temperature region between 17K and 48K i n which the magneto-resistance behaves roughly as -T . The magnitude of the magneto-resistance for a l l three samples, f a l l s below T~ i n the low . temperature end of this region. In the second region, from 48K to 54K, there i s a small (^1.4%) peak i n the magneto-resistance. F i n a l l y , above 54K the magneto-resistance i s zero within the resolution of the experiment (<0.1%). The conductivity curve can also be naturally divided into the same three temperature regions. Below 48K the material behaves as a semiconductor, between 48K and 54K i t undergoes a semiconductor to metal trans i t i o n , and above 54K, the material i s metallic. The nature of the s i m i l a r i t y between the magneto-resistance and conductivity may be i l l u s t r a t e d by taking the logarithmic derivative of the temperature dependence of the conductivity. An experimentally determined plot of 26b Figure 15 Resistivity derivative curve. The peaks near 38K and 52K were obtained by monitoring the crystal resistance,on a chart recorder as the sample temperature drifted slowly. The remainder of the curve was obtained by measuring the resistance change for a 20mK temperature change at 2K intervals. 27 [p(T+AT) - p(T)]/p(T) which is proportional to the logarithmic derivative, is shown in Figure 15. In this curve T i s 20mK. The peak in this derivative curve near 52K corresponds to the metal-semiconductor transition. The large peak near 38K corresponds to a sharp decrease i n the conductivity that i s probably due to a second phase transition.t The two peaks were measured by continuously monitoring the crystal resistance and the temperature as the system drifted very slowly (IK in 5 minutes) through the steep sections of the conductivity curve. A comparison of Figure 15 with Figure 8 (Bottom) seems to indicate that the slow d r i f t measurement may have broadened the 38K peak. Also preliminary measurements near 38K show about a IK hysteresis between heating and cooling curves. If further measurements verify this hysteresis, i t can be interpreted as evidence for a f i r s t order phase transition. Although the magnetoresistance shows a well-defined peak at 52K, there i s no corresponding peak in the data near 38K. however the results described here are somewhat inconclusive near 38K, since the feature at this temperature appears to be narrower than the one at 52K. The question of whether or not there i s a peak in the magnetoresistance near 38k can only be resolved by more detailed measurements. t The sharp decrease in conductivity at 38K has also been studied by S.Etemad, T.Penny and E.M.Engler at IBM, Yorktown Heights. 29 2.4 Interpretation In this section we interpret the magnetoresistance measurements i n terms of what i s presently known about the material. Fi r s t an argument is given for why the magnetoresistance should be zero in the metallic regime. Then for the semiconducting regime, we show how a -T dependence for the magnetoresistance results from a consideration of charge carrier densities. Finally we present a simple model of the metal-semiconductor transition which reproduces the observed peak i n the magnetoresistance. In the metallic phase (above 54K) since there i s no electronic energy gap, the magnetic f i e l d i s not expected to produce any significant changes in the carrier density. The most important remaining source of magneto-resistance should then be the orbital motion of the charge carriers in the magnetic f i e l d . The relevant parameter in determining the size of this effect i s U C T where o)c i s the cyclotron frequency and x i s the charge carrier relaxation time. In the weak f i e l d limit (ui^r < 1), a reasonable 2 estimate of an upper bound on the magnetoresistance i s (w T) (A.C.Beer, 1963). Assuming the effective mass to be a minimum of five times the free < -14 electron value and x^ 3x10 sec (Bright et al 1974, Grant et al 1973), we 2 -5 find, in a magnetic f i e l d of 50k0e , that the magnetoresistance ^(a^x) <10 This i s much smaller than our experimental resolution limit of 10 Since the relaxation time is not expected to change very much between 80K and 20K (Grant et a l , 1973), i t i s reasonable to neglect this effect in the semiconducting regime as well. 30 Below 45K, TTF-TCNQ may be regarded as a small gap (0.01 - 0.04eV) semiconductor. A very simple theory accounts for the observed -T dependence of the magneto-resistance, in this region. When a magnetic f i e l d is applied, the degenerate spin up and spin energy bands move r i g i d l y , and the energy gap does not change in the presence of a f i e l d one obtains the following expression for the magneto-resistance: This effect results from the small increase in the charge carrier density caused by the band s p l i t t i n g . The dashed curves in Figure 3 were computed from [1] with H = 50k0e. Both the magnitude and the temperature- dependence of the dashed curve are in good agreement with the data below 45K. Below 30K, the agreement between the dashed curve and the data for the other two crystals is not as good as for sample 1. These differences probably result from the fact that samples 2 and 3 are of lower quality, as evidenced by the conductivity data l i s t e d i n T a b l e n . Also below 17K the conductivity of a l l three samples begins to be dominated by a temperature independent component, which probably results from crystal defects and impurities. This component of the conductivity i s expected to be relatively insensitive to the magnetic f i e l d , so that down bands s p l i t into two bands separated by ^UgH • Assuming the 31 one anticipates a vanishing magnetoresistance at very low temperatures. The calculation of the magnetoresistance close to the metal-semiconductor transition is more complicated. We have seen that at low temperatures the most important effect of a magnetic f i e l d i s to 2 change the number of carriers by an amount proportional to (uDH/kT) . At these temperatures the energy gap A is much larger than uDH. This D latter condition w i l l not hold very close to T , where the energy gap approaches zero . Very close to T c where UgH/^ A , the energy gap may be sensitive to the small increase in the charge carrier density produced by the magnetic f i e l d . In fact, the calculation which foll ws shows at for a particular model of the metal- nsulator transition, a magnetic f i e l d does reduce the electronic energy gap. An alternative point of view is to look at the transition from the high temperature side. As we w i l l show for the Peierls type metal-insulator transition, the effect of the f i e l d i s to reduce the transition temperature by an amount, where y is a constant of order unity. Thus one expects a peak in the magneto-resistance at T with a width equal to a few times 6T. Rice and Strassler (1973)have published a particularly lucid description of the Peierls metal-insulator transition for a h a l f - f i l l e d tight binding band of electrons interacting with ±k c phonons where k is the Fermi wavevector. r r We will repeat their calculation for the case in which the electron spins interact with an external magnetic f i e l d . c 32 The unperturbed single p a r t i c l e energies may be written as, T3] = -£p cos kb where b i s the l a t t i c e constant for the chain, and £ i s half the r bandwidth, which i s also the Fermi energy i f we sp e c i a l i z e to the simple case of a h a l f - f i l l e d band. The Fro^Lich electron-phonon Hamiltonian for a li n e a r chain i n the presence of a magnetic f i e l d i s , 14] H = I I ( £ k + Sy BH) k s=±l + T fua b +b + k I I I g(q) a.1"^  a, (b + b + ) q q q q ^ £ s q  k q' S k s q " q t t where a^ g i s an electron creation operator and b^ i s a phonon creation operator. This Hamiltonian may be s i m p l i f i e d somewhat i f we assume that the inte r a c t i o n with the phonons i s unimportant except when q = ±2k . Then we fi n d that below some c r i t i c a l r temperature T £ , the l a t t i c e i s unstable against a d i s t o r t i o n i n which the size of the unit c e l l i s doubled. This s t a t i c " P e i e r l s " d i s t o r t i o n can be looked at as a macroscopic occupancy of the 2kp phonon mode. That i s , for t h i s p a r t i c u l a r phonon state, [5] <b > = <b+> - -£VN Sn q q 2 Qq where Q = 2kp, and we have introduced the dimensionless parameter u as a measure of the size of the s t a t i c d i s t o r t i o n . In the discussion that follows, we w i l l obtain an equation for u i n 33 The presence of a magnetic f i e l d , by requiring that the free energy be a minimum. Below the c r i t i c a l temperature |5] can be used to rewrite the Hamiltonian |4] as, ' *] H = Nr4 • 11 ( e + s y B H ) a + k s a k s k s + k I C ug ak +Q,s aks + U g ak-Q,s W where the non-interacting phonon part has been ignored, OJ=WQ, and g=g(Q) i s the electron phonon coupling parameter. The f i r s t term in ' £>] is the elastic strain energy produced by the static distortion. When the electronic part of the Hamiltonian is diagonalized the new Hamiltonian i s , H = N*<4 + I I E k s a k s where E k ^ = sgn (k 2 - k 2) (e£ + A 2) + sygH and gu ='A. If we define 6 = u^H the density of states for this band structure i s . D^(e)=D(0)eF|e±6|/ [(e 2 +A 2-(e-6) 2) ((e±6) 2 - A 2 ) ] 1 / 2 where +(-) is for the spin up (down) band and D(0) = ( i r C p ) * is the density of states in the middle of the band for each spin state in the absence of the magnetic f i e l d and the distortion. If the contri-bution of the non-interacting phonons to the free energy is ignored, the following expression can be written for the free energy of the insulating phase: [7] F_j = y"ntou2 N 1 (e) + Dj ( O ) l n ( l + e _ e e ) The f i r s t term i s the strain energy, per molecule, introduced by the static distortion. At this point i t is convenient to introduce a parameter A, defined by, x = 2g 2 p ( o ) hio which is identical to the electron-phonon interaction parameter used in the theory of superconductivity (Hopfield, 1970). If X and A are substituted for U and a partial integration i s performed on the integral in [7], one obtains, [8] ND(0) 04 + A 2 ) V 2 _ \ |F [ i n (cosh 3(A+ 6) + cosh3<5) + ln2] 2. 3 where 1(A) l X A2 + £2 ' 2 A F de sin 2e 2 - 2A2 - efe x Equation [8] for the free energy includes the assumption that 3 e F » l . This approximation is reasonable for T c = 52K since cp/k is expected to be ^2000K. Rice and Strassler have shown that there is always a non-zero value of A at T = 0 that leads to a lower free energy for the insulating state than for the metallic state. Since the magnetic f ie ld is not expected to change the gap much, i t makes sense to derive a gap equation by setting the derivative of Fj with respect to A equal to zero. The equation which results i s , [ 9 J 2 - - r . Ctanh ^ * - ^ /, i.i_ann 2 " tanh 2 ) d £ _ _ J -A ( e 2 -A 2 )" 2 (e F 2+A 2-e 2 )'2 A further simplification is possible i f we use the fact that close to the transition £p>>A and for a l l temperatures e p £ 10A. Using the assumption B£p>>l as well, we can make [9] look l ike the equation for the energy gap of a superconductor (Muhlschlegel 1959). With E = (e2 + A 2 )* 5 , we have, [ 1 0 ] ± - * n 2 = ^ § (tanh B C 4 ^ > + t a „ h H | l ^ + 0 ^ A i _ j 35b F i g u r e 16 E l e c t r o n i c energy gap near T c a l c u l a t e d from t h e model d e s c r i b e d i n t h e t e x t . The dashed c u r v e i s t h e gap i n t h e p r e s e n c e o f a m a g n e t i c f i e l d of 50k0e. provided that X£ 1. This equation may be written more simply as, CO [11] - i n f - f (E + 6) +• f (E. _ 6) with f(E) = Ce + 1) , and A q = A (T = 0). A graph of A(H,T) versus T i s shown in Figure IG for H = Q and H = 50 kOe, near the c r i t i c a l temperature. Notice that the magnetic f i e l d has reduced the electronic energy gap, and that the reduction i s greatest just below T . Also note that the effect of the f i e l d on the energy gap appears to be equivalent to a temperature rise of about 46 mK. This similarity suggests that we try to calculate the temperature s h i f t directly. Let a= ^ b e t n e fractional temperature s h i f t . Then from [10] we have, c [12] [ tanh B(E + S) 2 + tanh g(E -6) 2 ] 0 e F tanh 0 which may be solved for a. In the v i c i n i t y of the transition 36 i s small so that we can approximate the integrand on the l e f t by, [13] l- j t a n h A C L p O + t a n h | j t a n h M _ ( | i ) 2 s e c h 2 3 | tanh ^ | + OtCe-S)4] Similarly for a « l the integrand on the right band side of [12] can be expanded using, [14] tanh = tanh i f - M a sech 2 *| + 0 (a 2) By substituting [13] and [14] into [12] one obtain an equation for ct, OO f ~ tanh X sech 2 X [15] a = (Mo2\Q__ + 0 ( a 2 ) + o i m ^ J du sech 2X 0 where j I 2 86 2 In the case A = 0, a is .85 (—-) . This result i s consistent with the shift in the Peierls transition temperature calculated by Dieterich and Fulde (1974) in a different way. Equation [15] Apparent Temperature S h i f t of the Energy Gap i n a Magnetic F i e l d of 50k0e, for T Q = 52.8K. T (K) 15.8 31.7 42.2 47.5 51.7 52.8 AT (mK) 55.7 51.2 48.2 46.8 45.8 45.5 40 has been used to calculate the apparent temperature s h i f t of the energy gap curve for several temperatures using A(0,T) in the integrand. The result of the calculations i s l i s t e d in Table III. It is' clear from the table that for temperatures close to T , the •apparent temperature shift i s p r a c t i c a l l y constant, and that the shift i n T of 45.5 mK for H = 50 kOe i s identical to the c value found earlier in the calculation of A(H,T). If the increase in carrier density produced by the spin s p l i t t i n g of the conduction band i s neglected then the observed magneto-resistance should be nearly identical to the effect of a 45.5 mK temperature r i s e . Based on the conductivity derivative measurements discussed e a r l i e r , a temperature r i s e of 45.5 mK would cause a maximum increase of 1.9% in the conductivity of sample 1 near 52K. On the other hand the peak value of the magneto-resistance at about the same temperature i s -. 1.4±.2%. In fact a curve of the form [p(T+AT) - p(T ) ] / p(T) with T = 33mK gives a good f i t to the magneto resistance peak. It is interesting to note that with the possible exception of a small region near 38K, the magnetoresistance of sample 1 below 50K can also be f i t by [p(T+AT) - p(T ) ] / p(T) except that in this case T = 20mK. 41 Now that the energy gap as a function of temperature is known for both zero f i e l d and 50 kOe, we can calculate the conductivity and the magnetoresistance a s a function of temperature for the model. An expression for the e l e c t r i c a l conductivity of a one-dimensional conductor in the relaxation time approximation i s , us] o - / v 2 rff) dk where x is the charge carrier relaxation time, v i s the carrier • -velocity, b i s the la t t i c e constant and ft i s the volume of a unit c e l l . The variable of integration can be changed from k to E where the two variables are related by, 2 (E ± 6 ) 2 = £ F cos 2kb + A 2 With this substitution, and using the definition of v, 1 3E v = fT ak and integrating by parts, one obtains, [17] a(H) = / dE D(E)M(E)[f(E + 6) + f(E - 6)] when A + 0. The f i r s t factor in the integrand i s the zero f i e l d density of states given by, 42 a D(E) - DCO) |E| e FCE 2-A 2) _ r l (e 2 + A 2 -E 2 )' -1 with D(0)=(Tr£p) as before. The second factor i n the integrand of [17] is the inverse effective mass defined by, 1 3 2E M C E ) -b 2 8k 2 h 2 E 3 2e 2 - E 4 +A4 ] When the energy gap i s zero, that i s for T > T , an extra term appears from the integration by parts of [17] . In this case a(H) = 2e2T ft + f dE D(E) MCE) I f (E+<5) + f (E-6)| r r h 2 J I [ 42b Figure 17 Theoretical magnetoresistance for the model described in the text. The dashed curve i s an expansion of the solid curve near T„. 43b Figure 18 Magnetoresistance theory ( s o l i d l i n e ) and experimental data f o r sample 1. 44 45 2.5 Discussion We have calculated the magnetoresistance for this model using the conductivity expressions that are derived above, and the values of A(H,T) and AfO,T) that were calculated e a r l i e r . The model magneto-resistance curve is shown in Fig.17.The same curve i s plotted again in Fig.18 along with the experimental data for sample 1. The c r i t i c a l temperature has been adjusted to 52.8K for a best f i t . Notice that near the transition the model predicts a magnetoresistance that i s three or four times larger than the experimental value. Also the model predicts that the magnetoresistance i s zero above T £ whereas there i s some t a i l i n g above T £ in the experimental data. These discrepancies probably result from the i n a b i l i t y of the mean f i e l d theory to describe the details of the transition very close to T , because fluctuations have been ignored. The agreement between the model and the experiment i s also quite good below 38K even though the model does not take into account the sharp expansion of the electronic energy gap near 38K. The reason for the good agreement is that the magnetoresistance i s insensitive to the size of the energy gap, once the gap i s 46 greater than several kT. Thus, even though the energy gap in the model is too small below 38K by a factor of V2, the model s t i l l f i t s the magnetoresistance reasonably well. The model does not f i t the data for sample 2 quite as well, particularly between 48K and 40K, and below 25K. A variety of explanations are possible, linked either with a d i f f e -rence in samples, or in sample orientation in the magnetic f i e l d . In addition the near zero magnetoresistance points between 40K and 48K were taken after many thermal cyclings of the sample, and they seem to be contradicted by a much earlier measurement that i s in line with the model. It i s possible that the discrepancy may somehow result from damage to the crystal. In conclusion, we have succeeded in f i t t i n g the main features of the magnetoresistance of TTF-TCNQ by using a simple mean f i e l d model of a Peierls type metal-semiconductor transition with a T £ of 52.8K. The effect of fluctuations i s assumed to be small and restricted to a narrow temperature range near T . The nature of these fluctuations is presumably res i s t i v e and one thus expects that a peak i n the conductivity (observed at 58K) should.occur when these fluctuations increase at rate which i s faster than that at which the scattering due to phonons i s decreasing. Of course the fact that this particular model f i t s the data does not prove that a Peierls distortion actually occurs since there are conceivably other mechanisms for a second order metal-insulator transition for which the energy gap has a similar f i e l d and temperature dependence. It does seem however that the temperature depen-47 dence which we observe is rather different from the model of Lee, Rice and Anderson (1973) in which one-dimensional charge density waves freeze into a three dimensional ordered state. In that model, just below the three dimensional ordering temperature one expects to see quite a large gap that is related to the ordering temperature for the individual chains. On the other hand the transition which we see i s a sharp transition, and thus we expect the insulating state to exhibit three dimensional long range order. 48 Appendix: Checks for Systematic Error The obvious source of systematic error is a f i e l d dependence of the capacitor temperature sensor. As has already been mentioned we do observe a small f i e l d dependence in the capacitor at the highest fields used; however, this d i f f i c u l t y has been avoided by keeping the capacitor far enough out of the f i e l d that the magneto-capacitance i s negligible. The maximum acceptable f i e l d for the capacitor was deter-mined by successively turning the magnet on and o f f and pulling the sensor out of the magnet in one centimeter steps, until the f i e l d had no measurable effect on the capacitance. The sensor was then pulled a further one centimeter out of the magnet, and this position was fixed as the operating point. The f i e l d dependence of the capacitor in this position was ver i f i e d to be less than .2 pf (corresponding to 1.6 mk at 50 K) at 48K, 51K, 64K and 98K. The separation between the capacitor and the samples was then adjusted (8.6 cm) so that the crystals were near the centre of the magnet. The small dewar enclosing the probe (see Fig.11) reduces the thermal gradients between the sensor and samples. A second possible source of systematic error i s eddy current heating of either the copper probe or the sample i t s e l f , by the ripple on the magnetic f i e l d . To check this p o s s i b i l i t y we lowered a small pick-up c o i l into the centre of the solenoid to measure the f i e l d ripple directly. The magnitude of the induced voltage on the c o i l was found to be independent of the power supply current from i t s minimum of 1.5A to i t s maximum output of 100A. Since the magnet was frequently operated 49 at the 1.5A level for some time before switching on the sweep, without observing any change in the crystal resistance, i t i s f a i r to assume that the f i e l d ripple cannot cause any significant systematic temperature ris e . Moreover i f one uses the measured f i e l d ripple and estimates the heat input to the copper probe via eddy current heating one finds that i t is much too small to account for an apparent temperature s h i f t of 20 or 30 mK, even with a worst case heat flow. Furthermore the current, that i s induced in the sample because i t forms part of a loop with the current and voltage leads, is only a small fraction of the test current used to measure the sample conductivity. Finally the induced voltage generated in the pick-up c o i l by the linear f i e l d sweep for typical sweep rates i s about six times greater than the ripple voltage. Thus the eddy current heating should be much greater during the sweep than in constant current operation. However no crystal temperature rise due to the f i e l d sweep has been observed except possibly for the very fastest sweeps that the magnet w i l l handle. The operation of the magnet has a marked effect on the helium b o i l - o f f rate, and i t i s conceivable that this change may introduce systematic errors. While the magnetic f i e l d i s being increased the helium b o i l - o f f rate rises steadily u n t i l i t is about twice the zero f i e l d value, presumably because of the power dissipated in the magnet. When the f i e l d is held constant the b o i l - o f f drops to close to the zero f i e l d value. The same thing happens during the sweep down in f i e l d . By introducing a small heater into the helium bath one can simulate the thermal behaviour of a magnetic f i e l d sweep. Although the temperature controller heater power drops (from 250 mW to 200 mK for example) when the helium b o i l - o f f increases, and a slight kink i s introduced into the 50 baseline established by the crystal voltage, no shift in crystal resistance comparable to the magneto-resistance is observed. As a further check, the magnetic f i e l d was turned on with the sample probe outside of the magnet. After a baseline had been established, the probe was pushed very gently down into the magnet. Once the new baseline had been established, the probe was pulled back out of the f i e l d . The magneto-resistance measured in this way was identical to that measured by sweeping the f i e l d and keeping the probe fixed. Finally the position of the probe was adjusted so that one crystal was above the top of the magnet and one crystal was partly inside the bore. The difference in the observed magneto-resistance in the two crystals was consistent with the estimated difference in the f i e l d and a quadratic f i e l d dependence. In conclusion, we believe that the temperature change in the probe induced directly or indirectly by the magnetic f i e l d i s less than 1 mK and the measured change in resistance results from magneto-resistance and not from spurious heating. 51 bridge. 220KSL A/vV 2.2 Mn. — V v W — V\eaTer I IOXL Ke/><:p OPS 20-/B PovJer Supply Figure 19 Circuit diagram of proportional/integral temperature controller used with the capacitance sensor. 0 1 |2: 2v pfep. 5 H * . 1 —~3 o © 1 ZZ6£L ft ere nee Hammond 585J Transformer Figure 20 Capacitances-Inductance bridge used for measuring capacitance. BIBLIOGRAPHY Allender, D., Bray, J. W., and Bardeen, J. 1974. Phys. Rev. B9, 119. Beer, A. C. 1963. Solid State Physics suppl. 4, ed. F. Seitz and D. Turnbull (Academic Press, New York). Bright, A. A., Garito, A. F., and Heeger, A. J. 1973. Solid State Comm. 13_, 943. Cohen, M. J., Coleman, L. B., Garito, A. F. and Heeger, A. J. 1974. Phys. Rev. BIO, 1298. Coleman, L. B., Cohen, M. J., Sandman, D. J., Yamagishi, F. G., Garito, A. F., and Heeger, A. J. 1973. Solid State Comm. 12, 1125. Dieterich, W. and Fulde, P. 1973. Z. Physik 265, 239. Elbaum, C. 1 9 7 1 ) - . Phys. Rev. Lett. 3 2 , 377. Epstein, A.J., Etemad, S., Garito, A. F., and Heeger, A. J. 1972. Phys. Rev. B5, 952. Ferraris, J. P., Cowan, D. 0., Walatka, V. and Perlstein, J. H. 1973. J. Amer. Chem. Soc. 95_, 948. Grant, P. M., Greene, R. L., Wrighton, G. C. and Castro, G. 1973. Phys. Rev. Lett. 31, 1311. Groff, R. P., Suna, A., and Merrifield, R. E. 1974. Phys. Rev. Lett. 33, 418. Hopfield, J. J. 1970. Comments Solid State Phys. 3^, 52. Kistenmacher, T. J., P h i l l i p s , T. E., and Cowan, D. 0. 1974. Acta Cryst. B30, 763. Landau, L. D. and L i f s h i t z , E. M. 1969. S t a t i s t i c a l Physics , §152, (Addison-Wesley, Don M i l l s , Ontario). 53 Lawless, W. N. 1972. Temperature Its Measurement and Control in Science and Industry, pll43(Instrument Society of America, Pittsburgh). Lee, P. A., Rice, T. M., and Anderson, P. W. 1973. Phys. Rev. Lett. 31, 462. L i t t l e , W. A. 1964. Phys. Rev. A134, 1416. Luther, A. and Peschel, I. 1974. Phys. Rev. B9, 2911. Montgomery, H. C. 1971. J. Appl. Phys. 42_, 2971. Mlihlschlegel, B. 1959. Z. Physik, 155, 313. Reprinted in The Theory of Superconductivity, ed. N. N. Bogoliubov (Gordon and Breach, New York, 1962). Patton, B. R., and Sham, L. J. 1973. Phys. Rev. Lett. 31, 631. Peierls, R. E. 1955. Quantum Theory of Solids (Oxford University Press, London). Rice, M. J. and Strassler, S. 1973. Solid State Comm. 13, 125. Rubin, L. G. and Lawless, W. N. 1971. Rev. Sci. Instruments 42^ 571. Sample, H. H.,,Neuringer, L. J. and Rubin, L. G. 1974. Rev. Sci. Instruments 45, 64. Schegolev, I. F. 1972. Phys. Stat. Solidi (a) 12, 9. Thompson, A. M. 1958. IRE Trans. Instr. 1-7, 245. van der Pauw, L. J. 1961. Philips Res. Repts. 16, 187. Zeller, H. R. 1973. Festkorperprobleme 13_, 31. Ziman, J. M. 1972. Principles of the Theory of Solids , Ch. 7, (Cambridge University Press, London). 

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