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SOVIET PHYSICS JETP

VOLUME 33, NUMBER 6

DECEMBER 19?1

TEMPERATURE PATHS

DEPENDENCE

OF THE ELECTRON

AND HOLE

MEAN

FREE

IN ANTIMONY

V. F. GANTMAKIIER and V. T. DOLGOPOLOV Institute of solid state physics of the u.s.s.R. Academy of sciences Submitted January Zl, lgTL Zh. Eksp. Teor. Fiz. 60,2260-2269 (June, 19?1) The temperature dependencesof the mean free paths for electrons and holes in antimony are measured sepa.rately on the basis of the amplitudes o] the radio frequency size effect lines. In both cases the temperature-dependent part of the path was found io be proportional to T-2, the proportionality coefflcient for holes beingthree times greater tlan that for electrons. It is shown ttlat if the Fermi surface resembles a long, narrow cylinder, the term quadratic with respect to temperature in the carrier scattering probability may be ascribed to inteiaction between electrons and phonons and that it is precisely this case which is encountered in bismuth and antimony. The magnitudes of the deformation potentials in antimony are estimated for both electrons ancrholes. SO tr" there is rather litile information concernine the connection between the cross section for scattering-of electrons in a metal and their position on the Fermi surface. Kinetic processes widely used for the investi_ gation of electron relana.tion times, such as electric or thermal conductivity, are determined by quantities aver_ aged over the entire Fermi surface. At the same time. there already exist methods for separating and investi_ gating individual groups of carriers in a metal. Such methods include, in particular, the measurement of tlte amplitude of the lines of the radio frequency size effect (SE).t1,21 The SE line is formed by electrons located in the vicinity of a definite extremal section of the Fermi surface. Therefore the line amplitude is determined bv the probability of scattering of precisely these'eiec_ a trons. The purpose of the experiments described here was to compare t}te cross sections for the scattering of the carriers of both types by phonons, using one sample of antimony (in one experiment), by measuring the temper_ ature dependenceof the amplitude of the SE lines from the electronic and hole parts cf the Fermi surface. EXPERIMENT An antimony sample in the form of a disk of thickness d = 0.32 mm and diameter 10 mm was grown from the melt in a dismountable quartz mold. The initial ma_ terial was antimony with a resistance ratio Proom/p4.t 6= 2700. The angles betweenthe normal tothe plane of the sample and the axes C, and C. were 46'10' and, 44oZO, The experiments werjperformed in . the temperature range from 1.1? to 4.2"K. The ampli_ tudes of the SE lines were measured on the plots of the derivatives, with respect to ttre magnetic field, of the imaginary part of the surface impedance, OX/'OHor azx/AHz, at a frequency L2.d UHz. We used the usual modulation,pr ocedure for the measur ements_see, for example, t1l. To record the secondderivative, the de_ tecting part of the apparatus was tuned to a frequency equal to double the modulation frequency, A considerable line width ( tH/Hx 6/dx lSVo:6_

depth of skin layer) made it necessary to take into ac_ count in the data reduction the monotonic part of the function AX/AH. There was no monotonic part in ttre second derivative, but this did not lead to an appreciable increase of the measurement accuracy. Sample plots are shown in Fig. 1. As is well knownrtsl the Fermi surface of antimony consists of three electron and six hole surfaces of nearly-ellipsoidal shape. The number of hole surfaces that cannot be made congruent by parallel transfers in k-space is only 3, and each pair of surfaces capable of being made congruent gives one SE line. We observed the SE lines from the three electron surfaces and two pairs of hole surh.ces. Within the lim_ its of the experimental accuracy, the dimensions which we succeeded in measuring, and the slopes of ttre sur_ faces relative to the crystallographic axes, fully coin_ cide with those indicated in ts I The deviations of the projections of the Fermi surfaces on the plane of the sample from ellipticity, while noticeable (especiallv for the hole surfaces), were at the borderlin"'of th" ,""o_ racy of our experiments. To measure these deviations, we propose to carry out in the friture a special series of experiments on sampl.eswith different orientations. For the time bei.ng, in discussing all the e: otx/dH2

FIG. l. Sample plots of SE lines for antimonv The left line in the lower plots pertainsto the hole ellipsoid, for which 0 = 30', and the right one to the electronic ellipsoid, which lies in the plane of the sample (f = 37"). The arrows show the value chosen for the line amplitude in this plot.

H, Oe

rzt5


t216

V. F. GANTMAKHER and V. T. DOLGOPOLOV Since the electron can, generally speaking, execute several revolutions on the trajectory, it is more accurate to write in place of (1) the sum : A(r): C)', e-'nr' C(e"/'-1)-': Cf\exp(ndT"12p)- (3) 1l-', ='i where C = A(0) ( ?i = t) and 4 = exp (rd/21). we obtain the expression n l c +r 1): d t d r +ifl-'r^, \A(1) % From (3)

urements of the free paths, we used a quadratic model with the following values of the principal semiaxes of the ellipsoids and cyclotron masses at the minimal cross sectionstsl (k is in cm-r):
l0 7trr
io-i k2

Holes: Electrons:

0.42 0.42

0.45 0.50

lo-7h r.4 2.7

^f;inl*o 0.069 0.08,1

To measure the mean free path I with the aid of the SE it is necessary to know the path length A of the electron from one surface of ttle plate to the other. In the general case A is a rather complicated function of the angles, which may, furthermore, dependstrongly on small deviations of the Fermi surface from ellipticity. If, however, the magnetic field H and the major semiaxis of the ellipsoid lq lie in the plane of the sample, then, recognizing that for electrons and holes k, n: k, ( kr, we can use the formula Iy = nd/Z in a Large interval of angles { between H and k . This formula is valid up to those values of { for which we can neglect the deviation of the Fermi surface from cylindrical shape. For a circular cylinder whose axis makes an angle {r with the surface of the metal we have Iy= nd/Z cos 4,, regardless of the value of the angle g ({+ r/2). In this connection, all tie main measurements were performed on one of the electron ellipsoids and on one of the pairs of hole ellipsoids, for which i/ 5 5". If the electron traverses ttre path A between the two sides of the plate only once, then the line amplitude is A - exp -It /l and
ln.4 (2,; : qsn51

(4)

in which, unlike in (1), the logarithm contains the parameter C, defined by the value of lo. A reduction of the experimental data by means of formula (4), even with C corresponding to lo = 0.3 mmr did not lead to a change of the erponent n. The coefficient B increased at this value of C by L\Vo. Larger values of lo have low probability, since no SE was observed on samples 0.5 mm thick, made of the same material, down to the Iowest temperatures. DISCUSSION Since the experimentally obtained degree T2 is characteristic of the contribution of electron-electron scattering, we first make some estimates of the protrabitity of the electron-electron collision. To this end we use the formula for the differential cross seetion for the scattering of an electron from the state k to the state k' by a screened potential in the form U(r) =(e2/rcr) exp -gr (r is the static dielectric constantand g-1 the Debye radiusl see t4l Ch. 6):
t 2mc' o": \+) t2

- TA: c o n s t - nd( r; -| u ) r

7n r

(1)

The measurementresults processed in accordancewith this formuLa gave for the function la =(l-t -lft;-t = BT-n the values (see Fig. 2) t,el:0.221r" Icm] z,h:0.6a/r'Icm] (2)

_r (lk_ k,l, s")-".

(here and throughout T is in degrees K and lo is the mean free path at T = 0). For the hole ellipsoid, the value of F was measured at 12 points in the angle interval from 20 to 70". The scatter of the values of 6 was limited to +7V0. For the electron eltipsoid, the angle interval was 35o s 4 = 80" and the scatter xL2Vo.It was impossible to perform measurements at smaller angles {, owing to the overLap of the lines from different ellipsoids. (Even the plots of Fig. 1 show a partial overlap, but measurements of A(T) are still possible if one uses for A the dimensions indicated in the figure by arrows.)

If it is recognized that the Fermi wave vector is kp: g both for an ordinary metal (kp = g= l/a, where a is the interatomic distance) and for a semimetal (in bismuth kp x gx L00/a, and in antimony kp r g* 10/a), then ttte total scattering cross section at k = kp is expressed in terms of the Fermi energy ep:
I I me'\' o : 16n ) k,t+*,'+ s')l-'- \; \fi e'\'

)

.

(5)

Multiplying by the number of soatterers, taking into account the limitations imposed by the Pauli principle on tle possible collisions in the electron ga.s,we obtain for the mean free path lss
.-t tdN t..'=aI*l;-lE" 1re\ o t' l r-rF

t,"=N"(+) (+l GBr)', \ e'l \ de

wrtr,lV

\-'l

l ,,\

(*)l k}

co \t,-. l

(6)

3 2

o

,w, z

i

l

t

l

t

u

t

r

"

FIG. 2. Reduced amplitude of the SE line from the hole ellipsoid (in logarithmic scale)as a function of T (points +), T2 (points O), and T3 (points O). We see that the filled points fit the straight line well. Its slope gives the value of 0, see( 1).

(Ne is the total number of electrons per unit volume and kg is Boltzmann's constant), For antimony, Ne = 10mcm-3, 0Nu/|e r 1033 erg-r cm-3,tst rcr 100,t61and lss x l0fT2. We see therefore that the coefficient of T-2 exceeds by 10 times the corresponding value from our measurements of 11. Furthermore, as is known from calculations for sodium,tTJ the estimate obtainedfor lss from (6) is underestimated by several times because of numerical factors. The entire further discussion will therefore be devoted to electron-phonon interaction. For an isotropic electron dispersion law, the value of lT measured with the aid of the SE and determined


ELECTRON AND HOLE MEAN FREE PATHS IN ANTIMONY by the scattering of the electrons by the phonons should be proportional to (7) I,aT'-u fot q'lk,46ld<'<1, l,aT-" for 6/d{Q, 1k"41 (8)

l2l7

(qt = ksT/hs, s is the speed of sound; dependingon the type of the SE, the inequalities (7) and (8) may contain other expressions in lieu of 0/d--for details see trl), I, a T-t for q,1k"71 (9)

(the high-temperature case; under these conditions, however, the SE is no Ionger observed). In (8) and (9), the measured value of 11 is the true path length traversed by the electron between two collisions with phonons(If =le.Dh). Of course, the experi'l'a a l-z can be regarded mentally obtained relation as a transition from (B) to (9), since at 4" K q1 = 0.3 x 10?cm-r in antimony (compa.rewith the values of kr listed in ttre table above). However, as indicated intsl , in the case of a sharply anisotropic electron dispersion Lawthe relation le, ph - T-2 should be obtained in a definite rather broad temperature interval. lndeed, we shall use a model in which the probability of scattering of an electron with a wave vector K is given by the expression
v T: L' r. t c[zo(t - /*+o)d(e" hqs ex+e) EfrM, J * (zo* t) (l - l"-,)6(e * - hqs - e"-)ld'q. (10)

where q, is that component of the vector q which is perpendicular to the cylinder axis, and E is the angle between K and qt. Equating the expression in the parentheses in (11) to zero, we obtain precisely the equation of a cylinder. Expression (10) in terms of the function f6 1q depends on the energy of the scattered electron, so that this expression must be averaged over the initial values of the electron energ'y along the normal to the Fermi surface- It is necessary to bear in mind here that the SE lines are formed only by the nonequilibrium electrons determined by the increment Af to the equilibrium distribution function f(01 The value of Af is proportional, as is well known, to the derivative afo'/ae; the proportionality coefficient, which can be regarded as a constant near the Fermi surface during the course of integration along the normal to this surface, drops out upon normalization, so that it is necessary to make in (10) the substitution

r-1.*,*-j ffo-li?,)a..
The expression in the square brackets in (10) tlen goes over into Ioo( - luo) I @"+ l) (l - l"-")I + xe'(e" l)-2, r:hqs/ksT, and (10) is transformed into u: t",,0 av L' lksT\'f \ fi, , "J7;4"y 2n,NMush y":2QlT, where Q =hKs/kg is the characteristic temperature, which replaces in this problem the Debye temperature @. If the cylinder is sufficiently thin so that r;e'dt i tY $2) )'llr'- ak-

Here A is tJredeformation potential, M the mass of the lattice ions, N the number of atoms per unit volume' v the electron velocity, and nn and f6 +q are the occupation numbers for the phonons and electrons; the two terms in the square brackets take into account.redpectively the absorption and production of the phonon, the integration is carried out over all wave vectors q, and the 0 function ensures satisfaction of the conservation Iaws (see t4l, Ch. 5). The deformation potential, generally speaking, is a tensor quantity: ai5(k). In our experiments, however, we measured the total scattering of the electrons by phonons of all polarizations, and the effect of the action of different phononswas automatically averaged. It is therefore natural to reprd A as a scalar quantity and to take it outside the integral sign in (10). The averaging of A over k occurs along the orbit of the electron i.n k-space, and therefore, naturally' ael+ Ah. Let the Fermi surface be an infinite ci-rcular cylinder of radius K and, as always at low temperatures, Iet the conditions e (K) ) hqs and Vp ) s hold. Owing to the latter inequalities, the distance Ak between two equipotential surfaces and +hqs in k-space is Ak =qs/v (( q, and it can be assumed that the electron remains after scattering on the same surface. The regicn of integration in q-space in (10) reduces to a cylinder of the same radius K, passing through the origin, with an axis parallel to the Fermi-cylinder axis. lndeed, the argument of the 0 functions is ,. e*tfiqs- 6"*e: t hts-fi(af i2Kqlcosq) (11)
h'Kq, ' -\zx t q' \ * "o"q ),

r>Q,
then we obtain from (12)

(13)

#fr#"(;f(14 *:,"#(# )(#)':

(q6 = On"ll is ttre Debye wave r/ector). It is precisely this case which is reatrized in bismuth for electrons.tsl In the opposite limiting case T ( Q' we obtain the same formula as for the Fermi sphere:

*:snt',#(+)':ry#(+)
((3):1.20.

(15)

For the intermediate region of temperatures' the integral in (12) was evaluated with a computer (see Fig. 3). Antimony falls precisely in the intermediate region. In order for the experimental points to fall in the region I q T-z on the curve of Fig. 3 (the right-hand side of the curve, correspondingto T/Q > 1), it would be necessary to assume that the average sotrndvelocity is s 5 105cm,/sec. This contradicts the experimental data.ts r It is more correct to assume the average value to be s rlv2 x 105 cmfsec. In this connection, we can note the following. In the region T nvQ, a change takes place in the char-


r2t8

V. F. GANTMAKHER a n d V . T . D O L G O P O L O V effect, cyclotron resonance, or the size effect it is very difficult to observe the sections that pass through the major semiaxes of ttte Fermi ellipsoids. Thus, it is quite probable that at helium temperatures lo ( lerph for some carriers and lo ) ls,ph for others. Under these conditions, strictly speaking, the electric resistivity cannot be subdivided into residual and ideal components, since the Matthiessen rule does not hold. It is therefore very difficult to compare our coefficients in (2) with the results of measurements of the electric resistance, and no great significance should be attached to the rather good agreement between our coefficients in (2) and t}te eoefficient obtained from the uJ 1ro* the quadratic term prf in the redata of tr01 sistance in accordance with the formula Ip =hK/Nee,pT. The quantities ls, ph measured by us make it possible to calculate the constant A. Putting in (1a) [l[tl =,6.7g/cm3, s = 2 x 105cm,/sec, vel = 5.8 x10?cm/sec, vn = 6.? x 10?cm/sec,and K =4.4 x106 cm-r, we obtain : ev, lAhl 1.g ev. lA"rl: 2.9 (16)

tt

2 3qV,"Ktt

z 3t\t."t\

,,fr, FIG. 3. The double integral I in expression(12) as a function of the parameter TiQ. The vertical dashed lines show the measurement regions for antimony and bismuth, respectively. The scale corresponds for antimcny to the sound velocity s = 2 X | 05 cm/sec, and for bismuth to s = 1.5 X 105 cm/sec.

0.05 0.t

0.t

,

t

acter of the scattering: a major role is assumed by colIisions tlnt change ttle direction of the electron momentum through angles y -1. For such collisions, the matrix element of the transition in (10) should contain one more factor-a certain function of the scattering angle G(y) (see r+:, Ch. 5). This function does not changJthe power-law relations in the asymptotic formulas (14) and (15). For T ((Q, the function G(Z =0) = 1 by definition, and for T >> Q the scattering is isotropic, so that the contribution of G reduces to the appearance of some additional numerical factor. But in the region T n; Q the function G(y) is quite capable of extending the quadratic dependenceinto the region of low temperatures. Inasmuch as under the condition T i,Q each act of scattering by a phonon leads to a change of the electron mornentum by a large angle, formula (14) can be used directly to calculate the qr.rantity/o which enters in the electric resistivity of semimetals'in the corresponding temperature range. Indeed, in tro, ul a quadratic dependenceof the temperature-dependent part of tl-r.e electric resistivity of antimony on T was observecl in the region from 1.5'K to hydrogen temperatures. Let us note one peculiarity of electron-phonon scattering. As seen from formulas (12), (14), and (lb), l^ nr - v2. Since the velocity^along the principal directiorid'" of the ellipsoid is v2 = ri - ki', the mean free patl ls,ph for electrons located near the major semiaxes turns out to be much smaller than near the minor ones. For eleetrons in bismuth, for enample, where In antimony this ratio is approximately 30 for electrons and 10 for holes. (These, of course, are only estimates, since no account was taken of either the anisotropy of the phonon spectrum or of the dependenceof the integration region in (12) on k.) The values of /s.p5 measured by us for antimony, and also in 16l for ele'c^trons bisin muth, pertain to carriers with minimum values of the momentum within the limits of the given ellipsoid, i.e., - max close to / e, ph. Little is known concerning the anisotropy From numerical estimates for the averaged it follows that near the points lkl = kmax at peratures tfie scattering by phonons prevails impurity scattering, and the resultant mean of lo. value of lo helium temover the free pa.th

nv kmax/kmin 14,theratio I ?,X\trflif1, reacnes 200.

These values can be compared, first, with nleasurements of the deformation potential as determined from the amplitude of the quantum oscillations of sound absorption:t12J by averaging the squares of the diagonal elements of the tensor Aij, we obtain for electrons A^r4.1 eV,while all that is known for holes is thatthe components of the tensor are smaller than those for electrons. Second,we note that the defqrmati.onpotential obtainedunder the assumption ,oeli = lahl = a from measurements of the thermal conductivity of antimony amountsto 1.8 "Y.tB l Violation of condition (13) makes it necessary to consider the experimental values (16) as approximate. However, in view of the fact that for electrons and holes in antimony the transverse dimensions of the ellipsoids are practically the same, and the difference between the dimensions of the long semiaxes in theinvestigated temperature interval is negligible, since kgT/hs ( k' the "geometric factor" connectedwith the integration is the same for electrons and holes..Therefore the inaccuracy of formula (14) and the uncertainty in the choice of s should not aJfectthe ratio ltell/ 1ah1= f.S. This ratio permits another'comparison with an independentexperiment. According to ttnl, hydrostatic compression increases the number of carriers: ANe/Ne xlVo per kbar. Assuming for estimating purposes
^ o dN" 3 6h_6el

N":T

E

'

I = /f{ir << lo. This can probably exptain why in such experifrrents on semi metals as the de Haas-van Alphen

where E is the band overlap and B is the bulk modulus of elasticity (B nv.400 kbar), we obtain either AeI = -0.3 eV and An = 0.2 eV (the extrema shift in oppo.site directions at decreasing specific volume), or Ael = -1.6 eV and Ah = -1.0s eV (both extrema shift downward). The second assumption agrees much better with (16). A similar analysis can also be carried out with respect to the results of t8 I for bismuth. Although the lss calculated from (6) differs much less from the mean l free path 11 measured in 18 for electrons, there is every reason for assuming that in bismuth the tempera-


ELECTRON AND HOLE MEAN FREE PATHS IN ANTIMONY ture-dependent scattering is determined by electronphonon collisions. This is confirmed, in particular, by the character of the dependenceof the coefficient of the temperature-dependent part of the electric resistivity t rsl on Ut" carrier density in bismuth-antimony alloys . The data of tsl reduced in accordance with formuta (14) lead to aRl = Z.OeV' whereas the diagorpl elements of the tensor"'ai5 for bismuthrrel yield a[!. = 3.8 eV. The authors are grateful to E. M. Rodina and R. R. Ponomareva for help with the computer calculations and to S. I. Za\tsev for taking part in the numerical reduction of the experimental results.

LzL9

1V. F. Gantmakher,Progr. in Lour Temp. Phys. 5, Ed. C. J. Gorter, North-Holl. Pulcl. Co., Amsterdam, p. 196?, 181. 2V. S. Tsoi and V. F. Gantmakher' Zh. Eksp. Teor. Fiz. 56, 1232 (1969)[Sov. Phys.JETP 29, 663 (1969)]. 3 L. B. Windmiller, Phys. Rev. t49,472 (1966). 4J. M. Ziman, Electrons and Phonons, Clarendon Press, 1960(Russ. Transl. IIL' 1962), 5A. P. Cracknell, Adv. in Phys. 18, 681 (1969).

6M. Cardona and D. L. Greenaway,Phys. Rev. 133A, 1685(1964). ?E. Abrag'ams, Phys. Rev. 101, 544 (1956). sV. F. Gantmakherand Yu. S. Leonov, ZhETF Pis. Red. 8, 264 (1968)[.lnte Lett. 8, 162 (1968)]. eH. B. Huntington,Solid State Phys' ?, Acad. Press, New York, 1958,P. 214. bJ. R. Long, C. G. Grenier, and J. M. Reynolds, Phys. Rev. 140, A187 (1965). 1rB. N. Aleksandrov, V. V. Dukin, L. A. Maslova, and S. V. Tsivinskii, Paper presented at Sixteenth Conf. on Low-temperature Physics, Leningrad, 1970. 12Sh.Mase, Paper at Soviet- JapaneseConf. on Lovvtemperature Physics, Novosibirskr 1970. 13R.S. Blewer, N. H. Zebouni, and C. G. Grenier, Phys. Rev. 1?4, 700 (1968). 14N.B. Brandt and N. Ya. Minina, ZhETF Pis. Red. ?, 264 (1968)[JETP Lett. 7, 205 (1968)]. bE. W. Fenton, J.-P. Jan, A. Karlsson, and R. Singer, Phys. Rev. 184, 663 (1969). roK. Walther, Phys. Rev. 174, 782 (1968). Translated by J. G. Adashko 244