THE INFLUENCE OF ELASTIC SHEAR STRAINS ON THE CONDUCTIVITY AND THERMO-ELECTRIC FORCE OF CUBIC METALS BY J. SMIT

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• Collection: General CIA Records
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• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 11
• Document Creation Date: November 9, 2016
• Document Release Date: February 22, 1999
• Sequence Number: 9
• Publication Date: August 1, 1952
• Content Type: REPORT

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SEPARAAT
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LABORATORIA
N.Y. PHILIPS' GLOEILAMPENFABRIEKEN
EINDHOVEN (HOLLAND)
THE INFLUENCE OF ELASTIC SHEAR STRAINS ON
THE CONDUCTIVITY AND THERMO-ELECTRIC
FORCE OF CUBIC METALS
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Physica XVIII, no 8-9 Augustus-September 1952
THE INFLUENCE OF ELASTIC SHEAR STRAINS ON
THE CONDUCTIVITY AND THERMO-ELECTRIC
FORCE OF CUBIC METALS
by J. SMIT
Philips Research Laboratories N.V. Philips' Glocilampenfabrieken Eindhoven-Nederland
Synopsis
Elastic shear strains cause a change in the shape of the Fermi surface
of metals. The influence that such a change has upon the electrical conducti-
vity and the thermo e.m.f. has been calculated for monovalent f.c.c. metals,
and the results of it have been compared with the experimental values for
Cu, Ag and An. The conclusion can be drawn that the Fermi surface of
gold touches the Brillouin zone boundary, whereas that of copper does
not, whilst silver is an intermediate case. The theoretical results account
also qualitatively for the experimental values of the trivalent Al.
1. Introduction. Even in cubic monovalent metals the Fermi
surface in wavevector space for the conduction electrons is not
spherical. For sodium it is generally assumed that the deviations
from the spherical shape are negligible, but for the noble metals this
is probably not the case. The electrical conductivity is isotropic for
cubic crystals, but if it could be made anisotropic by artificial means,
it might be possible to obtain information about the anistropy of the
Fermi surface. This can indeed be done by applying elastic shear
strain.
The linear dependence of the resistance on the strain can be
described by a fourth-order tensor, since it relates two second order
tensors. In the case where the coordinate axes coincide with the
cubic axes of the crystal one obtains :
4exxIQO = -Op40xx = al1r + al2(#vy + ezz)
Jo,/C)o = - eodaxa = 2a44Exy \ !
and similar expressions for the other directions. Here eo is the
resistance in the undeformed state, and cc the conductivity. The
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constants ay are analogous to the elastic constants, and therefore
the same abbreviations are used. The three combinations of interest
are all + 2a12, a,, - a12 and 2a44. The first quantity gives the
change under hydrostatic pressure and the last two the changes
under shear strains, leaving the volume constant. For isotropic
materials the last two quantities are equal: a,I - a12 = 2a44. The
theory for all + 2a12 has been given by M o t t 1). Under hydro-
static pressure the elastic constants increase due to the anharmonic
term in the potential energy, which also gives rise to the thermal
expansion. Thus the amplitude of the lattice vibrations is decreased,
giving a smaller resistance and al, + 2a12 > 0. The theory accounts
satisfactorily for the experimental values. Here we shall be con-
cerned only with al, - a12 and 2a44.
Another property strongly depending on the conduction electrons
near the Fermi surface is the absolute thermo-electric power,
being also a second-order tensor and thus isotropic for cubic crystals.
The changes under the application of strain are given by equations
analogous to (1): Aesr = llE. + fl
f'12(e + Ell) ('7\
Jec.. = 2#44ezv,
/1;, being the components of a fourth-order tensor, giving the change
in the thereto e.m.f. e;,. In this case the fl;, are not dimensionless
quantities, as in (1). We did not divide by the values for zero strain,
as this quantity may be either positive or negative, as well as zero.
The clement /344 gives rise to effects as observed by P c r r i c r 2)
(transverse currents in a twisted tube in which a longitudinal
temperature gradient exists), though the mechanism is quite differ-
ent for ferromagnetic metals. It will be shown below that the /3's
are closely related to the as, and that both can give us qualitative
information as to the shape of the Fermi surface.
2. Derivation of basic formulae. The electrical current density is
given by
2e/h(2,z)3 f f J=~ CE(k),Tk; /(k) dksdkl,dks, (3)
where -e is the electronic charge, /(k) is the probability that an
electron is in the state characterized by the wave vector k, and E(k)
the energy of that state. In the presence of an electrical fieldstrength
F and a temperature gradient, for /(k) we can take
/(k) = /0(E) + dfoj`dE, (cF-f- (E -')IT. VT + VC). X(k), (4)
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INFLUENCE OF SHEAR ON CONDUCTIVITY OF CUBIC METALS 589
with A(k) satisfying the integral equation
rk E(k) = f f 7i(k) - A(k')} P(kk') (aE/akn)-' dS' (5)
Fermi
f0(E)is the Fermi-Dirac function f o(E) . (exp. (E - ')/kT -+- 1)-',
and C the thermodynamic potential of the electrons. B(kk') is,
apart from a numerical factor, the transition' probability. In (5) it
is assumed that the energy is conserved during a transition (T > OD),
the integration' lias to be taken over the Fermi surface in k' space.
The introduction of the vector field A(k) is in general not equivalent
to that of a relaxation time r(k). The use of this last quantity means
that A(k) = z(k) (1 /h) vk E(k), but for the general case of non-
spherical Fermi surface and a transition probability which does not
depend solely upon the angle between k and. k', X(k) will not always
be parallel to Vk E(k), and the use of z(k) is not correct.
The conductivity is given by
aij = 2e2/h(2n)3 . f cos (n, i) ~1(k) dS (6)
Fermi
where for dfo/dE has been taken -o(E - ~), this being valid for not
Sij(E) = 2e2/h2(27r)3. f cos (n, i) 21(k) dS (8)
where the integration in (8) has to be taken over the surface with
constant energy E. Comparing (8) with (6) it follows that aij = Sij(4)
Making use of the wellknown series expansion for integrals containing
dfo/dE, one finds for the absolute thermo-electric force tensor
ee. ? - 7c2k2T/3e.Pjk(dSk;(E)/dE)E=I (9)
In order to determine the change of (9) under elastic strain we
make use of the generalized equations (1) :
L9ij = e,(6ij+ aijk5ekl) Sj(E) = S0(E) Oil' - aijkl(E) -'hl)
and by means of (9) find
aeij/aEkl = Nijkl = 3L21i,2T/3e. (daijkl(E)IdE)E_I (10)
3. The change of the Fermi surface under shear strain. One of
the most characteristic features of the influence of elastic shear strain
too high temperatures. After multiplying (3) by the resistance tensor
?ij we find
1 aT ?? dfo 1 a~
Fj= iQij+ - Sik(E)dE - (7)
T k f(E dE e axj
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is the change in the shape of the non-spherical parts of the Fermi
surface. This is most easily seen in the one-dimensional case (Fig. 1).
A strain eki of the specimen corresponds to a strain -eki of the k
space. We shall use the nearly-free-electron approximation. The
energy gap is 2V,, VA being the Fourier component of the periodic
potential corresponding to the repetency vector g of the zone
boundary. Moreover we assume that electrons near one particular
energy gap are only affected by that gap. Strains which leave
the volume constant do not alter . In fig, 1 there is an extension
a -.k ko
Fig. 1. One-dimensional energy curve for nearly free electrons. The change
in it due to strain is dashed.
(E = e .> 0) in the drawn direction, and compression in directions
perpendicular to it (e,,, = e? e/2). The change in V. is immate-
rial, since Vg is small compared with Eo in this approximation.
Moreover it is constant for the "deformierbare" potential of
13 1 o c h a.o., as used in the theory of the interactions between the
lattice vibrations and the conduction electrons. B a r d e e n 3) has
shown that this is a good approximation for monovalent metals.
But also for the rigid ions of N o r d h c i in the change in the ener-
gy gap will be small as compared with that of E0. Consequently we
shall ignore it. The point A in the first Brillouin zone goes over
into A', much nearer to the boundary and having a smaller slope
dEdk. W 'e can say that the electron is less free than in A, as a conse-
quence of the increased lattice parameter in that direction. This fact
increases the resistance in this direction. Since it corresponds to
ail - a1z or 2am, these quantities should he positive. In the case of
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INFLUENCE OF SHEAR ON CONDUCTIVITY OF CUBIC METALS 591
sxx = & , = e,, = e > 0 the same picture of fig. 1 applies, but then C
is no longer constant, and A' will be lower than A. These effects
are therefore more important for purely shape changes than for
volume changes. It is easily seen that much the same applies for
holes within the first Brillouin zone. The contrary is true in a
point B of the second Brillouin zone, where B' is nearer to the
free electron parabola than B, leading to negative all - a12 or
2a44. It may be stressed that for free electrons these effects vanish,
and that the largest changes maybe expected for points very near a
Brillouin zone boundary, which for f.c.c. monovalent metals lie in
the (111) directions. Extensions therein and equal compression
perpendicular to it, leaving the volume constant, gives in the (111)
direction a relative change in resistance of 2a44s. From these
very simple considerations one should expect for the monovalent
f.c.c. metals 2a44 > all - a12 > 0, and for those of higher valency
with electrons in the second zone 2a44 < a11 - a12 < 0. These quali-
tative results are now in striking agreement with the experimental
values published by D r u y v e s t e i n 4) for the noble metals Cu,
Ag and Au and for the trivalent Al. Moreover it is seen that a will be
larger, the higher the F e r m i surface as long as it does not touch
the zone boundary. According to (10) then fl11 - #12 or 2(344 will be
positive. Following the same reasoning as before we get 2/44 > /311-
- fl 12 > 0. But the contrary applies for holes within the first Bril-
louin zone, since these holes will become more "free" for higher
energy. It will be shown that this is also in accordance with the
experiments, and therefore it seems worth-while to investigate this
aspect of the problem more quantitatively.
From (6) and (5) it follows that
Aaii = 2e 2/h2. (27r)3 A f cos (n, i) gi(k) dS
(1 1)
ask
4 a?i~ = A ~,(k) . f P(kk') (-)'ds'+
~/ n
a -1
+ 2(k) ? ,l f P(kk') ()s' - A Vf 7i(k') P(kk') (akE
) ds (12)
For shape changes under constant volume the first two terms of
(12) are the most interesting. For an isotropic transition probability
the last two terms should vanish. They contain the influence of the
change in the lattice vibrations and depend also on the change in
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the properties of quantities at the Fermi surfaces, but to a lesser
degree th2n the first two terms of (12), because of the integration.
Also in thV present case of non-isotropic I'(kk') we shall ignore the
last two integrals in (12). In the same approximation we shall
assume the coefficient of J2;(k), i.e. JI'(kk') (aE/Ck')-' dS' to be
isotropic. In general X(k) will not be parallel to pkE(k), but in this
approximation the changes in it are parallel. Z 'c now get for the
relative change in the conductivity
l Qtr h (fEE 1E dSj 6 ck; rkF
from which also the change in thermo c.m.f. cat, be derived. For zero
strain. we shall take the free electron value. Calculations are then
made for the f.c.c. monovalent metals by dividing the k space into
pyramids formed by the intersections of the discontinuity planes
and with their apex in the origin as indicated in fig. 2, and after-
Fig. 2. T1 c partition of the Brillouin zone into pyramids.
wards replacing these pyramids by cones of revolution having the
same solid angle. The points within one cone arc assumed to be
affected only by their own energy gap. For the f.c.c. lattice there. are
8 (111) discontinuity planes and 6 (100) ones, having energy gaps
2V,,i and 217100 respectively. The energy in this approximation is
given by
lc2 (k - g)2 -I- (k) 2 f 1 /{ h2 (k - g)2 - (k)212
Z k _ Y --- -- - + Vg 2
( 2m 2 2m 2
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INFLUENCE OF SHEAR ON CONDUCTIVITY OF CUBIC METALS 593
and we find for f 3.E/aki.dSi in the direction of the axis of the cone
O and perpendicular to it (2) respectively
27rm fkmax aE 2 ~h2 rhmax 2
aee - k2 kmin \ 31a ) dk and a~,~= m rzmi~ k dk .
The contributions for each cone towards the changes in conducti-
vity and thermo e.m.f. due to a shape change could now be calculated
and then summed up. The results are plotted in fig. 3 for all - a12
0 0,1 0,2 0,3
/a22
8m0gao
Fig. 3. The calculated curve for all-e12 (lower curve) as 2e44 (upper curve)
as a function of 8 mV10olk2gio0 and 8 mV111/h2g 11 respectively. The
experimental points are indicated.
and 2a44 and in fig. 4 for (31 '- N 1 (T = 300?K). For the change in
the thermo-electric force only experimental values for polycrystal-
line samples were known, and therefore we have to average. The
result is 011 - fl _L = 0,4 ( 311 - P12) -H 1,2 fl 44. The calculated posi-
tive values are practically only due to j344 since N11 - (312 is very
small or negative. In plotting the figures it has been assumed for
100 but this is immaterial.
simplicity that V111/gi11 V100/92
4. Discussion and conclusions. In the table the experimental
values of D r u y v e s t c y n 4) are given for the resistance effect
and those reviewed by B or e l i u s 5) for the thermo-electric force
an
an
2a44
au aiz
a,1+2alz (311.10?17J?K f 1.10?VJ?K
1) ?
0110?V ?K
0, in agree
ment with the experiments. Since the larger part of all + 2a12 is due.
to the change in the lattice vibrations, it may be expected that the
ratio between the shape-change effect. and that of the volume-
change effect (e.g. (aii a1)/{(tll + 2a12)) is much larger .for 'the:
thermo e.m.f. than for the resistance (by a factor 5.4 and 2.2 for Cu
and Ag respectively). For gold we now find an interesting exception:
The difference (iii - /3_L - 0.6) is smaller than for copper and silver
by a factor 10. According to fig. 4 this indicates that gold touches
the Brillouin zone boundary with its Fermi surface in the (111)
direction. The value of 2a44 for copper being rather small, we may
conclude that copper certainly does not touch the zone boundary.
Silver with its larger 2a44, and still rather large (fin - /31) may be an
intermediate case. Hence copper has the smallest energy gaps, and
gold the largest. This corresponds to what one might expect from
their sequence in the periodic table, for large nuclear charges give
deep potential troughs. In fig. 3 and 4 we plotted the presumable
points for Cu, Ag and An, but this has only qualitative value. In
order of magnitude the computed effects agree with the effects
observed, better so for the thermo e.m.f. than for the resistance
effect, probably due to the closer approximation.
The considerations of P e i e r l s 8) on the "Umklappprocessen"
in the theory of conductivity led to the result that most of the mono-
valent metals should touch the zone boundary .with their Fermi
surface. This point. has been greatly clarified by the recent paper of
K 1 e in e n s 0), who showed that P e i e r 1's calculation does not hold
and that the problem of touching is still an open question. Our
result for gold agrees with the conclusion arrived at by K o h-
1 e r 10), who considered the magneto-resistance in high fields; this is
another way of making the resistance anisotropic. K o h l e r showed:
that this magneto-resistance should be isotropic if the Fermi surface.
does not touch the Brillouin zone boundary and anisotropic if it does
touch. Experiments") showed that for.gold it is highly anisotropic.
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596 INFLUENCE OF SHEAR ON CONDUCTIVITY OF CUBIC METALS
For copper and silver no experimental values are available.
We have seen that electrons in the second Brillouin zone have a
negative (a11-a12) or 2a44and this can perhaps account qualitatively
for the experimental values of Al. Also the negative value of (ail-fl )
bears this out. The fact that al I - a12 < 0 indicates that there is
also an overlap of electrons in the second zone in the (100) direction.
L e i g h 12) in his work on the elastic constants of Al arrives at the
same conclusion.
This work was started at theTechnische Hogeschool at Delft under
the auspices of Professor M. J. f) r u y v e s t c y n, to whom the
author wishes to express his indebtedness.
Eindhoven, 7 April 1952.
1) Cf. M o t t, N. F. and J o n c s, 11. The theory of the properties of metals and alloys.
Oxford 1936, p. 271.
2) P c r r i e r, A. et A z a d, A. A., Ilelvetica physica Acta 17 (1944) 463.
3) Bar d e c n, J., Phys Rev. 52 (1937) 688.
4) Druyvestein, M. J., Physica0 (1951) 748.
5) B o r e 1 i u s, G., Handbuch der Mfetallphysik I. Leipzig (1935) p. 410.
6) D r u y vest e i n, M. J., Philips Res. Rep. 1 (1946) 77.
,
7) M o t t, N. F. and j o n c s, I L, The theory of the properties of metals andalloys
Oxford 1936, p. 312.
8) P e i e r 1 s, R., Ann. Physik Leipzig, 12 (1932) 154.
9) K I e m e n s, P. Proc. phys. Soc. A 04 (1951) 1030.
10) K o h I c r, M., Ann. Phys. Leipzig 5 (1948) 99.
11) Justi, E. and Scheffcrs, H.,Phys.Z.37(1936)383,475.
12) Leigh, R. S., Phil. Mag. 42 (1951) 139.
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