Low-temperature magnetothermoelectric power of aluminum

Solid State Communications,

Vol. 11, pp. 1109—1113, 1972. Pergarnon Press.

Printed in Great Britain

LOW-TEMPERATURE MAGNETOTHERMOELECTRIC POWER OF ALUMINUM* R.S. Averbackt Department of Physics, Michigan State University, East Lansing, Michigan 48823 and D.K. Wagner Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14850 (Received 6 July 1972 by N.B. 1-lannay)

Measurements of the low-temperature magnetothermoelectric power of aluminum and a number of aluminum alloys show that the electrondiffusion component Sd increases and then saturates with increasing magnetic field. The difference ~Sd between the high- and zero-field values of Sd is observed to be independent of the type of impurity in the aluminum and has the value (2.2 ±0.2)T x 108 V/K. These observations are interpreted in terms of a realistic semiclassical analysis of the thermopower. This treatment gives an excellent account of the qualitative behavior of the thermopower in a magnetic field but yields a value of L~Sdwhich is 30% lower than the observed value. Possible reasons for this descrepancy are discussed.

IN THIS letter we present measurements of the low-temperature magnetothermoelectric power MTEP of aluminum and dilute aluminum alloys, We observe that (1) the electron-diffusion component Sd of the MTEP saturates in a strong magnetic field; and (ii) the high- and zero-field values of the electron-diffusion component Sd depend rather sensitively on the impurities in the aluminum, but their difference does not. By

extending the semiclassical treatment of transport processes in metals in a magnetic field developed by Lifshitz, Azbel, and Kaganov,’ we show that this behavior is characteristic of uncompensated metals having closed electronic orbits.

The specimens were prepared from aluminum which had a measured residual resistance ratio *Work supported by the U.S. Atomic Energy Commission under Contracts AT(11-1)-1247 and AT(11-1)-3 150, Technical Report No. COO3 150-3. Additional support was received from the Advanced Research Projects Agency through the Materials Science Center at Cornell University, Report No. 1752.

RRR [defined as R(300K)/R(42K)} of over 8000 and solute materials which were of 69 grade. The alloys were all chill cast, cold rolled into strips 10mm thick, and air annealed. A control specimen which was prepared ident-

tPresently at the Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14850.

ically to the alloys but to which no impurities were intentionally added, had a RRR of 4600. Details of the experimental procedure are des2 cribed in a previous publication.

.

1109

.

.

1110

Vol. 11, No. 9

MAGNETOTHERMOELECTRIC POWER OF ALUMINUM 24~-

~

“a

-I

3— 0

2

4

6

~

.0

2

4

6

IS

20 22 24 26

4~////

20 0

I wcr

FIG. 1. Temperature dependence of the thermopower of specimen ~_~-3for several values of the magnetic field. A straight line indicates a thermopower of the form S .4(H)T B(H)T The magnetic fields are denoted by: (a) 0, (b) 0.5, (c) 1.5, (d) 3.0, (e) 5.0, and (f) 10.0kG.

FIG. 2. Variation of the electron-diffusion thermopower coefficient .4(H) with ~ for

The measured thermopowers were all found to fit an equation of the form,

Table 1. The measured residual resistance ratio RRR and ~A — A(H ,~) — A(H 0) for each of

3. (1) S - A(H)T + B(H)T The separation of the thermopower S into terms linear and cubic in temperature is illustrated in Fig. 1 for the pure aluminum specimen (Al-3). The therrnopowers of the other specimens behaved in a similar fashion. The linear term A(H)T is generally identified as the electron-diffusion contribution Sd and the other term B(H)T3 as the phonon-drag contribution. We are concerned here only with S 5 the phonon drag contribution will be discussed in greater detail in a later

the specimens

publication. In Fig. 2 the dependence of the coefficient A(H) on magnetic field is shown for each specimen, and in Table 1 the quantity 14 — i-lW -. -~)— .4W 0) and the residual resistance ratio are listed for each specimen. From these data it is observed that: (i) .4(11) saturates in the high-field limit; and (ii) 14 is approximately the same for all specimens and has the value (2.2 ±0.2) x 10 ~ V/K’. ~ This behavior can be understood in terms of a semiclassical analysis of the thermopower. In the presence of a static magnetic field ~i, an

various aluminum alloys. Here ~o.r H/p nec, where p is the residual resistivity in zero magnetic field and a is the electron number density computed on the basis of 3 electrons atom. The alloys are (a) Al: Cu-i, (b) Al : Cu-2, (c) Al-3, (d) Al:Tl-1, (e) Al:Sn, and (f)Al:Cd-i.

Specimen Al: Cu-i Al: Cu-2 AU3

RRR 127 650 4600

14 x 103(V/K2) 2.10 2.11

Al:Tl-1 2600 2.49 Al: Sn 1200 2.26 Al Cd-i 2200 2.11 ____________________________________________ *High~fieldlimit not obtained; see Fig.2.

applied electric field ~, and a uniform temperature gradient ~T, the electric and thermal currents f and ~ are given by .~,

~.÷

J

L~W)

~,

..

E

+

LE~~). ..

(2)

U LTEW). ~ L~~) VT. The electrical conductivity tensor ci is LEE, while the thermopower tensor S is —L~ LET, anfl,, the thermal conductivity tensor .< is 4—~. . ~ (LTT LrEL~LEy). With the magnetic field

Vol. 11, No. 9

MAGNETOTHERMOELECTRIC POWER OF ALUMINUM

1111 -.

~ along the z axis, the thermal current U along

relation LET(H)

conditions, the thermopower will be given by the x axis, and the usual adiabatic boundary

-s

s

—

—

—

LET(fl)

(3)

,

—

.~

1<210/

provided that all xz, zx, yz, and zy tensor elements can be neglected. This is likely to be a very good approximation for a polycrystalline specimen. In the following calculation we will assume that the essential behavior of polycrystalline aluminum is similar to that of a single crystal, oriented with the magnetic field parallel to a twofold or higher symmetry axis; in this situation the xz, zx, yz, and zy tensor elements all vanish by symmetry. Further, we shall neglect phonon-drag effects altogether. Thus, the following calculation applies only to the electrondiffusion component.

—

—LTE(—H)/T, one can write

—eL0T [d

(_Th]

(8)

,

s”.

where ‘~ is the transpose of the tensor It should be emphasized that equation (8) does not depend upon the existence of a relaxation time and holds for arbitrary scattering mechanisms. Using the preceding results, the zero-field thermopower can be written S~(H— 0)

=

eLoT[~ ln ci~~W 0)] -

.

(9)

To calculate the thermopower in a strong field we make use of the fact that a~()

[ne()

n~(E)]ec H

—

(iOa)

and The tensor coefficients in equation (2) can be calculated by a method similar to that used by Azbel, Kaganov, and Lifshitz in their original semiclassical treatment6ofFollowing thermoelectric phenomtheir treatena in a magnetic field. ment the perturbed electron distribution function ía

Idax21()J

[

d

j

~ ~ (~) I

y \ Iec\

(lOb)

where n3(E) and nh(E) are the number of electrons and holes, e(k) respectively, within theof surfaces — , and ycontained is the coefficient the electronic specific heat.7 Using equations

for the ath band can be written (3) and (10) and the fact that —

f~+

(_ ~f2\

.

l/JE.a

—

kB

VT .

cl/Ta],

=

j

I

de (-. ~[2 )~(e)

—

~~l7~)

—

with

I

Jde

(_ ~!2~

e

0

~

ac)

(E

~

j

)

-

(11)

L0 —~ (L~1

eL0T

ln

a~(H

There are several features of equations (9)

(6)

cr,~(H Consequently, both Sa~(H 0) and Sd(H oo) can be expected to be sensitive to the

—

eLoT2f.~..’~’(E)l + (k~’\3 [dE ~ 2 rdSa’’ cr(E) — ~ ‘~~~Vacl’Ea e

—

~ ii

and (ii) that should be emphasized. First, both equations are valid for a general scattering mechanism and depend on the detailed nature of this mechanism through the quantities L212, and the energy derivatives of a~~(H= 0) and

and =

L

—

(5)

0

LTE

L

L2121T a~ the high-field thermopower becomes S~H ~ yT I IL0 (n3 nh)e ~

“.

a)

LEE

“-

0T a,~and

[eE

(4) where 10 is the equilibrium distribution function (1 + eu)_l with u = (— p.)/kBT (e is the electron energy and p- is the chemical potential). By using equation (4) to calculate the electrical and thermal currents I and ~J,one obtains 03

~

-‘

~3).

-.

(7)

and L 2/K2. The sum in equation 0 — 2.44 x i0~ V (7) extends over the bands and the integration is over the portion of the constant energy surface e(~)= in the band. By using the Onsager

results Second,asequation (9) holds for nature ofindicate. thewhereas scattering the(ii) experimental all metals, equation holds only for

uncompensated closed-orbit metals. Finally, inspection of equation (11) shows that Sd(H -~ does not depend on the strength of the magnetic field; this corresponds to saturation of the MTEP

1112

MAGNETOTHERMOELECTRIC POWER OF ALUMINUM

\~1 11, No. 9

in the high-field regime. Both the magnitude and sign of the saturation value depend upon details of the band structure and the nature of the scattering mechanism.

this value with the value of the firs’ term gives the value 1.6T x 10 V/K for \S~. This is about 30% lower than the observed value of (2.2 ±0.2)T < 10 V K.

The difference .\S~ Sd(H -‘ oc) — Sd(H 0) can be computed simply from equations (9) and (ii). In the temperature range of the experiment the scattering was dominated by elastic impurity scattering for which L 21,, — L0. Thus,

We offer three possible explanations for this discrepancy. Firstly, the free electron value for 7 probably underestimates the actual value of y in aluminum due to distortion of the free electron

2yT (n~— n5)e

e

L T~~ in a10/(H [d~

d in cr~.Pjl 0)]

.

~) ~l2~

di It is important to note that AS~is largely inde pendent of the scattering mechanism because -. -.-) and a~P.H 0) tend to have opposite dependences on the scattering mechanism. ~ This is especially evident when a single relaxation time ~-. exists, for in that case a~~(~H -, ~) iS proportional to 1/T(e), ci~~(H 0) is proportional to ‘r(<), and the dependence on this relaxation time cancels in the second term of equation (12). This explains the insensitivity of ASd to the impurity content of the aluminum. To compute the value of AS~for aluminum, we note that the first term in equation (12) has the value 1 .9T ~< 10 \‘/K. a Since, as we have indicated, the second term in equation (12) is not sensitive to the scattering mechanism, it is simplest to use the relaxation-time approximation to calculate its value. Using the relaxation-time 10 approxifor the mation and the Harrison construction Fermi surface of aluminum, we obtain a value of 0.3T 10 V/K for this term. ~ Combining

1.

sphere and reduction of the electron velocity in the vicinity of the Bragg planes. 12 Secondly, there still does not exist a treatment of the effect of phonon drag on the MTEP, so that the separation of the phonon-drag component from the electron-diffusion component in the MTEP is purely empirical. Therefore, although we feel that we have correctly identified S~with the actual electron-diffusion component, it is impossible to be certain. Lastly, the experiments were performed on polycrystalline specimens, while the calculations assume a single crystal with the magnetic tield along a symmetry axis. These two situations are probably equivalent if the MTEP of a single crystal is not very anisotropic. Although we expect this to be the case, future experiments on single crystal specimens would be useful in making further comparisons between theory and experiment. Acknowledgements — We are grateful to B. Schumacher for preparing the alloys used in the experiments and to NW. Ashcroft, C.H. Stephan, and J.W. Wilkins for helpful discussions concerning this work. One of the authors (R.S.A.) wishes express his encourgratitude to his thesis advisor, J. toBass, for his agement and suggestions during this study.

REFERENCES LIFSHITZ I.M., AZBEL M.I. and KAGANOV MI., Zh. Eksp. Teor. Fiz. 31, 63 (1956) [Soviet Phys. JETP 4, 41(1957)1.

J.,

2.

AVERBACK R.S. and BASS

3.

AVERBACK R.S., Ph.D. Thesis, Michigan State University (1971).

Phys. Rev. Lett. 26, 882 (1971).

4.

Another interesting feature of the data is the sign reversal of .4(11). However, it can be seen from Fig. 2 that this is not a property of all of the alloy systems.

5.

ZIMAN J.M., Electrons and Phonons Section 12.4. Oxford University Press, Oxford (1960~.

6.

AZBEL MI., KAGANOV MI. and LIFSHITZ I.M., Zh. Eksp. Teor. Fiz. 32, 1188 (1957) [Soviet Phys. JETP 5, 967 (1957)].

Vol. 11, No.9

MAGNETOTHERMOELECTRIC POWER OF ALUMINUM

1113

7.

This value of 7 pertains only to the band structure and is not the same as that which would be measured in a specific heat experiment. The latter is invariably larger due to many-body effects arising from the interaction of the electron and phonon systems. See PRANGE R.E. and KADANOFF L.P., Phys. Rev. 134, A566 (1964).

8.

10.

For a discussion of this point, see WAGNER D.K., Phys. Rev. B5, 336 (1972). 2. For aluminum the quantity The value of y used was the free-electron value of 0.91 mJ/mole-K (n~— nh) is — 1 electron/atom. HARRISON W.A., Phys. Rev. 118, 1190 (1960).

11.

We estimate this value to be accurate to better than 20%.

12.

An upper bound on the value of LSSd can be estimated by using the value of y obtained in a specific heat experiment (see reference 7). Using the value of y of 1.35 mJ/mole-K~, measured by PHILLIPS N.E., Phys. Rev. 114, 676 (1959), we obtain ASd ~ 2.5T x iO~V/K.

9.

Tieftemperaturmess ungen der magnetothermischen E . M .K. an Aluminum und einigen Aluminumlegierungen ergaben, daB der elektronendiffusive Beitrag Sd mit der magnetischen Feldstärke zunimmt und schliesslich eirien Sdttigungswert erreicht. .~Sd,die Differenz zwischen Null und Sättigung, hat den Wert (2.2 ±O.2)T x 1O~V/K und ist unabhängig vom Reinheitsgrad des Aluminums. Dieses Ergebnis wird mit einem semiklass ischen Modell des thermoelektrischen Effektes diskutiert. Die qualititative Ubereinstimmung mit dem beobachteten Verlauf der thermoelektrischen E.M.K. ist ausgezeichnet, aber der Wert von liegt 30% unter dem beobachteten. Mdgliche Ursachen für diese Diskerpanz werden diskutiert.