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Hall coefficient and transverse magneto-resistance in HgTe at 4·2°k and 77°k
J. Phys.
Chem. Solids
HALL
Pergamon
COEFFICIENT RESISTANCE
AND TRANSVERSE
Great Britain.
MAGNETO-
IN HgTe AT 4*2”K AND 77°K
T. C. HARMAN, Lincoln Laboratory,t
Printed in
Press 1967. Vol. 28, pp. 1995-2001.
Massachusetts
J. M. HONIG*
and
Institute of Technology, (Received
2 March
P. TRENT Lexington,
Massachusetts
02173
1967)
and Abstract-Using a computer optimization routine, it was ascertained that magneto-Hall magnetoresistance data for HgTe at 4*2”K could be fitted by a three-carrier model involving two sets of electrons and one set of holes. This quantitative examination of the number and types of carriers establishes the occurrence of off-axis maxima [see Fig. l(d)] associated with the inverted band structure model for HgTe. Under special crystal growth conditions the electron Hall mobility at 4.2”K of HgTe exceeded 640,000 ems/V-sec.
INTRODUCTION
WHILE much work has been published in the last few years pertaining to the thermoelectric, thermomagnetic, and optical properties of HgTe, there is as yet no unanimity of opinion concerning the details of the band structure of thismaterial. In 1961, it was concluded(l) that HgSe and HgSe,., Tea., are semimetals and it was suggested that HgTemightbelikewise. This was later corroborated in measurementsof theHal coefficient and of electrical resistivity(2) on HgTe and on its alloys, at low concentration, with CdTe. These and related experiments were ultimately interpretedc3) in terms of an inverted band structure model derived from that proposed originally by GROVES and PAUL(~) for grey tin. Independently of Ref. 3, GROVESand PAUL(~) suggested that the inverted band model applies to HgTe as well as a-Sn. As illustrated in Fig. l(a), the band structure normally encountered in Group IV elements involves a Is conduction band separated by a set of l?s bands that are degenerate at the origin. The dispersion relations for grey tin are shown in Fig. l(b) in terms of the l?s bands lying above the Is band; relative to the diamond structure, the order of the bands is now inverted. * Guest Scientist, National Magnet Laboratory, MIT, Cambridge, Mass. t Operated with support from the U.S. Air Porte. 1995
The success of the band inversion procedure in explaining the semimetallic properties of grey tin suggested that a similar scheme be tried for HgTe. The parent band structure in this case, which
Vrs
5
AL ‘;,
rs i\
+ (a)
(b) GREY
Ge
Sn
JL rs /\ Cc) InSb FIG.
1.
(d) PROPOSED
Various band structure
HgTe
models
(see
text).
corresponds to the zinc blende crystal structure, is shown in Fig. l(c) and is labelled as the InSb structure. Owing to lack of inversion symmetry, the energy dispersion relations now contain a term linear to the wave number vector, k. As a consequence, a small split is predicted, away from k = 0, in the I’s band of Fig. l(c), with shallow extrema
1996
T. C. HARMAN,
J. M. HONIG
and P. TRENT
located away from the origin of the Brillouin zone. 77°K for small x he provisionally identified a This deviation from l(a) is so slight that it has been second set of electron-like carriers. On the other observed only recently in InSb,@) experimentally. hand, TOVSTYUK and co-workers(13) concluded The problem of inverting the InSb model has been that their measurements indicated the presence considered by GORZKOWSKI,(~)who indicates that of one set of electrons and two sets of holes in their a quantitative specification of c(k) would be a comsamples. KOLOSOV and SHARAVSKII(~~)reach a plex undertaking. In the absence of a simple similar conclusion on the basis of Seebeck coeffiquantitative theoretical inversion scheme, we cient measurements at 188°K in a transverse therefore investigated the suitability of the band magnetic field. VERIB(~~)bases the interpretation structure model previously proposed in the literaof his electrical measurements in the range turec3e5) for HgTe as sketched in Fig. l(d). 77-353°K on a two-band model, though he does If Fig. l(b) represents the band structure of sketch an overlap band model similar to that in HgTe, then it should be possible to explain all Fig. l(d). Finally, YAMAMOTOand FUKUROI(~~)on electrical measurements down to liquid helium the basis of the Shubnikov-de Haas effect conclude temperatures, in terms of one set of holes and one that HgTe is characterized by two distinct conducset of electrons, so long as the Fermi level remains tion bands involving electrons of different mass. close to the degenerate band edge energy. By conPart of the difficulty in arriving at a clear decision concerning the applicability of the various trast, the model represented by Fig. l(d) predicts band structures stems from the fact that a threethat two sets of electrons and one set of holes participated in conduction processes under carrier model yields very complex expressions for these conditions. This point is brought out in the transport coefficients,(i7) so that it becomes difficult to make comparisons between theoretical Appendix A. predictions and experimental observations. The The interpretation of experimental results with respect to this problem is still rather confused: present work was undertaken to clarify aspects concerning the applicability of the three-carrier In a review(*) of work completed through 1964, it model to HgTe. Towards this end we accumulated was pointed out that a two-carrier model is ina large body of experimental results on the Hall adequate to account for the transverse magnetocoefficient and the magnetoresistance of undoped resistance and magneto-Hall data measurements HgTe. Selected representative runs were then at 77°K and 4~2°K for a variety of doping levels, subjected to optimal fitting procedures by a comand this was attributed to an unusually complex puter to ascertain whether a three-carrier model valence band structure. More recent work has furnished further corroboration: IVANOV-OMSKII was indeed required to fit the data, and if so, whether one could distinguish between the alterand co-workers(g) have reported systematic measnatives of requiring two sets of electrons and one urements on a series of samples which varied over set of holes as against the inverse possibility. a wide range of net donor and acceptor concentration. Their attempts to fit the data to a twoPREPARATION OF MATERLUS carrier model, using the method of HILSUM and The three samples reported on in this investigaBARRIE(~~) proved unsuccessful. By contrast, tion were prepared by the technique discussed in STRADLINGand ANTCLIFFE(~~)have reported quanRef. 8 with the following differences in detail: tum oscillations in magnetoresistance at 4.2”K in Sample 4A was an as-grown sample cut from the high magnetic fields on a sample of HgTe in which first-to-freeze part of an ingot grown at a rate of the Hall coefficient reversed sign with increasing 3 mm/hr from a stoichiometric 100 g charge which magnetic field. They interpreted their results in had been sealed, along with 2.7 mg of Al, in a terms of a two-carrier model and they report an quartz tube. The temperature of the coolest part of overlap energy in excess of 0.0007 eV between the the tube in contact with vapor during growth was valence and conduction bands. No attempt was 560”, corresponding to a mercury pressure of 15 made to determine if their data were also consistent atm. Sample 300-2A was cut from the first-towith a three-carrier model at low temperatures. freeze part of an ingot grown at a rate of 3 mm/hr, DELVE#~) investigated the properties of single from a stoichiometric 400 g charge containing crystals of Hg,_ ,Mn,Te; from measurements at
HALL
COEFFICIENT
AND
TRANSVERSE
10.8 g of excess Hg. The temperature of the coolest part of the tube in contact with vapor during growth was 590”. The as-grown sample was sealed in an evacuated quartz tube and the tube placed in the first zone at a temperature of 350”, while the temperature of the coolest part or second zone was 300”. The sample was annealed for 336 hr and quenched in water. Sample 50 LB was cut 3 mm from the first-to-freeze part of an ingot, which was grown from a stoichiometric 400 g charge containing 15 g of excess Hg. The growth rate was 0.1 mm/hr, and the material was quenched in water. The temperature of the coolest part of the tube in contact with vapor during growth was 590”. Since only 10 g of excess Hg was required to make up for the loss of mercury to the vapor, this ingot crystallized from a liquid composition that was slightly rich in mercury. Table 1. Experimental
Dopant
50 LB 300-2A 4A
None None Al
1.
-27
2.
1.95x10-”
IN
1997
HgTe
analyzed according to the method of HILSUM and BARRIE.(~~)Plots were made of the Hall coefficients vs. magnetoresistance and of the magneto-Hall and magnetoresistance vs. the inverse square magnetic field. These should result in straight lines on the basis of a two-carrier model: they were, in fact, found to exhibit serious departures from linearity. This clearly required a more sophisticated analysis in terms of at least a three-carrier model. The transport theory involving such a model was first worked out by WILLARDSON and coworkers(19) for the case of P-type Ge. The final results rested on two assumptions, namely that classical statistics was applicable and that the mean free path of the charge carriers was independent of energy. Even so, the final results were rather complex, involving exponential integrals and error functions. In HgTe the situation is further com-
values of some electrical properties of HgTe samples
300°K Sample
MAGNETORESISTANCE
1. -94 -99 -14.5
4.2”K
77°K 2.
3.
1.
9.6 x~O-~ 2.1 x10-3 1.29x10-a
10.7 5.9 0.137
-2700 -1210 -4.2
2. 4.40~10-~ 3.33x10-2 1.43x10-a
3. 191 4.9 0.076
4. 6.4 x lo5 3.6 x lo4 -
1. Hall coefficient W(cm3C-‘) at H = 4 kG at 300°K and 77°K and H = 0.4 kG at 4=2”K. 2. Electrical resistivity in zero magnetic field, p(O) (a-cm). 3. Transverse
maanetoresistance.
A~Io(01. at H = 21 kG.
4. Hall mobility, &‘/p(O), at H =‘O.h’k:’ EXPERIMENTAL RESULTS Representative data for the Hall coefficient in low magnetic field, for the electrical resistivity in the absence of magnetic fields, for transverse magnetoresistance, and for Hall mobility at 300, 77 and 4.2”K are assembled in Table 1. An interesting feature of the data is the unusually large value of the electron Hall mobility of unannealed sample 50 LB at 4*2”K. The value of 640,000 cm2/V-set is significantly higher than the previously higher value of 360,000 cm2/V-see reported in Ref. 8. In fact, unannealed specimens from ingot 50 have exhibited unusually well defined quantum oscillatory effects in both magnetoresistance and magneto-optical measurements.(l*) ANALYSIS
OF RESULTS
The magneto-Hall and magnetoresistance measurements taken at 77°K and 4~2°K were first
plicated by the fact that for semimetals classical statistics is inapplicable. Accordingly, it was decided to adopt a thermodynamic formulationC1’) for isotropic materials in which the overall conductivity c and Hall coefficient g are expressed in terms of the one-band contributions al and Wt (i = 1, 2, 3). That these relations should apply to HgTe is substantiated by unpublished work of the authors on the angular dependence of magnetoresistance of oriented single crystals of HgTe in high magnetic fields; no significant anisotropy effects were encountered even in the conduction region. For ease of interpretation, the substitutions gi = n;ept and Wi = l/n,e were used in the overall expressions for $!J?and o. Here, e is the electronic charge, rzl the carrier density, and ~_l~the carrier mobility of the ith species. In adopting these expressions we neglect the variation of W, with the magnetic field, H, which is usually considerably
T. C. HARMAN,
1998
J. M. HONIG
less than the maximum value of 50 per cent. We also neglect the dependence of ut on H, which is usually considerably less than the maximum factor of 2,4.(20*22) Since changes in&%? and o of many orders of magnitude are observed, it is believed that the conclusions reached in this publication are not altered by use of these simplifications. As FISCHER@~) was first to show, in the approximation scheme discussed above, the dependence of the overall resistivity p and of the overall Hall coefficient 9 on H may be expressed as 1 A+Bx+-Lx2 (1)
f4x) = ;KDy+-M> 1 E+Fx+Nx2 g(x)
(2)
= ;=$-
where, with e in coulombs, x = 10T8 Hz (ir in gauss), and where A, B, D, E, F, L, M* and N are eight nonlinear functions of the three-charge carrier densities ni (in cme3) and of the three mobilities pt (in cm2/V-set) participating in conduction. The parameters ni and pi were determined with the aid of a computer program described in further detail in Appendix B. In essence, the experimentally determined p(O) and Z!2?(HE) were used as input variables for a program which yielded values of n, and ,ui that provided an optimal fit of equation (2) to the experimental Hall measurements. With the ni and pi determined in this manner, the magnetoresistance was then calculated from equation (1) according to the relation Gus - 1. The parameters determined in this fashion for three representative specimens are listed in Table 2. The following points should be noted: (1) The signs for the ni and p”iwere determined by the computer; from the tabulation one concludes that an optimal fit of the data requires that two of the three
and P.
TRENT
carriers be electron-like. (2) A variation of any of the carrier density or mobilities by + 10 per cent about the listed value resulted in a noticeably poorer fit of the theory to the data, as judged from least-squares calculations. This was also the case when one or more of the parameters were simultaneously altered by f 10 per cent. The values listed in Table 2 as providing the optimal fit are thus known with reasonable precision. A comparison between the calculated and the observed magneto-Hall effects is provided in Fig. 2. Correlations between the results of Table 2 and the observations in Fig. 2 are readily apparent: Sample 50 LB exhibits a negative Hall coefficient because the two sets of electrons outweigh the holes in concentration. In passing one should note the distinct maximum (actually, a minimum) in the magneto-Hall curve for this sample; on the basis of elementary arguments it may be shown that this cannot occur in a two-carrier model under the assumption that the one carrier gi and ui are independent of El. In samples 4A and 300-2A the sign reversal of &? in passing from low to high magnetic fields reflects the preponderance in concentration of low-mobility holes over intermediate and high-mobility electrons. As is evident from Fig. 3, there is considerable discrepancy between the calculated and the observed magnetoresist~ce values, although the general trend of the experimental results is correctly brought out in the calculations. The fit could be considerably improved by using both W(NJ and P(HJ as input parameters to the computer program. The discrepancies are due in part to the simplifying assumption which neglects the variation of &Z$?{ and cri on H. It should be noted that in sample 50 LB the magnetoresistance attained values in excess of 600 without showing any saturation effects at 110 kG. The results cited above are consistent with the band model sketched in Fig. l(d). As has been
Table 2. CaIcuEated carrier concentrations and carrier mobilities in HgTe at 4=2”K
Sample No.
n,(cm - 3,
50 LB 300-2A 4A
- 2.1 x 1015 -1-2 x 1015 -1.2 x 1015
n,(cm - s) -7*0x10= -3.3x1015 -1~0x10~~
ns(cm - “) 6.2 x IO’* 6.4 x 1O1” 2.5 x 701%
~,(cm2/V-see)
&cmz/V-set)
-6.5 x lo5 -1~0x10~ -1*2x10*
- 9.5 x 102 -1*3x109 -1.7x103
&cm2/V-see) 7.1 x 104 420 160
HALL
COEFFICIENT
AND
TRANSVERSE
MAGNETO~ESISTANCE
_/4A
-
.
EXPERIMENT
(riqhl
se&t
IN
_
HgTe
1999
c
POINTS
HtkG)
2. Theoretica of HgTe at 4-2°K.
FIG.
1
1
16
8 tf!!i~l
!
3 1
id’
and experimental Hall coefficient Samples 50 LB, 4-A and 300-2A.
St1211
too
1
11
111111
to’
’
1
‘)1111’
v.3
’
’
?IQiJ
iu3
&‘P(O) FIG. 3. Theoretical and experimental of HgTe at 4.2”K. Samples 50LB,
pointed out in Ref. 8, it is generally rather difficult to determine the thermal overlap energy Et with precision. A lower limit to this quantity is calculated from the value of ni for sample 50 LB; here, E, > EF where .EF is the Fermi level relative to the bottom of the conduction band. Using an effective electron mass of m/m, = 0.023, as determined from the direct gap(18) energy value of E, = O-283 eV and from the Kane parameter P = 8.3 x 10e8 eV-cm, one finds that -
=
0.0028
eV;
(3)
magnetoresistance 4-A and 300-2A.
an upper limit(*) to Et of 0.02 eV is set by calculations involving the two-band model. CONCLUSIONS
~a~eto-~a11 and transverse mag~etoresist~ce data on a large number of HgTe single crystals were analysed at 4*2”K. The number and types of carriers were determined to be two sets af electrons and one set of holes. These findings are shown to be consistent with a previously proposed inverted band structure model involving off-axis maxima as depicted in Fig. l(d). The lower limit to the overlap energy E, is 0.~28 eV. An unusually high
2000
T. C. HARMAN,
J. M. HONIG
electron mobility of 640,000 cm2/V-set was obtained at 4*2”K as a result of special conditions of crystal growth, Acknowledgements-The authors are grateful to A. E. PALADINO and J. R. MCNALLY for their assistance in preparing the materials and for making many of the electrical measurements, and to S. HILSENRATH for performing some calculations. They are also pleased to acknowledge the helpful comments of Dr. J. 0. DIM~IOCK and Dr. S. H. GROVES. Finally, they express their appreciation to the National Magnet Laboratory for use of the high magnetic field facilities in carrying out the Hall and resistivity measurements.
and P.
20. WILSON A. H., The Theory of Metals, pp. 236-240, Cambridge Univeaity Press, 2nd ed. (1958). 21. FISCHER G., Helv. phys. Acta 33, 463 (1960). 22. HARMAN T. C. and HONIG J. M., J. Phys. Chem. Solids 23, 913 (1962); HARMAN T. C., HONIC J. M. and TARMY B. M., ibid. 24, 835 (1963). 23. NELDERJ. A. and MEAD R., Comput. J. 7,308 (1965).
THE
1. 2.
3.
4. 5. 6. 7. 8.
9.
10. 11.
12. 13.
14. 15. 16. 17. 18. 19.
APPENDIX QUARTIC
A BAND
In this appendix we describe briefly the electrical characteristics to be associated with a quartic band model of the form E -_
REFERENCES HARMAN T. C. and STRAUSSA. J., J. Appl. Phys. Suppl. 32, 2265 (1961). STRALW A. J., HARMAN T. C., MAVROIDE~J. G., DICKEY D. H. and DRE~SELHAUS M. S., PYOC.Int. Conf. Physics of Semiconductors, Exeter, p. 703 ff, Institute of Physics and The Physical Society; London (1962). HARMAN T. C., KLEIN~R W. H., STRAUSS A. J., WRIGHT G. B., MAVROIDE~J. G., HONIG J. M. and DICKEY D. H., Solid State Commun. 2, 305 (1964). GRU~E~ S. and PAUL W., Phys. Rev. Lett. 11, 194 (1963). GROVESS. and PAUL W., Proc. 7th Int. Conf. Physics of Semiconductors, p. 41 ff, Dunod, Paris (1964). ROBINSONM. I,. A., Phys. Rev. Lett. 17,963 (1966). GORZI
TRENT
= x =
--Alr4+B/r2
kT
(A-1)
which is sketched in Fig. 4. Here E is the energy, K is Boltzmann’s constant, A and B are constants, and .&r is the radial component of the wave number vector. It is seen that a band of the form (A.l) simulates one of the Ps bands in the proposed band scheme for HgTe, Fig. l(d). It is important to note that for such a band with spherical symmetry the current density may be split into two contributions, namely
J = rJ’ + i
7
0
-n/2
=
I0
~.Zevjl
ig,” sin B dJ.dtM’@k
0
J1+J2
(A.21
where e is the electronic charge Z = It 1, Z, = fi- ‘Cke, fi the deviation of the nonequilibrium distribution function from the Fermi-Dirac distribution function f,_, 3 [exp(x- 7) -I- 11 -I, 7 the reduced Fermi level relative to the appropriate band edge; 0~ and +r are the appropriate angular coordinates in k-space, and quantities z) and fi are functions of x alone and are thus independent of t?d, &. One may therefore treat the band in Fig. 4 as a combination of two sub-bands, each of which may be considered separately. Focusing on the inner band, 0 < /, < /a, and switching to the formulation of transport effects in terms of irreversible thermodynamics we now write: J1” =
or3 115 +
31” = oysv,+
C2Y VY
(A.31
oyvvly
(A.41
where Vh 3 Vh(c/e), 1 being the Fermi level relative to the electron at rest in free space, and where az,(Hz2) = ay,(Hz2) are the a,,(-%) = o&H,)> components of the conductivity tensor. Setting J1y = 0 and eliminating I’, from (A.3) we then obtain (VT = 0, Jly = 0) W,
=
V,lJl= = ~yz/(~zy~yz-~zt~yy) (A.51
HALL
COEFFICIENT
AND
TRANSVERSE
As the band under consideration is filled with charge carriers, thereby increasing &‘rfrom 0 onwards & the denominator in (A.5) remains unaltered in sign. The sign of WI is thus determined by that of oYI. As shown
MAGNETORESISTANCE
IN
HgTe
2001
These findings are substantiated by a detailed study in which (A-1) was solved for k(x) and the results used in the explicit evaluation of transport integrals according to the procedures of Ref. 22.
APPENDIX B OPTIMIZATION ROUTINE The charge carrier densities and mobiiities listed in Table 2 were determined by an optimization routine due to NELD~R and MEAD. In essence, the program is designed to minimize the function d(n2, %,Pl,P2r
113)
i=l
BAND
STRUCTURE
where the subscript E and T designate respectively the Hall coefficients determined experimentally and those calculated according to equation (2). In equation fB.1) the variabie nr has been ehminated by use of equation (l), rewritten as
CHARACTERIZED
BY j+=-Aibk2 FIG. 4. The quartic band structure model,
elsewhere, for bands af spherical symmetry, and for cases where the Boltzmann transport equation in the relaxation time formalism applies, ko
ni2
2x tg”72
Qyt = 0 -n12 0
8fo
r+
x lTg,”sin Ok d k,dOk’d$L
(-4.6)
where I is the relaxation time, ZI, = tc -rVkYe, and where UJZZ -eH,jm,*. The sign of cry. is thus governed by that of the first derivatiwe effective masPa)
(-4.7) With 6 being the radius vector relative to the origin of coordinates at the point of spherical symmetry, the effective mass, according to (A.l), remains positive for 0 < L, < lo, but is negative for A> A,,. It follows that insofar as the Hall coefficient is concerned, the inner portion of the band in Fig. 4 is always populated with electron-like carriers, regardless of the degree of filling, while the outer portion of the band in Fig. 4 is populated with holes. It is of interest that a similar argument pertaining to the Seebeck coefficient leads to the conclusion that the thermoelectric power changes sign as the inner band is progressively filled with carriers.
A = nlpl + n2p2 +n3p3
=
lle~(O)
(B.2)
The method requires an initial guess of na, x3, pr, pa, pa which locates a point in five-space : the computer then generates five additional points in the vicinity, according to a prescribed routine, and determines the center of gravity of the resulting convex polyhedron in five-space, called a simplex. The function r# is then calculated according to equation (B.l) for each of the vertices of the simplex; the vertex (extremum) corresponding to the highest Q is singled out, and a new point is located at a prescribed distance along the line passing through the vertex in question and the center of gravity. The old extremum is now discarded, and a new simplex generated from the new point and the remaining vertices of the original simplex. The procedure is repeated until the final simplex consists of six points spaced very closely together. The center of gravity now corresponds to the optimal values of the parameters; however, this set under certain conditions may correspond to a local minimum in Q rather than to the true minimum. To guard against this possibility, each of the points of the final simplex is distended in turn by S-10 percent from its value, and the process of finding a minimum in 4 is repeated. The advantage in the approach lies in the fact that it is readily programmed in Fortran, that it is free from erratic loops, and that it does not require programmi~ of the derivatives of (b. The total machine time required to run through a set of experimental points on a SDS 930 computer was of the order of five minutes.
Chem. Solids
HALL
Pergamon
COEFFICIENT RESISTANCE
AND TRANSVERSE
Great Britain.
MAGNETO-
IN HgTe AT 4*2”K AND 77°K
T. C. HARMAN, Lincoln Laboratory,t
Printed in
Press 1967. Vol. 28, pp. 1995-2001.
Massachusetts
J. M. HONIG*
and
Institute of Technology, (Received
2 March
P. TRENT Lexington,
Massachusetts
02173
1967)
and Abstract-Using a computer optimization routine, it was ascertained that magneto-Hall magnetoresistance data for HgTe at 4*2”K could be fitted by a three-carrier model involving two sets of electrons and one set of holes. This quantitative examination of the number and types of carriers establishes the occurrence of off-axis maxima [see Fig. l(d)] associated with the inverted band structure model for HgTe. Under special crystal growth conditions the electron Hall mobility at 4.2”K of HgTe exceeded 640,000 ems/V-sec.
INTRODUCTION
WHILE much work has been published in the last few years pertaining to the thermoelectric, thermomagnetic, and optical properties of HgTe, there is as yet no unanimity of opinion concerning the details of the band structure of thismaterial. In 1961, it was concluded(l) that HgSe and HgSe,., Tea., are semimetals and it was suggested that HgTemightbelikewise. This was later corroborated in measurementsof theHal coefficient and of electrical resistivity(2) on HgTe and on its alloys, at low concentration, with CdTe. These and related experiments were ultimately interpretedc3) in terms of an inverted band structure model derived from that proposed originally by GROVES and PAUL(~) for grey tin. Independently of Ref. 3, GROVESand PAUL(~) suggested that the inverted band model applies to HgTe as well as a-Sn. As illustrated in Fig. l(a), the band structure normally encountered in Group IV elements involves a Is conduction band separated by a set of l?s bands that are degenerate at the origin. The dispersion relations for grey tin are shown in Fig. l(b) in terms of the l?s bands lying above the Is band; relative to the diamond structure, the order of the bands is now inverted. * Guest Scientist, National Magnet Laboratory, MIT, Cambridge, Mass. t Operated with support from the U.S. Air Porte. 1995
The success of the band inversion procedure in explaining the semimetallic properties of grey tin suggested that a similar scheme be tried for HgTe. The parent band structure in this case, which
Vrs
5
AL ‘;,
rs i\
+ (a)
(b) GREY
Ge
Sn
JL rs /\ Cc) InSb FIG.
1.
(d) PROPOSED
Various band structure
HgTe
models
(see
text).
corresponds to the zinc blende crystal structure, is shown in Fig. l(c) and is labelled as the InSb structure. Owing to lack of inversion symmetry, the energy dispersion relations now contain a term linear to the wave number vector, k. As a consequence, a small split is predicted, away from k = 0, in the I’s band of Fig. l(c), with shallow extrema
1996
T. C. HARMAN,
J. M. HONIG
and P. TRENT
located away from the origin of the Brillouin zone. 77°K for small x he provisionally identified a This deviation from l(a) is so slight that it has been second set of electron-like carriers. On the other observed only recently in InSb,@) experimentally. hand, TOVSTYUK and co-workers(13) concluded The problem of inverting the InSb model has been that their measurements indicated the presence considered by GORZKOWSKI,(~)who indicates that of one set of electrons and two sets of holes in their a quantitative specification of c(k) would be a comsamples. KOLOSOV and SHARAVSKII(~~)reach a plex undertaking. In the absence of a simple similar conclusion on the basis of Seebeck coeffiquantitative theoretical inversion scheme, we cient measurements at 188°K in a transverse therefore investigated the suitability of the band magnetic field. VERIB(~~)bases the interpretation structure model previously proposed in the literaof his electrical measurements in the range turec3e5) for HgTe as sketched in Fig. l(d). 77-353°K on a two-band model, though he does If Fig. l(b) represents the band structure of sketch an overlap band model similar to that in HgTe, then it should be possible to explain all Fig. l(d). Finally, YAMAMOTOand FUKUROI(~~)on electrical measurements down to liquid helium the basis of the Shubnikov-de Haas effect conclude temperatures, in terms of one set of holes and one that HgTe is characterized by two distinct conducset of electrons, so long as the Fermi level remains tion bands involving electrons of different mass. close to the degenerate band edge energy. By conPart of the difficulty in arriving at a clear decision concerning the applicability of the various trast, the model represented by Fig. l(d) predicts band structures stems from the fact that a threethat two sets of electrons and one set of holes participated in conduction processes under carrier model yields very complex expressions for these conditions. This point is brought out in the transport coefficients,(i7) so that it becomes difficult to make comparisons between theoretical Appendix A. predictions and experimental observations. The The interpretation of experimental results with respect to this problem is still rather confused: present work was undertaken to clarify aspects concerning the applicability of the three-carrier In a review(*) of work completed through 1964, it model to HgTe. Towards this end we accumulated was pointed out that a two-carrier model is ina large body of experimental results on the Hall adequate to account for the transverse magnetocoefficient and the magnetoresistance of undoped resistance and magneto-Hall data measurements HgTe. Selected representative runs were then at 77°K and 4~2°K for a variety of doping levels, subjected to optimal fitting procedures by a comand this was attributed to an unusually complex puter to ascertain whether a three-carrier model valence band structure. More recent work has furnished further corroboration: IVANOV-OMSKII was indeed required to fit the data, and if so, whether one could distinguish between the alterand co-workers(g) have reported systematic measnatives of requiring two sets of electrons and one urements on a series of samples which varied over set of holes as against the inverse possibility. a wide range of net donor and acceptor concentration. Their attempts to fit the data to a twoPREPARATION OF MATERLUS carrier model, using the method of HILSUM and The three samples reported on in this investigaBARRIE(~~) proved unsuccessful. By contrast, tion were prepared by the technique discussed in STRADLINGand ANTCLIFFE(~~)have reported quanRef. 8 with the following differences in detail: tum oscillations in magnetoresistance at 4.2”K in Sample 4A was an as-grown sample cut from the high magnetic fields on a sample of HgTe in which first-to-freeze part of an ingot grown at a rate of the Hall coefficient reversed sign with increasing 3 mm/hr from a stoichiometric 100 g charge which magnetic field. They interpreted their results in had been sealed, along with 2.7 mg of Al, in a terms of a two-carrier model and they report an quartz tube. The temperature of the coolest part of overlap energy in excess of 0.0007 eV between the the tube in contact with vapor during growth was valence and conduction bands. No attempt was 560”, corresponding to a mercury pressure of 15 made to determine if their data were also consistent atm. Sample 300-2A was cut from the first-towith a three-carrier model at low temperatures. freeze part of an ingot grown at a rate of 3 mm/hr, DELVE#~) investigated the properties of single from a stoichiometric 400 g charge containing crystals of Hg,_ ,Mn,Te; from measurements at
HALL
COEFFICIENT
AND
TRANSVERSE
10.8 g of excess Hg. The temperature of the coolest part of the tube in contact with vapor during growth was 590”. The as-grown sample was sealed in an evacuated quartz tube and the tube placed in the first zone at a temperature of 350”, while the temperature of the coolest part or second zone was 300”. The sample was annealed for 336 hr and quenched in water. Sample 50 LB was cut 3 mm from the first-to-freeze part of an ingot, which was grown from a stoichiometric 400 g charge containing 15 g of excess Hg. The growth rate was 0.1 mm/hr, and the material was quenched in water. The temperature of the coolest part of the tube in contact with vapor during growth was 590”. Since only 10 g of excess Hg was required to make up for the loss of mercury to the vapor, this ingot crystallized from a liquid composition that was slightly rich in mercury. Table 1. Experimental
Dopant
50 LB 300-2A 4A
None None Al
1.
-27
2.
1.95x10-”
IN
1997
HgTe
analyzed according to the method of HILSUM and BARRIE.(~~)Plots were made of the Hall coefficients vs. magnetoresistance and of the magneto-Hall and magnetoresistance vs. the inverse square magnetic field. These should result in straight lines on the basis of a two-carrier model: they were, in fact, found to exhibit serious departures from linearity. This clearly required a more sophisticated analysis in terms of at least a three-carrier model. The transport theory involving such a model was first worked out by WILLARDSON and coworkers(19) for the case of P-type Ge. The final results rested on two assumptions, namely that classical statistics was applicable and that the mean free path of the charge carriers was independent of energy. Even so, the final results were rather complex, involving exponential integrals and error functions. In HgTe the situation is further com-
values of some electrical properties of HgTe samples
300°K Sample
MAGNETORESISTANCE
1. -94 -99 -14.5
4.2”K
77°K 2.
3.
1.
9.6 x~O-~ 2.1 x10-3 1.29x10-a
10.7 5.9 0.137
-2700 -1210 -4.2
2. 4.40~10-~ 3.33x10-2 1.43x10-a
3. 191 4.9 0.076
4. 6.4 x lo5 3.6 x lo4 -
1. Hall coefficient W(cm3C-‘) at H = 4 kG at 300°K and 77°K and H = 0.4 kG at 4=2”K. 2. Electrical resistivity in zero magnetic field, p(O) (a-cm). 3. Transverse
maanetoresistance.
A~Io(01. at H = 21 kG.
4. Hall mobility, &‘/p(O), at H =‘O.h’k:’ EXPERIMENTAL RESULTS Representative data for the Hall coefficient in low magnetic field, for the electrical resistivity in the absence of magnetic fields, for transverse magnetoresistance, and for Hall mobility at 300, 77 and 4.2”K are assembled in Table 1. An interesting feature of the data is the unusually large value of the electron Hall mobility of unannealed sample 50 LB at 4*2”K. The value of 640,000 cm2/V-set is significantly higher than the previously higher value of 360,000 cm2/V-see reported in Ref. 8. In fact, unannealed specimens from ingot 50 have exhibited unusually well defined quantum oscillatory effects in both magnetoresistance and magneto-optical measurements.(l*) ANALYSIS
OF RESULTS
The magneto-Hall and magnetoresistance measurements taken at 77°K and 4~2°K were first
plicated by the fact that for semimetals classical statistics is inapplicable. Accordingly, it was decided to adopt a thermodynamic formulationC1’) for isotropic materials in which the overall conductivity c and Hall coefficient g are expressed in terms of the one-band contributions al and Wt (i = 1, 2, 3). That these relations should apply to HgTe is substantiated by unpublished work of the authors on the angular dependence of magnetoresistance of oriented single crystals of HgTe in high magnetic fields; no significant anisotropy effects were encountered even in the conduction region. For ease of interpretation, the substitutions gi = n;ept and Wi = l/n,e were used in the overall expressions for $!J?and o. Here, e is the electronic charge, rzl the carrier density, and ~_l~the carrier mobility of the ith species. In adopting these expressions we neglect the variation of W, with the magnetic field, H, which is usually considerably
T. C. HARMAN,
1998
J. M. HONIG
less than the maximum value of 50 per cent. We also neglect the dependence of ut on H, which is usually considerably less than the maximum factor of 2,4.(20*22) Since changes in&%? and o of many orders of magnitude are observed, it is believed that the conclusions reached in this publication are not altered by use of these simplifications. As FISCHER@~) was first to show, in the approximation scheme discussed above, the dependence of the overall resistivity p and of the overall Hall coefficient 9 on H may be expressed as 1 A+Bx+-Lx2 (1)
f4x) = ;KDy+-M> 1 E+Fx+Nx2 g(x)
(2)
= ;=$-
where, with e in coulombs, x = 10T8 Hz (ir in gauss), and where A, B, D, E, F, L, M* and N are eight nonlinear functions of the three-charge carrier densities ni (in cme3) and of the three mobilities pt (in cm2/V-set) participating in conduction. The parameters ni and pi were determined with the aid of a computer program described in further detail in Appendix B. In essence, the experimentally determined p(O) and Z!2?(HE) were used as input variables for a program which yielded values of n, and ,ui that provided an optimal fit of equation (2) to the experimental Hall measurements. With the ni and pi determined in this manner, the magnetoresistance was then calculated from equation (1) according to the relation Gus - 1. The parameters determined in this fashion for three representative specimens are listed in Table 2. The following points should be noted: (1) The signs for the ni and p”iwere determined by the computer; from the tabulation one concludes that an optimal fit of the data requires that two of the three
and P.
TRENT
carriers be electron-like. (2) A variation of any of the carrier density or mobilities by + 10 per cent about the listed value resulted in a noticeably poorer fit of the theory to the data, as judged from least-squares calculations. This was also the case when one or more of the parameters were simultaneously altered by f 10 per cent. The values listed in Table 2 as providing the optimal fit are thus known with reasonable precision. A comparison between the calculated and the observed magneto-Hall effects is provided in Fig. 2. Correlations between the results of Table 2 and the observations in Fig. 2 are readily apparent: Sample 50 LB exhibits a negative Hall coefficient because the two sets of electrons outweigh the holes in concentration. In passing one should note the distinct maximum (actually, a minimum) in the magneto-Hall curve for this sample; on the basis of elementary arguments it may be shown that this cannot occur in a two-carrier model under the assumption that the one carrier gi and ui are independent of El. In samples 4A and 300-2A the sign reversal of &? in passing from low to high magnetic fields reflects the preponderance in concentration of low-mobility holes over intermediate and high-mobility electrons. As is evident from Fig. 3, there is considerable discrepancy between the calculated and the observed magnetoresist~ce values, although the general trend of the experimental results is correctly brought out in the calculations. The fit could be considerably improved by using both W(NJ and P(HJ as input parameters to the computer program. The discrepancies are due in part to the simplifying assumption which neglects the variation of &Z$?{ and cri on H. It should be noted that in sample 50 LB the magnetoresistance attained values in excess of 600 without showing any saturation effects at 110 kG. The results cited above are consistent with the band model sketched in Fig. l(d). As has been
Table 2. CaIcuEated carrier concentrations and carrier mobilities in HgTe at 4=2”K
Sample No.
n,(cm - 3,
50 LB 300-2A 4A
- 2.1 x 1015 -1-2 x 1015 -1.2 x 1015
n,(cm - s) -7*0x10= -3.3x1015 -1~0x10~~
ns(cm - “) 6.2 x IO’* 6.4 x 1O1” 2.5 x 701%
~,(cm2/V-see)
&cmz/V-set)
-6.5 x lo5 -1~0x10~ -1*2x10*
- 9.5 x 102 -1*3x109 -1.7x103
&cm2/V-see) 7.1 x 104 420 160
HALL
COEFFICIENT
AND
TRANSVERSE
MAGNETO~ESISTANCE
_/4A
-
.
EXPERIMENT
(riqhl
se&t
IN
_
HgTe
1999
c
POINTS
HtkG)
2. Theoretica of HgTe at 4-2°K.
FIG.
1
1
16
8 tf!!i~l
!
3 1
id’
and experimental Hall coefficient Samples 50 LB, 4-A and 300-2A.
St1211
too
1
11
111111
to’
’
1
‘)1111’
v.3
’
’
?IQiJ
iu3
&‘P(O) FIG. 3. Theoretical and experimental of HgTe at 4.2”K. Samples 50LB,
pointed out in Ref. 8, it is generally rather difficult to determine the thermal overlap energy Et with precision. A lower limit to this quantity is calculated from the value of ni for sample 50 LB; here, E, > EF where .EF is the Fermi level relative to the bottom of the conduction band. Using an effective electron mass of m/m, = 0.023, as determined from the direct gap(18) energy value of E, = O-283 eV and from the Kane parameter P = 8.3 x 10e8 eV-cm, one finds that -
=
0.0028
eV;
(3)
magnetoresistance 4-A and 300-2A.
an upper limit(*) to Et of 0.02 eV is set by calculations involving the two-band model. CONCLUSIONS
~a~eto-~a11 and transverse mag~etoresist~ce data on a large number of HgTe single crystals were analysed at 4*2”K. The number and types of carriers were determined to be two sets af electrons and one set of holes. These findings are shown to be consistent with a previously proposed inverted band structure model involving off-axis maxima as depicted in Fig. l(d). The lower limit to the overlap energy E, is 0.~28 eV. An unusually high
2000
T. C. HARMAN,
J. M. HONIG
electron mobility of 640,000 cm2/V-set was obtained at 4*2”K as a result of special conditions of crystal growth, Acknowledgements-The authors are grateful to A. E. PALADINO and J. R. MCNALLY for their assistance in preparing the materials and for making many of the electrical measurements, and to S. HILSENRATH for performing some calculations. They are also pleased to acknowledge the helpful comments of Dr. J. 0. DIM~IOCK and Dr. S. H. GROVES. Finally, they express their appreciation to the National Magnet Laboratory for use of the high magnetic field facilities in carrying out the Hall and resistivity measurements.
and P.
20. WILSON A. H., The Theory of Metals, pp. 236-240, Cambridge Univeaity Press, 2nd ed. (1958). 21. FISCHER G., Helv. phys. Acta 33, 463 (1960). 22. HARMAN T. C. and HONIG J. M., J. Phys. Chem. Solids 23, 913 (1962); HARMAN T. C., HONIC J. M. and TARMY B. M., ibid. 24, 835 (1963). 23. NELDERJ. A. and MEAD R., Comput. J. 7,308 (1965).
THE
1. 2.
3.
4. 5. 6. 7. 8.
9.
10. 11.
12. 13.
14. 15. 16. 17. 18. 19.
APPENDIX QUARTIC
A BAND
In this appendix we describe briefly the electrical characteristics to be associated with a quartic band model of the form E -_
REFERENCES HARMAN T. C. and STRAUSSA. J., J. Appl. Phys. Suppl. 32, 2265 (1961). STRALW A. J., HARMAN T. C., MAVROIDE~J. G., DICKEY D. H. and DRE~SELHAUS M. S., PYOC.Int. Conf. Physics of Semiconductors, Exeter, p. 703 ff, Institute of Physics and The Physical Society; London (1962). HARMAN T. C., KLEIN~R W. H., STRAUSS A. J., WRIGHT G. B., MAVROIDE~J. G., HONIG J. M. and DICKEY D. H., Solid State Commun. 2, 305 (1964). GRU~E~ S. and PAUL W., Phys. Rev. Lett. 11, 194 (1963). GROVESS. and PAUL W., Proc. 7th Int. Conf. Physics of Semiconductors, p. 41 ff, Dunod, Paris (1964). ROBINSONM. I,. A., Phys. Rev. Lett. 17,963 (1966). GORZI
TRENT
= x =
--Alr4+B/r2
kT
(A-1)
which is sketched in Fig. 4. Here E is the energy, K is Boltzmann’s constant, A and B are constants, and .&r is the radial component of the wave number vector. It is seen that a band of the form (A.l) simulates one of the Ps bands in the proposed band scheme for HgTe, Fig. l(d). It is important to note that for such a band with spherical symmetry the current density may be split into two contributions, namely
J = rJ’ + i
7
0
-n/2
=
I0
~.Zevjl
ig,” sin B dJ.dtM’@k
0
J1+J2
(A.21
where e is the electronic charge Z = It 1, Z, = fi- ‘Cke, fi the deviation of the nonequilibrium distribution function from the Fermi-Dirac distribution function f,_, 3 [exp(x- 7) -I- 11 -I, 7 the reduced Fermi level relative to the appropriate band edge; 0~ and +r are the appropriate angular coordinates in k-space, and quantities z) and fi are functions of x alone and are thus independent of t?d, &. One may therefore treat the band in Fig. 4 as a combination of two sub-bands, each of which may be considered separately. Focusing on the inner band, 0 < /, < /a, and switching to the formulation of transport effects in terms of irreversible thermodynamics we now write: J1” =
or3 115 +
31” = oysv,+
C2Y VY
(A.31
oyvvly
(A.41
where Vh 3 Vh(c/e), 1 being the Fermi level relative to the electron at rest in free space, and where az,(Hz2) = ay,(Hz2) are the a,,(-%) = o&H,)> components of the conductivity tensor. Setting J1y = 0 and eliminating I’, from (A.3) we then obtain (VT = 0, Jly = 0) W,
=
V,lJl= = ~yz/(~zy~yz-~zt~yy) (A.51
HALL
COEFFICIENT
AND
TRANSVERSE
As the band under consideration is filled with charge carriers, thereby increasing &‘rfrom 0 onwards & the denominator in (A.5) remains unaltered in sign. The sign of WI is thus determined by that of oYI. As shown
MAGNETORESISTANCE
IN
HgTe
2001
These findings are substantiated by a detailed study in which (A-1) was solved for k(x) and the results used in the explicit evaluation of transport integrals according to the procedures of Ref. 22.
APPENDIX B OPTIMIZATION ROUTINE The charge carrier densities and mobiiities listed in Table 2 were determined by an optimization routine due to NELD~R and MEAD. In essence, the program is designed to minimize the function d(n2, %,Pl,P2r
113)
i=l
BAND
STRUCTURE
where the subscript E and T designate respectively the Hall coefficients determined experimentally and those calculated according to equation (2). In equation fB.1) the variabie nr has been ehminated by use of equation (l), rewritten as
CHARACTERIZED
BY j+=-Aibk2 FIG. 4. The quartic band structure model,
elsewhere, for bands af spherical symmetry, and for cases where the Boltzmann transport equation in the relaxation time formalism applies, ko
ni2
2x tg”72
Qyt = 0 -n12 0
8fo
r+
x lTg,”sin Ok d k,dOk’d$L
(-4.6)
where I is the relaxation time, ZI, = tc -rVkYe, and where UJZZ -eH,jm,*. The sign of cry. is thus governed by that of the first derivatiwe effective masPa)
(-4.7) With 6 being the radius vector relative to the origin of coordinates at the point of spherical symmetry, the effective mass, according to (A.l), remains positive for 0 < L, < lo, but is negative for A> A,,. It follows that insofar as the Hall coefficient is concerned, the inner portion of the band in Fig. 4 is always populated with electron-like carriers, regardless of the degree of filling, while the outer portion of the band in Fig. 4 is populated with holes. It is of interest that a similar argument pertaining to the Seebeck coefficient leads to the conclusion that the thermoelectric power changes sign as the inner band is progressively filled with carriers.
A = nlpl + n2p2 +n3p3
=
lle~(O)
(B.2)
The method requires an initial guess of na, x3, pr, pa, pa which locates a point in five-space : the computer then generates five additional points in the vicinity, according to a prescribed routine, and determines the center of gravity of the resulting convex polyhedron in five-space, called a simplex. The function r# is then calculated according to equation (B.l) for each of the vertices of the simplex; the vertex (extremum) corresponding to the highest Q is singled out, and a new point is located at a prescribed distance along the line passing through the vertex in question and the center of gravity. The old extremum is now discarded, and a new simplex generated from the new point and the remaining vertices of the original simplex. The procedure is repeated until the final simplex consists of six points spaced very closely together. The center of gravity now corresponds to the optimal values of the parameters; however, this set under certain conditions may correspond to a local minimum in Q rather than to the true minimum. To guard against this possibility, each of the points of the final simplex is distended in turn by S-10 percent from its value, and the process of finding a minimum in 4 is repeated. The advantage in the approach lies in the fact that it is readily programmed in Fortran, that it is free from erratic loops, and that it does not require programmi~ of the derivatives of (b. The total machine time required to run through a set of experimental points on a SDS 930 computer was of the order of five minutes.