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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  i n the Department of PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA March, 1975  In p r e s e n t i n g t h i s  thesis  an advanced degree at  further  agree  fulfilment  of  the  requirements  the U n i v e r s i t y of B r i t i s h Columbia, I agree  the L i b r a r y s h a l l make it I  in p a r t i a l  freely  available  for  t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f  of  this  representatives. thesis for  It  this  thesis  financial  gain s h a l l  of  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada  Date  1,  or  i s understood that copying o r p u b l i c a t i o n  written permission.  Department  that  reference and study.  f o r s c h o l a r l y purposes may be granted by the Head of my Department by h i s  for  ? 9 7 < r  not be allowed without my  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 c r y s t a l s of tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ). Although the data i s mainly f o r the c r y s t a l l o g r a p h i c b axis conductivity, some less complete a axis data i s also presented. The temperature dependence of the e l e c t r o n i c energy gap i s calculated from the b axis conductivity data. The magnetoresistance f o r currents along the  B axis of TTF-TCNQ, has been measured as a function of  temperature Between 17K and 98K i n s t a t i c f i e l d s 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 i s small and negative and i s described reasonably w e l l By the formula Ap/p = -(1/2)(ugH/kT)  . At a l l temperatures Ap/p  was found to Be approximately independent of the o r i e n t a t i o n of the applied f i e l d with respect to the current. The high temperature behaviour i s consistent with that expected f o r a metal i n the short scattering time limit.(w T<<1). We a t t r i b u t e the peak at 52.8K to c  the suppression of the metal-insulator t r a n s i t i o n By the magnetic, and we show why such Behaviour would Be expected f o r a P e i e r l s  iii  t r a n s i t i o n . In the low temperature region the c r y s t a l acts as a small gap semiconductor for which the T 2 dependence of A p / p i s e a s i l y understood.  iv  TABLE OF CONTENTS Page Abstract  i i  Table of Contents  iv  L i s t of Tables  v  L i s t of Figures  vi  Acknowledgements  ix  CHAPTER I 1.1 Background  1  1.2 Conductivity Measurements  8  CHAPTER I I 2.1 Introduction  14  2.2 Experiment  17  2.3 Experimental Results  26  2.4 Interpretation  29  2.5 Discussion  45  Appendix: Systematic  Bibliography  Error Checks  48  52  V  LIST OF TABLES Table I II  Page 9  Summary of conductivity data. Conductivity of the samples used i n the magnetoresistance  15  experiments. III  Apparent temperature s h i f t of the electronic energy gap i n a magnetic f i e l d of 50k0e for T = 52.8K. c  39  LIST OF FIGURES gure 1  TTF and TCNQ molecules.  2  C r y s t a l structure looking along the a axis.  3  One dimensional, h a l f - f i l l e d , e l e c t r o n i c energy bands with (a) no gap at the Fermi l e v e l and (b) a non-zero gap at the Fermi l e v e l .  4  Density of states f o r the one dimensional bands shown i n Figure 3.  5  One dimensional l a t t i c e s corresponding to the band structures shown i n Figure 3.  6  Contact configurations used f o r (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 i n the semi-conducting phase.  9  Top: Conductivity along the c r y s t a l l o g r a p h i c a axis. Bottom: Conductivity along the a axis i n the semi-conducting phase.  vii Figure 10  Page Electronic energy gap determined by f i t t i n g the b axis conductivity.  11  13  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 d r i f t e d steadily up a t o t a l of about 0.03K i n 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. 15  25  R e s i s t i v i t y derivative curve. The peaks near 38K and 52K were obtained by monitoring the c r y s t a l 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  viii Figure 16  Page E l e c t r o n i c energy gap near  calculated from the  model described i n the text. The dashed l i n e i s the gap i n the presence of a magnetic f i e l d of 50k0e. 17  36  Theoretical magnetoresistance for the model described i n the text. The dashed curve i s an expansion of the s o l i d curve near T . c  18  Magnetoresistance theory ( s o l i d l i n e ) and experimental data f o r sample 1.  19  44  C i r c u i t diagram of p r o p o r t i o n a l / i n t e g r a l temperature c o n t r o l l e r used with the capacitance sensor.  20  43  51  Capacitance-inductance bridge used f o r 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 u s e f u l  suggestions  had a strong influence on the outcome of the project, e s p e c i a l l y with regard to the i n t e r p r e t a t i o n of the experimental  results.  W. I. Friesen's generous assistance i n obtaining a numerical s o l u t i o n to the energy gap equation i s g r a t e f u l l y acknowledged. F i n a l l y , 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 i n 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 g r a t e f u l to the National Research Council for f i n a n c i a l support  i n 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 i n recent years. The material i s i n t e r e s t i n g f o r 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)  i n which one might expect to  observe e f f e c t s which are peculiar to one dimensional e l e c t r o n i c systems (Allender et a l 1974, Lee et a l 1973, Luther and Peschel 1974, Patton and Sham 1973). The term "quasi" i s usually interpreted to mean that the material can be viewed as a c o l l e c t i o n of i d e n t i c a l l i n e a r chains which are (1) s u f f i c i e n t l y weakly coupled that a one dimensional band structure accurately describes the i n d i v i d u a l chains, and  (2) s u f f i c i e n t l y strongly coupled that the  behaviour of the system i s not dominated by the e f f e c t s of thermodynamic fluctuations as are t r u l y one dimensional systems (Landau and L i f s h i t z 1969). A second reason for i n t e r e s t i n the material i s that i t i s 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  Figure 2  TCNQ  C r y s t a l structure looking along the a a x i  3  et a l 1974)  i s higher than any organic s o l i d s p r e v i o s l y synthesized.  In p r i n c i p l e a good organic conductor opens up the p o s s i b i l i t y of designing new materials with desireable e l e c t r o n i c properties, because of the nearly l i m i t l e s s f l e x i b i l i t y of organic In fact L i t t l e  chemistry.  (1964) has suggested a possible new mechanism for  high temperature superconductivity, which might be r e a l i z e d i n a system of conducting  organic chains with polarizeable side chains.  F i n a l l y i t should be mentioned that anomalously high c o n d u c t i v i t i e s have been measured i n 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 c r y s t a l l i n e TTF-TCNQ r e s u l t s from the stacking behaviour of the r e l a t i v e l y large and f l a t TTF and molecules (Figure 1). In the s o l i d the molecules form TTF and TCNQ stacks (Kisterrmacher et a l 1974)  CNQ  segregated  with r e l a t i v e l y  strong coupling between molecules on the same stack and weak coupling Between molecules on d i f f e r e n t stacks (Figure 2). As a result , the c r y s t a l l o g r a p h i c B axis, which i s the d i r e c t i o n with 1  strong intermolecular coupling, i s the highly conducting  direction.  (b)  (a)  One dimensional, h a l f - f i l l e d , e l e c t r o n i c energy bands with (a) no gap at the Fermi l e v e l and (b) a non-zero gap at the Fermi l e v e l .  (a)  (b)  Density of states for the one dimensional bands shown i n Figure 3.  U—  •  -*-|  •  •  •  • •  (a)  2b |*-  e o  • e  • «  (b)  One dimensional l a t t i c e s corresponding to the band structures shown i n 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 i s 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 i s unstable with respect to a particular type of l a t t i c e d i s t o r t i o n . This i n s t a b i l i t y can be explained by considering a one dimensional, h a l f - r f i l l e d , tight-binding, electronic energy band as shown i n Figure 3(a). The corresponding density of states i s shown i n Figure 4(a). The average electronic energy may be lowered by creating an energy gap at the Fermi l e v e l . 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 i s simply a dimerization of the one dimensional chain as shown i n Figure 5(b). Clearly, for an energy  Figure 6  Contact configurations used f o r (a) b axis and (b) a axis conductivity measurements.  6fe  Figure  7  Photograph  o f mounted TTF-TCNQ  Scale:  actual  7X  size.  sample.  8  band which i s not exactly h a l f - f i l l e d the d i s t o r t i o n which lowers the energy of the electronic system w i l l have a p e r i o d i c i t y determined by the position of the Fermi l e v e l . In principle this I n s t a b i l i t y can also exist i n three dimensional systems. However the l a t t i c e d i s t o r t i o n which i s required i n order to create an energy gap over the entire Fermi surface, i s i n 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 d i s t o r t i o n , and i t i s a possible mechanism for the metal-^insulator t r a n s i t i o n i n TTF-TCNQ.  1.2 Conductivity Measurements Detailed d.e. b axis conductivity measurements were made on nineteen single c r y s t a l samples of TTF-TCNQ from room temperature to about 10K. The measurements were made with a four probe technique using s i l v e r paint to make e l e c t r i c a l contact to the crystals. A photograph of a sample mounted for conductivity measurements i s shown i n 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 c r y s t a l as shown i n 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 (ncm)  1  RT  °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  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  660±110  15.9  177-190±4O  23.8-9.9  430-480±130  4.7-8.3  * 50 •k  51  8.8  These samples were measured with current leads on the corners of the ends of the c r y s t a l . The apparent c o n d u c t i v i depends on which current connections are used.  9b  Figure  8  Conductivity  along  the c r y s t a l l o g r a p h i c b  Bottom: C o n d u c t i v i t y  along  semiconducting  the b  phase.  axis  axis.  i n the  TTF-TCNQ b AXIS CONDUCTIVITY -I d - 66O(0.CM] JV1J Rr  SAMPLE 4-8  IOO  150 T(K)  200  b AXIS  250  CONDUCTIVITY  SAMPLE 1 8  4  6 I O 0 / T  00  8  10b  Figure 9  Top: Conductivity along the c r y s t a l l o g r a p h i c a a x i s . Bottom: Conductivity along the a a x i s i n the semiconducting phase.  11  • « e  a AXIS CONDUCTIVTY  o  <L = i A (-acM)"' SAMPLE  0  100  53  300  200  T(K)  0  a AX/S  SAMPLE  -2  RT  CONDUCTIVITy 53  -4  -6  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 i n Table I. The b axis conductivity of a typical crystal i s shown i n Figure 8, and the a axis conductivity of another crystal i s shown in Figure 9. Notice that there i s a rapid change i n 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 i n 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 calculated from the b axis conductivity data i n the following  be 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  Figure 10  (K)  E l e c t r o n i c energy gap determined by f i t t i n g the b axis conductivity.  54K can be f i t t e d  using the expression  (Ziman 1972),  o(T) = with the band structure given by, / E  where  K  =  ±  / A ( T )  +  e  -  i s the energy band structure above the t r a n s i t i o n temperature.  This expression  for  w i l l be derived l a t e r . The constants appearing  i n the conductivity expression lattice  2  are the relaxation time T , the.  constant b, the volume of a unit c e l l ft, and the charge  c a r r i e r v e l o c i t y v^. A graph of A ( T ) which gives a f i t to the conductivity i s shown i n Figure 10.  CHAPTER I I 2.1 Introduction This thesis project was o r i g i n a l l y undertaken i n an attempt to duplicate the anomalously high conductivity measurements mentioned e a r l i e r , and then measure the e f f e c t of a magnetic  field  on t h i s conductivity. Although no anomalously high c o n d u c t i v i t i e s were found, and no magnetic f i e l d e f f e c t s were observed i n the v i c i n i t y of the peak conductivity at 58K, the magnetic  field  measurements at lower temperatures do give some i n t e r e s t i n g  information about the nature of the low temperature i n s u l a t i n g phase. The remainder of t h i s work i s concerned with the descript i o n and i n t e r p r e t a t i o n of some measurements of the e f f e c t of a magnetic f i e l d on the b axis conductivity.  The data i s presented  as magnetoresistance defined by [p(H)-p(0)]/p(0) where p(H)  is  the sample r e s i s t i v i t y i n the presence of a magnetic f i e l d H. The magnetoresistance of three c r y s t a l s of TTF-TCNQ has been measured as a function of temperature between 17K and 98K i n f i e l d s up to 50k0e. The c r y s t a l s were mounted i n three mutually orthogonal orientations r e l a t i v e 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 P e i e r l s t r a n s i t i o n at 52.8K to a semiconducting state. Both the model c a l c u l a t i o n and the experimental r e s u l t s indicate that the magnetic f i e l d lowers the metal-semiconductor t r a n s i t i o n temperature by an amount,  The magnetoresistance that i s calculated  f o r the model i s found  to be i n close agreement with the experimental data.  Table II  Conductivity of the Samples Used i n the Magnetoresistance a R T (ftcm)  Sample l  a  1  Experiments. a^^Cficm)  740±200  910012500  2  a  540190  82001300  3  b  390+100  34001900  a  1  decreased ^6% on repeated (>15) thermal cyclings. RT  k  0  M A V  MAX  Note:  decreased ^25% on repeated c y c l i n g .  Samples 1, 2, and 3 l i s t e d here correspond to sampL 45, 44, and 41 respectively, l i s t e d i n Table I.  17 a 2.2  Experiment  The s i n g l e c r y s t a l samples of TTF-TCNQ used i n the experiment were prepared i n the Chemistry Department of t h i s u n i v e r s i t y . Approximately 2.5 mm by 0.1 mm  by 0.02  mm needle - l i k e c r y s t a l s  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 c o n f i g u r a t i o n used to measure the e l e c t r i c a l c o n d u c t i v i t y along the h i g h l y conducting d i r e c t i o n o f the c r y s t a l  (needle a x i s ) .  were a l l < 5ft .  The s i l v e r paint contact r e s i s t a n c e s  Indium solder was used to connect the gold  wires to the copper current source and voltmeter l e a d s .  The  room temperature and peak (near 58K) b axis c o n d u c t i v i t i e s o f the three c r y s t a l s on which the most extensive measurements were taken, are shown i n Table I I.  Less d e t a i l e d magneto-resistance  measurements, that were made on two a d d i t i o n a l samples, were consistent with the r e s u l t s from the f i r s t three samples. the c r y s t a l c o n d u c t i v i t i e s dropped  Although  s l i g h t l y a f t e r repeated thermal  c y c l i n g , no discontinuous jumps i n the c o n d u c t i v i t y i n d i c a t i v e o f cracking i n the sample or contacts were observed.  The sample  holder mounting c o n f i g u r a t i o n i n the cryostat permitted three c r y s t a l s to be measured at the same time, with three mutually orthogonal o r i e n t a t i o n s o f the long axis o f the c r y s t a l . The three sample holders were mounted on a probe made of a high thermal c o n d u c t i v i t y copper.  A c a l i b r a t e d s i l i c o n diode  temperature  sensor (Lake Shore C r y o t r o n i c s , Type DT 500) was mounted c l o s e to the samples, and another s i l i c o n diode and capacitance thermometer  Figure 11  Low temperature probe used f o r conductivity and magnetoresistance measurements.  18a  PUMPING  LINE  He E X C H A N G E GAS  HEATER OUTER  DEWAR  T R A C E OF He CAPACITOR Si  DIODE  COPPER  HEAT  GAS  SENSOR SENSOR  PROBE  SINK  POST  SUPERCONDUCTING SOLENOID INNER  DEWAR  VACUUM Si DIODE  TTF-TCNQ BAKELITE HOLDER  SENSOR  SAMPLE SAMPLE  He E X C H A N G E  GAS  18b  Figure  12  Photograph  of low  temperature  probe.  20  were mounted 8.6 cm above the top c r y s t a l on the same piece o f copper (Figure 1).  Resistance wire was wrapped around the top  of the probe as a heater, and the bottom of the probe was enclosed i n a small s t a i n l e s s s t e e l dewar can.  This small can was placed  inside a larger s t a i n l e s s dewar, which i n turn was immersed i n a l i q u i d helium bath.  The end o f the outer dewar f i t t e d into the centre o f  a 50 kOe superconducting  magnet. (See Figure 11)..  In operation  the vacuum jacket on the inner dewar i s evacuated, and a f r a c t i o n of an atmosphere of helium exchange gas i s introduced into the space between the small inner can and the outer can, and into the space around the c r y s t a l s .  A much smaller quantity of helium  i s introduced  into  the jacket of the outer dewar to allow the system to c o o l . T y p i c a l l y 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 p r a c t i c e , since the  exchange gas can c i r c u l a t e from the inside to the outside o f the small dewar can through the holes f o r the sensor leads, there i s a small thermal gradient  ( <.1K) along the length of the probe.  Although the small dewar d i d not eliminate the thermal gradient e n t i r e l y , i t d i d reduce the thermal d r i f t to acceptable  levels.  The c a l i b r a t e d s i l i c o n diode was used to measure the-temperature i n 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 f i x e d temperature.  The  calibration  of the s i l i c o n diode was checked with a platinum r e s i s t a n c e thermometer (Thermal Systems, Type 5001-A), which was c a l i b r a t e d at 4.2K, the i c e p o i n t . temperature  i n turn  the hydrogen and nitrogen t r i p l e points and at  As a r e s u l t we b e l i e v e the absolute accuracy of the  measurements to be b e t t e r than 0.2K.  The capacitance  sensor was measured with a p r e c i s i o n v a r i a b l e capacitor (General Radio #1422CB) and a capacitance-inductance bridge of the type described by Thompson (1958).  The bridge was  capacitance changes of .1 pf with a 50 mV at 5kHz on the sensor.  s e n s i t i v e to  rms modulation voltage  At 64K and about 25kOe we were able to 4  measure a small "but s i g n i f i c a n t of the c a p a c i t o r .  (1 part i n 10 ) f i e l d dependence  This observation i s i n contrast to zero  magneto-capacitance (< 1 part i n 10^) reported at 4.2K 140 kOe  (Rubin et a l 1971).  i n up to  However both r e s u l t s are consistent  with the comments of Sample et a l (1974) which suggest that there i s a f i e l d dependence i n the v i c i n i t y of 60K which disappears as the temperature  i s lowered to 4.2K.  The f i e l d dependence of the  sensor was measured with the temperature and the probe temperature  d r i f t i n g very slowly.  that the f i e l d dependence may long time constant  c o n t r o l l e r s turned o f f , It i s conceiveable  relax to zero with a r e l a t i v e l y  (as suggested to us by P.M.  C h a i k i n ) . At any r a t e  by sweeping the f i e l d up and down and p r o g r e s s i v e l y p u l l i n g the sensor f a r t h e r out of the magnet we were able to determine that for  f i e l d s less than about 8k0e at 64K,  the sensor exhibits  22  negligible  magneto-capacitance  temperature change). feature.  ( < 1.0 mK  a  equivalent  The c a p a c i t o r has another inconvenient  A f t e r cooling from room temperature to the experimental  operating point  (between 18K and 53K) the capacitance e x h i b i t e d  an exponential r e l a x a t i o n o f about 200pf with a time constant o f about 45 minutes  (Lawless 1972).  TPte c r y s t a l r e s i s t a n c e was measured with a d.e. technique. A combination o f r e s i s t o r s and a mercury b a t t e r y provided constant currents o f luA, lOuA, and lOOuA to the sample and the r e s u l t i n g c r y s t a l voltage was measured with a K e i t h l e y 140 nanovoltmeter. For to  magneto-resistance measurements, the nanovoltmeter was zero, and then the output was a m p l i f i e d 30 times.  offset  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 t o s t a b i l i z e at the temperature o f i n t e r e s t  for at l e a s t an hour u n t i l the c r y s t a l temperature was s t a b l e and the  c a p a c i t o r d r i f t manageable.  At t h i s point the temperature  c o n t r o l l e r was switched from the c a l i b r a t e d diode to the capacitor and a new b a s e l i n e e s t a b l i s h e d .  , Then the magnetic  f i e l d was swept from zero t o 50 kOe i n four to ten minutes, held steady at 50 kOe f o r up to twenty minutes, and then swept back to zero,  A t y p i c a l chart recorder t r a c e of the c r y s t a l  22b  Figure 13  Chart recorder trace from a magnetoresistance measurement at 24K. The temperature has drifted  steadily  up a t o t a l 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 p r i n c i p a l source of uncertainty was  irregularity  i n the b a s e l i n e d r i f t . In a d d i t i o n 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 s e r i e s of checks f o r systematic temperature e r r o r s of t h i s type These checks are described i n the Appendix. A l s o included i n th Appendix are c i r c u i t diagrams of the capacitance b r i d g e and of the temperature c o n t r o l l e r that was b u i l t to operate with the capacitance sensor.  24b  Figure 14  Magnetoresistance measurements f o r three samplesThe dashed l i n e s are the function,. H = 50k0e. The symbols 0, A, x , « , experimental runs #1-5; respectively.  6; 7,8;  k,  9,10;  2"|j^^  with  and S i n d i c a t e 11;  12,13  25 T T  (a)  -l.6h  SAMPLE  I -I,b  -1.2-  axis  H • 50 kOe  -0.8-  H4  h  -0.4-  00.2-  T • £> 1  40  20  I  1  kl  O-  98 °K  -I  60 .  80  T (°K)  -1.6-  \  \  SAMPLE  2 •  \ \ \  -1.2-  g  (b)  -0.8- f  i  \  U  T  I  I,b  axis  H • 50 kOe \  II ^  -o.4[-  1  HflJ  -'-1  98 °K J  I  0-  1  »  0.2-  -l.6h  TI  \  T (°K)  <~  (c)  V  80  60  40  20  SAMPLE  3  -1.2-  • I,b  "1 3  -0.8 -  1  H •50  \  -0.4-  1  axis!  kOe  I 98 °K  0-  0.220  40  60 T (°K)  80  261  2.3  Experimental Results .  The measured magnetoresistance f o r three d i f f e r e n t c r y s t a l s i n three mutually orthogonal d i r e c t i o n s i s shown i n Figure 3.  Note  that the general features o f the r e s u l t s are independent o f sample o r i e n t a t i o n .  This i s o t r o p y suggests that the e f f e c t  r e s u l t s from an i n t e r a c t i o n with s p i n s r a t h e r than with o r b i t a l motion.  Furthermore, the magneto-resistance measurements f a l l  n a t u r a l l y i n t o three temperature r e g i o n s .  The f i r s t i s the  temperature region between 17K and 48K i n which the magnetor e s i s t a n c e behaves roughly as -T  .  The magnitude o f the magneto-  r e s i s t a n c e f o r a l l three samples, f a l l s below T~ temperature end o f t h i s region.  i n the low .  In the second r e g i o n , from  48K t o 54K, there i s a small (^1.4%) peak i n the magnetoresistance.  Finally,  above 54K the magneto-resistance i s  zero within the r e s o l u t i o n o f the experiment  (<0.1%).  The c o n d u c t i v i t y curve can also be n a t u r a l l y d i v i d e d i n t o the same three temperature regions.  Below 48K the m a t e r i a l  as a semiconductor, between 48K and 54K  behaves  i t undergoes  a semiconductor to metal t r a n s i t i o n , and above 54K, the m a t e r i a l is metallic.  The nature o f the s i m i l a r i t y between the magneto-  r e s i s t a n c e and c o n d u c t i v i t y may be i l l u s t r a t e d by t a k i n g the logarithmic d e r i v a t i v e o f the temperature dependence o f the conductivity.  An experimentally determined p l o t o f  26b  Figure 15  R e s i s t i v i t y d e r i v a t i v e curve. The peaks near 38K and 52K were obtained by monitoring the c r y s t a l resistance,on a chart recorder as the sample temperature d r i f t e d slowly. The remainder of the curve was obtained by measuring the resistance change f o r a 20mK temperature change at 2K  intervals.  27  [p(T+AT) - p(T)]/p(T) which i s proportional to the logarithmic derivative, i s shown i n Figure 15. In this curve  T i s 20mK. The  peak i n this derivative curve near 52K corresponds to the metalsemiconductor transition. The large peak near 38K corresponds to a sharp decrease i n the conductivity that i s probably due to a second phase t r a n s i t i o n . t The two peaks were measured by continuously monitoring the crystal resistance and the temperature as the system drifted very slowly (IK i n 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 i n 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 i n the magnetoresistance near 38k can only be resolved by more detailed measurements. t  The sharp decrease i n 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 t h i s section we i n t e r p r e t the magnetoresistance measurements i n terms of what i s presently known about the m a t e r i a l . F i r s t an argument i s given for why the magnetoresistance should be zero i n the m e t a l l i c regime. Then f o r the semiconducting regime, we show how a -T  dependence  for the magnetoresistance r e s u l t s from a consideration o f charge c a r r i e r d e n s i t i e s . F i n a l l y we present a simple model of the metal-semiconductor t r a n s i t i o n which reproduces the observed peak i n the magnetoresistance.  In the m e t a l l i c phase (above 54K) since there i s no e l e c t r o n i c energy gap, the magnetic f i e l d i s not expected to produce any s i g n i f i c a n t changes i n the c a r r i e r density. The most important remaining source of magnetoresistance should then be the o r b i t a l motion of the charge c a r r i e r s i n the magnetic f i e l d . The relevant parameter i n determining the s i z e o f t h i s e f f e c t i s U T where o) i s the c y c l o t r o n frequency and x i s the charge C  c  c a r r i e r r e l a x a t i o n time. In the weak f i e l d l i m i t  (ui^r < 1), a reasonable  2 estimate o f an upper bound on the magnetoresistance i s (w T) (A.C.Beer, Assuming the e f f e c t i v e  1963).  mass to be a minimum o f f i v e times the free  < -14 electron value and x^ 3x10 sec (Bright et a l 1974, Grant et a l 1973), we 2 -5 f i n d , i n a magnetic f i e l d of 50k0e , that the magnetoresistance ^(a^x) <10 This i s much smaller than our experimental r e s o l u t i o n l i m i t of 10 Since the r e l a x a t i o n time i s not expected to change very much between 80K and 20K (Grant et a l , 1973), i t i s reasonable to neglect t h i s e f f e c t i n the  semiconducting regime as w e l l .  30  Below 45K, TTF-TCNQ may be regarded as a small gap (0.01 - 0.04eV) semiconductor. -T  A very simple theory accounts f o r the observed  dependence o f the magneto-resistance, i n t h i s region.  a magnetic f i e l d  When  i s applied, the degenerate spin up and spin  down bands s p l i t into two bands separated by ^UgH • Assuming the energy bands move r i g i d l y , and the energy gap does not change in the presence o f a f i e l d one obtains the following expression for the magneto-resistance:  This e f f e c t r e s u l t s from the small increase i n the charge c a r r i e r density caused by the band s p l i t t i n g .  The dashed curves i n Figure  3 were computed from [1] with H = 50k0e.  Both the magnitude and  the temperature- dependence o f the dashed curve are i n good agreement with the data below 45K.  Below 30K, the agreement  between the dashed curve and the data f o r the other two c r y s t a l s i s not as good as f o r sample 1.  These differences probably  r e s u l t from the fact that samples 2 and 3 are o f lower q u a l i t y , 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 o f a l l three samples begins to be dominated by a temperature independent component, which probably r e s u l t s from c r y s t a l defects and impurities. This component of the conductivity i s expected to be r e l a t i v e l y i n s e n s i t i v e to the magnetic f i e l d , so that  31  one a n t i c i p a t e s a vanishing magnetoresistance at very low temperatures.  The c a l c u l a t i o n of the magnetoresistance close to the metalsemiconductor t r a n s i t i o n i s more complicated. We have seen that at low temperatures the most important e f f e c t of a magnetic f i e l d i s to 2 change the number of c a r r i e r s by an amount proportional to (u H/kT) . D  At  these temperatures the energy gap A i s much l a r g e r than u H. D  This  D  l a t t e r 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 s e n s i t i v e to the small increase i n the charge c a r r i e r densityfollows produced by that the magnetic f a c t , of the lculation which shows f o r a p a rf ti ie cl ud .l a rInmodel thec a metal-insulator t r a n s i t i o n , a magnetic f i e l d does reduce the e l e c t r o n i c energy gap. An a l t e r n a t i v e point of view i s to look at the t r a n s i t i o n from the high temperature side. As we w i l l  show f o r the P e i e r l s type metal-insulator  t r a n s i t i o n , the e f f e c t of the f i e l d i s to reduce the t r a n s i t i o n  temperature  by an amount,  where y i s a constant of order u n i t y . Thus one expects a peak i n the magnetoresistance at T  c  with a width equal to a few times 6T. Rice and S t r a s s l e r  (1973)have published a p a r t i c u l a r l y l u c i d d e s c r i p t i o n of the P e i e r l s metal-insulator t r a n s i t i o n f o r a h a l f - f i l l e d electrons i n t e r a c t i n g with ± k  c  t i g h t binding band o f  phonons where k  i s the Fermi wavevector. r  r  We w i l l repeat t h e i r c a l c u l a t i o n f o r the case i n which the e l e c t r o n spins interact with an external magnetic  field.  32  The unperturbed  s i n g l e p a r t i c l e energies may be w r i t t e n as,  = -£p cos kb  T3]  where b i s the l a t t i c e constant f o r the chain, and £  r  i s h a l f the  bandwidth, which i s a l s o the Fermi energy i f we s p 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 f o r a l i n e a r chain i n the presence of a magnetic f i e l d i s , 14]  H = I I ( k s=±l  + T fua q  + y H)  £ k  S  b b qqq +  B  +  I I I g(q) a."^ ^ £ s q ' k  1  k  q S  a, (b + b k s  q  "  +  q  )  t t where a ^ i s an e l e c t r o n c r e a t i o n operator and b^ i s a phonon g  c r e a t i o n operator. This Hamiltonian may be s i m p l i f i e d somewhat i f we assume that the i n t e r a c t i o n w i t h the phonons i s unimportant except when q = ±2k . Then we f i 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 s i z e of the u n i t 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 , f o r t h i s p a r t i c u l a r phonon s t a t e , [5]  <b > = <b > - -£VN S q q 2 Qq +  n  where Q = 2kp, and we have introduced the dimensionless parameter u as a measure of the s i z e of the s t a t i c d i s t o r t i o n . In the d i s c u s s i o n that f o l l o w s , we w i l l o b t a i n an equation f o r u i n  33  The presence of a magnetic f i e l d , by r e q u i r i n g 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  k I  +  Cu  • 11 k s  g k Q,s a  ( e  a  +  +  ks  +  sy H)a B  Uga  k-Q,s  + k s  a  k 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 i n =  ' £>] i s the e l a s t i c s t r a i n energy produced by the s t a t i c d i s t o r t i o n . When the e l e c t r o n i c part of the Hamiltonian i s diagonalized the new Hamiltonian i s ,  H = N*<4 + I I E  a  k s  k s  where E ^ k  = sgn ( k - k ) (e£ 2  2  A ) + sygH  +  2  and gu ='A. I f we define 6 = u^H the density of states f o r t h i s band structure i s . D^(e)=D(0)e |e±6|/ [ ( e A - ( e - 6 ) ) ((e±6) - A ) ] 2  F  2  2  2  2  1 / 2  +  where +(-) i s f o r the spin up (down) band and D(0) = ( i r C p )  * i s the  density of states i n the middle of the band f o r each spin state i n the absence of the magnetic f i e l d and the d i s t o r t i o n . I f the c o n t r i bution of the non-interacting phonons to the free energy i s ignored, the following expression insulating phase:  can be written for the free energy of the  [7]  F_j = y"ntou N  (e) + Dj ( O ) l n ( l + e  2  _ e e  )  1  The  f i r s t term i s the s t r a i n energy, per molecule, introduced  by the s t a t i c d i s t o r t i o n .  At this point i t i s convenient to  introduce a parameter A, defined by, x = g2p(o) hio 2  which i s i d e n t i c a l to the electron-phonon i n t e r a c t i o n parameter used i n the theory of superconductivity (Hopfield, 1970). X and A are substituted f o r U and a p a r t i a l i n t e g r a t i o n i s performed on the i n t e g r a l i n [7], one obtains,  [8]  04  A )V  +  2  2  ND(0) _ \ |F [ i (cosh 3(A+ 6) + cosh3<5) + ln2] 2. 3 n  where 1(A)  x  l  XA  2  A  2  2 +  £  '  F  de s i n  2e  2  - 2A  2  - efe  If  Equation [8] for the free energy includes the assumption that 3eF»l.  This approximation i s  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  change the gap  magnetic f i e l d is not expected to  much, i t makes sense to derive a gap equation  by setting the derivative o f Fj with respect to A equal to zero.  [ 9  J  The equation which results  2  - -  r  . ,  /  d  Ctanh i.i_ann  ^  is,  *" tanh -  2  £  A  (e2-A2)"2  (e F 2 +A 2 -e 2  2 ^  )  _ _ J -  )' 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 £ Using the assumption B£p>>l  10A.  as well, we can make [9] look l i k e  the equation for the energy gap of a superconductor (Muhlschlegel 1959). [10]  With E = (e 2 + A 2 )* 5 , we have, ±-*n2 =  ^ §  (tanh  B C  4^> ta„hH|l^ +  + 0  ^Ai_j  35b  Figure  16  Electronic  described  the  energy  gap n e a r  i n the text.  presence  T  c  alculated  The dashed  of a magnetic  field  from  the model  c u r v e i s t h e gap i n  of  50k0e.  provided that  X£  as,  1.  This equation may be written more simply  CO  -inf-  [11]  with  f (E + 6) +• f (E. _ 6)  f(E) = Ce  + 1)  , and  A  q  = A (T = 0 ) .  A graph of A(H,T) versus T i s shown i n Figure IG f o r 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 e l e c t r o n i c energy gap, and that the reduction i s greatest just below T .  c  Also note that the  effect of the f i e l d on the energy gap appears to be equivalent to a temperature r i s e of about 46 mK.  This s i m i l a r i t y suggests that  we t r y to c a l c u l a t e the temperature s h i f t d i r e c t l y . Let a= ^  b  e  t  n  e  f r a c t i o n a l temperature s h i f t .  Then from [10]  we have,  [12]  [ tanh 0 eF tanh 0  B(E + S) 2  + tanh  g(E -6) 2  ]  which may  be solved f o r a.  In the v i c i n i t y of the t r a n s i t i o n  36 i s small so that we can approximate the integrand on the l e f t  [13]  jtanhACLpO  -  l  +  jtanhM_(|i)  |  Similarly for a « l  t  2  a  n  by,  h  3|  2  s e c h  tanh ^ |  OtCe-S) ] 4  +  the integrand on the r i g h t band side of [12] can  be expanded using,  [14]  = tanh i f  tanh  By substituting  [13] and  - Ma  [14] into  sech  2  *|  (a ) 2  +  0  [12] one obtain an equation f o r ct,  OO  f  [15]  a =  ~  (Mo \Q__  J where  tanh X s e c h  2  X  2  +  0 ( a 2 )  +  o  i  m  ^  du sech X 2  0  j  I 2  86 In the case A = 0, a i s .85  (—-)  2  .  This r e s u l t i s consistent  with the s h i f t i n the P e i e r l s t r a n s i t i o n temperature by Dieterich  and Fulde  (1974) i n a d i f f e r e n t way.  calculated Equation  [15]  Apparent Temperature i n a Magnetic F i e l d  T  (K)  Shift  of the Energy  of 50k0e, f o r T  AT  15.8  55.7  31.7  51.2  42.2  48.2  47.5  46.8  51.7  45.8  52.8  45.5  (mK)  Q  Gap  = 52.8K.  40  has been used to c a l c u l a t e the apparent temperature s h i f t o f the energy gap curve f o r several temperatures using A(0,T) i n the integrand.  The r e s u l t o f the c a l c u l a t i o n s i s l i s t e d i n Table I I I .  It i s ' c l e a r from the table that f o r temperatures close to T , the •apparent temperature s h i f t i s p r a c t i c a l l y constant, and that the s h i f t i n T o f 45.5 mK f o r H = 50 kOe i s i d e n t i c a l to the c value found e a r l i e r i n the c a l c u l a t i o n o f A(H,T). If the increase i n c a r r i e r density produced by the spin s p l i t t i n g o f the conduction band i s neglected then the observed magneto-resistance should be nearly i d e n t i c a l to the e f f e c t of a 45.5 mK temperature r i s e .  Based on the c o n d u c t i v i t y  d e r i v a t i v e measurements discussed e a r l i e r , a temperature r i s e o f 45.5 mK would cause a maximum increase o f 1.9% i n the c o n d u c t i v i t y of sample 1 near 52K.  On the other hand the peak value o f 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. I t i s i n t e r e s t i n g 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 i n t h i s case  T = 20mK.  41  Now that the energy gap as a function o f temperature i s known f o r both zero f i e l d  and 50 kOe, we can c a l c u l a t e the  conductivity and the magnetoresistance for  the model.  An expression  a  s  a  function o f temperature  f o r the e l e c t r i c a l c o n d u c t i v i t y o f  a one-dimensional conductor i n the r e l a x a t i o n time approximation i s ,  o -  us]  / v  r f f ) dk  2  where x i s the charge c a r r i e r r e l a x a t i o n time, v i s the c a r r i e r • - v e l o c i t y , b i s the l a t t i c e constant and ft i s the volume o f a u n i t cell.  The v a r i a b l e of i n t e g r a t i o n can be changed from k to E  where the two variables are r e l a t e d by,  2 (E ± 6 )  2  =  £  cos 2kb +  F  A  2  With t h i s s u b s t i t u t i o n , and using the d e f i n i t i o n o f v,  1  and  [17]  3E  fT ak  v =  i n t e g r a t i n g by parts, one obtains,  a(H)  when A + 0.  =  /  dE D(E)M(E)[f(E + 6) + f ( E - 6)]  The f i r s t f a c t o r i n the integrand  density of states given by,  i s the zero  field  42 a  e CE -A )  D(E) - DCO) |E|  2  2  (e + A  _ r l  2  F  -1 with D(0)=(Tr£p) as before.  - E )'  2  2  The second f a c t o r i n the integrand  of [17] i s the inverse e f f e c t i v e mass defined by,  1 MCE)  b  3 E 2  -  8k  2  2  h E 2  2  e  2  -  E  4  3  +A  4  ]  When the energy gap i s zero, that i s f o r T > T , an extra term appears from the i n t e g r a t i o n by parts of [17] .  2e T 2  a(H) =  ft  f dE D(E)  +  rrh  2  J  In t h i s case  MCE) I f (E+<5) + f (E-6)| I  [  42b  Figure 17  Theoretical magnetoresistance f o r the model described i n the text. The dashed curve i s an expansion of the s o l i d curve near T„.  43b  F i g u r e 18  Magnetoresistance d a t a f o r sample  theory 1.  (solid  l i n e ) and  experimental  44  45  2.5  Discussion  We have calculated the magnetoresistance f o r t h i s model using the c o n d u c t i v i t y expressions that are derived above, and the values of A(H,T) and AfO,T) that were c a l c u l a t e d e a r l i e r . The model  magneto-  r e s i s t a n c e curve i s shown i n Fig.17.The same curve i s p l o t t e d again i n Fig.18 along with the experimental data f o r sample 1. The c r i t i c a l temperature has been adjusted t o 52.8K f o r a best f i t . Notice that near the t r a n s i t i o n the model p r e d i c t s a magnetoresistance that i s three or four times larger than the experimental value. Also the model p r e d i c t s that the magnetoresistance i s zero above T there i s some t a i l i n g above T  £  £  whereas  i n the experimental data. These  discrepancies probably r e s u l t from the i n a b i l i t y o f the mean f i e l d theory to describe the d e t a i l s of the t r a n s i t i o n very close to T , because f l u c t u a t i o n s 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 i n t o account the sharp expansion of the e l e c t r o n i c energy gap near 38K. The reason f o r the good agreement i s that the magnetoresistance i s i n s e n s i t i v e to the s i z e of the energy gap, once the gap i s  46  greater than several kT. Thus, even though the energy gap i n the model i s too small below 38K by a factor of V2, the model s t i l l  f i t s the  magnetoresistance reasonably w e l l . The model does not f i t the data f o r sample 2 quite as w e l l , p a r t i c u l a r l y between 48K and 40K, and below 25K. A v a r i e t y o f explanations are p o s s i b l e , linked e i t h e r with a d i f f e rence i n samples, or i n sample o r i e n t a t i o n i n the magnetic  field.  In addition the near zero magnetoresistance points between 40K and 48K were taken a f t e r many thermal cyclings of the sample, and they seem to be contradicted by a much e a r l i e r measurement that i s i n l i n e with the model. I t i s possible that the discrepancy may somehow r e s u l t from damage to the c r y s t a l .  In conclusion, we have succeeded i n 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 P e i e r l s type metal-semiconductor  t r a n s i t i o n with a T  £  o f 52.8K.  The e f f e c t o f f l u c t u a t i o n s i s assumed t o be small and r e s t r i c t e d to a narrow temperature range near T . The nature o f these f l u c t u a t i o n s i s presumably  r e s i s t i v e and one thus expects that a peak i n the  conductivity (observed at 58K) should.occur when these f l u c t u a t i o n s increase at rate which i s f a s t e r than that at which the s c a t t e r i n g due to phonons i s decreasing. Of course the f a c t that t h i s p a r t i c u l a r model f i t s the data does not prove that a P e i e r l s d i s t o r t i o n a c t u a l l y occurs since there are conceivably other mechanisms f o r a second order metali n s u l a t o r t r a n s i t i o n f o r which the energy gap has a s i m i l a r f i e l d and temperature dependence. It does seem however that the temperature depen-  47  dence which we observe i s rather d i f f e r e n t from the model of Lee, Rice and Anderson (1973) i n which one-dimensional freeze into a three dimensional below the three dimensional  charge density waves  ordered s t a t e . In that model, j u s t  ordering temperature one expects to see  quite a large gap that i s r e l a t e d to the ordering temperature f o r the i n d i v i d u a l chains. On the other hand the t r a n s i t i o n which we sharp t r a n s i t i o n , and thus we three dimensional  see i s a  expect the i n s u l a t i n g state to e x h i b i t  long range order.  48 Appendix:  Checks f o r Systematic  Error  The obvious source of systematic e r r o r i s a f i e l d dependence o f the capacitor temperature sensor.  As has already been mentioned  we do observe a small f i e l d dependence i n the capacitor at the highest f i e l d s used; however, t h i s d i f f i c u l t y has been avoided by keeping the capacitor f a r enough out o f the f i e l d that the magneto-capacitance i s negligible.  The maximum acceptable f i e l d f o r the capacitor was deter-  mined by successively turning the magnet on and o f f and p u l l i n g the sensor out of the magnet i n one centimeter steps, u n t i l the f i e l d had no measurable e f f e c t on the capacitance.  The sensor was then p u l l e d a  f u r t h e r one centimeter out o f the magnet, and t h i s p o s i t i o n was f i x e d as the operating point.  The f i e l d dependence o f the capacitor i n t h i s  p o s i t i o n was v e r i f i e d to be less than .2 p f (corresponding t o 1.6 mk at 50 K) at 48K, 51K, 64K and 98K. the samples was then adjusted the centre o f the magnet.  The separation between the capacitor and  (8.6 cm) so that the c r y s t a l s were near  The small dewar enclosing the probe (see Fig.11)  reduces the thermal gradients between the sensor and samples. A second possible source o f systematic e r r o r i s eddy current heating o f e i t h e r the copper probe or the sample i t s e l f , by the r i p p l e on the magnetic f i e l d . pick-up  To check t h i s p o s s i b i l i t y we lowered a small  c o i l i n t o the centre o f the solenoid to measure the f i e l d r i p p l e  directly.  The magnitude o f the induced voltage on the c o i l was found t o  be independent o f the power supply current from i t s minimum o f 1.5A t o i t s maximum output o f 100A. Since the magnet was frequently  operated  49 at the 1.5A  l e v e l f o r some time before switching on the sweep, without  observing any change i n the c r y s t a l r e s i s t a n c e , i t i s f a i r to assume that the f i e l d r i p p l e cannot cause any s i g n i f i c a n t systematic temperature rise.  Moreover i f one uses the measured f i e l d r i p p l e and estimates  the  heat input to the copper probe v i a eddy current heating one finds that i t i s much too small to account f o r an apparent temperature s h i f t o f 20 or 30 mK,  even with a worst case heat flow.  Furthermore the current,  that i s induced i n the sample because i t forms part of a loop with the current and voltage leads, i s only a small f r a c t i o n of the test current used to measure the sample conductivity. generated  F i n a l l y the induced voltage  i n the pick-up c o i l by the l i n e a r f i e l d sweep f o r t y p i c a l  sweep rates i s about s i x times greater than the r i p p l e voltage.  Thus  the eddy current heating should be much greater during the sweep than i n constant current operation.  However no c r y s t a l temperature r i s e  due to the f i e l d sweep has been observed f a s t e s t sweeps that the magnet w i l l  except p o s s i b l y f o r the very  handle.  The operation of the magnet has a marked e f f e c t on the helium b o i l - o f f r a t e , and i t i s conceivable that t h i s change may systematic e r r o r s .  introduce  While the magnetic f i e l d i s being increased the  helium b o i l - o f f rate r i s e s s t e a d i l y u n t i l i t i s about twice the zero f i e l d value, presumably because of the power d i s s i p a t e d i n the magnet. When the f i e l d i s 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 i n f i e l d .  By introducing a small heater i n t o the helium bath one can thermal behaviour of a magnetic f i e l d sweep. c o n t r o l l e r heater power drops (from 250  Although  mW to 200 mK  simulate  the  the temperature  f o r example) when  the helium b o i l - o f f increases, and a s l i g h t kink i s introduced i n t o the  50  baseline established by the c r y s t a l voltage, no s h i f t i n c r y s t a l resistance comparable to the magneto-resistance i s 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 i n t o the magnet.  Once the new baseline had been established, the probe  p u l l e d back out o f the f i e l d . t h i s way was  The magneto-resistance measured i n  i d e n t i c a l to that measured by sweeping the f i e l d and  keeping the probe f i x e d . adjusted  was  F i n a l l y the p o s i t i o n of the probe  was  so that one c r y s t a l was above the top o f the magnet and  one c r y s t a l was p a r t l y inside the bore.  The difference i n the  observed magneto-resistance i n the two c r y s t a l s was  consistent with  the estimated d i f f e r e n c e i n the f i e l d and a quadratic f i e l d dependence. In conclusion, we believe that the temperature  change i n  the probe induced d i r e c t l y or i n d i r e c t l y by the magnetic f i e l d i s less than 1 mK and the measured change i n resistance r e s u l t s from magneto-resistance and not from spurious  heating.  51  2.2 Mn. —VvW— 220KSL A/vV  bridge.  V\eaTer I IOXL  Ke/><:p OPS 20-/B PovJer Figure 19  Supply  C i r c u i t diagram of p r o p o r t i o n a l / i n t e g r a l  temperature  c o n t r o l l e r used with the capacitance sensor.  0  1  1  |2:  ere nee  o ©  1  2v pfep.  5H*.  —~3  ZZ6£L  ft Hammond  585J  Transformer Figure 20  Capacitances-Inductance bridge used f o r 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. D i e t e r i c h , W. and Fulde, P. 1973. Z. Physik 265, 239. Elbaum, C.  197 )-. 1  Phys. Rev. L e t t .  3 2 ,  377.  Epstein, A.J., Etemad, S., Garito, A. F., and Heeger, A. J . 1972. Phys. Rev. B5, 952. F e r r a r i s , J . P., Cowan, D. 0., Walatka, V. and P e r l s t e i n , J . H. 1973. J . Amer. Chem. Soc. 95_, 948. Grant, P. M., Greene, R. L., Wrighton, G. C. and Castro, G. 1973. Phys. Rev. L e t t . 31, 1311. Groff, R. P., Suna, A., and M e r r i f i e l d , R. E. 1974. Phys. Rev. L e t t . 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 I t s Measurement and Control i n Science  and Industry, pll43(Instrument Society of America,  Pittsburgh). Lee, P. A., Rice, T. M., and Anderson, P. W. 1973. Phys. Rev. L e t t . 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 i n The Theory of Superconductivity, ed. N. N. Bogoliubov (Gordon and Breach, New York, 1962). Patton, B. R., and Sham, L. J . 1973. Phys. Rev. L e t t . 31, 631. P e i e r l s , R. E. 1955. Quantum Theory of Solids (Oxford University Press, London). Rice, M. J . and S t r a s s l e r , S. 1973. Solid State Comm. 13, 125. Rubin, L. G. and Lawless, W. N. 1971. Rev. S c i . Instruments 42^ 571. Sample, H. H.,,Neuringer, L. J . and Rubin, L. G. 1974. Rev. S c i . Instruments 45,  64.  Schegolev, I. F. 1972. Phys. Stat. S o l i d i (a) 12, 9. Thompson, A. M. 1958. IRE Trans. I n s t r . 1-7, 245. van der Pauw, L. J . 1961. P h i l i p s Res. Repts. 16, 187. Z e l l e r , H. R. 1973. Festkorperprobleme 13_, 31. Ziman, J . M. 1972. P r i n c i p l e s of the Theory of Solids , Ch. 7, (Cambridge University Press, London).  

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