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Quantum theory of the electrical conductivity of metals in a magnetic field
J. Phyo. Chem. Solids
Pergamon
QUANTUM
Press 1958. Vol. 4. pp. 11-18.
THEORY
CONDUCTIVITY
OF THE ELECTRICAL
OF METALS IN A MAGNETIC FIELD E. M. LIFSHITS
Physical-Technical
Institute,
Academy of Science of the Ukranian S.S.R., (Received
26 December
Kark’off, U.S.S.R.
19.56)
Abstract-The
paper gives a consistent quantum-mechanical theory of the electrical conductivity of metals in magnetic fields. A kinetic equation is deduced for the statistical operator with general assumptions on the energy spectrum of electrons, and the limit of the transition to a classical case is analysed. Asymptotes of the kinetic coefficients in strong magnetic fields are studied. Starting from the established relationship between the solutions of quantum and classical equations, quantum corrections to the kinetic coefficients, which include resistance and conduction oscillations, are determined.*
1. INTRODUCTION
quantum oscillation effects mentioned above and to find the transition to the classical description. As shown in t4) the energy levels en of a particle, which obeys the law of dispersion (l), in a magnetic field H directed along the z-axis are determined in the quasiclassical approximation by the conduction
IT is well known that the correct interpretation of electrical conduction in metals is basically quantum in character. Nevertheless, since the electron wavelength is small compared with its free path, and, in the presence of the magnetic field, also compared with the radius of curvature of its trajectory, it is usually sufficient to employ only a semiclassical treatment. This means that the electron may be regarded as a classical quasi-particle, which obeys Fermi statistics with a dispersion law E = &z,
P,, Pz)
S(~,pe) =
+(n+y) (n B
1)
(2)
(0 < 61)
where S(E, pB) = area of a section of the surface (1) by a plane p, = const. in quasimomentum space. The separation of the levels, according to (2), is:
(1)
where l r(p) = a periodic function with the period of the reciprocal lattice, j = band number. On this basis a semiclassical theory of galvanomagnetic effects in metals was developed in.(l) In such a treatment, however, certain specifically quantum effects (e.g. resistance oscillations in a magnetic field), connected with the quantization of levels, are lost. These effects were found experimentally@) but their theoretical treatmentt3) does not seem to be satisfactory or consistent. The present treatment aims to develop a consistent quantum theory of metallic conduction in magnetic fields. The theory enables us to study the
Ac, =.s~+~-E~
eH
a* = -;
m*c
=Aw*
r)2* =-.-*
i
2rr
=/.PH
as
ae'
/.L*=- eh
m*c
(3)
m*= m*(e,pZ). (4)
We can expect some intrinsic
effects at Aw* 2
kT. On the other hand to use the quasiclassical approximation and the dispersion law (1) we must have hw < 5 (6 = chemical potential). We shall take the latter condition to be fulfilled in this treatment.
* Some of the results obtained were briefly reported on in reference (5). 11
E.
12
2. KINETIC
EQUATION OF MATRIX
THE
M.
DENSITY
LIFSHITS
es,s =
If the electron is in a uniform magnetic field H and a weak electric field E, we can represent its Hamiltonian as
s=fl;
4pz)fs~oH; eii p-J
=-----.
2mac
(51
Later, unless it will be unavoidable, we shall omit the indices j, s. The kinetic equation for f^’ may be written as
where &‘, is the kinetic energy operator of the electron in a magnetic field. This operator, in the Iimiting case of the quasiclassical approximation, is obtained from c(p) by replacing components of the momentum p by operators of the kinetic momentum 6, and by appropriate symmetrization the exact meaning of which is not important in our case. Since the kinetic momentum operators i,, py, fiz do not commute in a magnetic field H, = H
Here fi/t, denotes a certain linear transformation corresponding to the collision integra1, and te is a relaxation time. To make our discussion clearer we shah start from the simplest case @ = 1. Since clearly [p, &J = 0, and [p, Tj] which is a quadratic function of the electric field, can be neglected, we have
&=
fi=
J&-i-G
--eEr”
[A we shall describe the electron gas state not by a distribution function, but by a density matrix5 In the absence of the eIectric field (7? = 0), the operator f^ = f^” is equal to the equilibrium Fermi function :
fi
=fO~$To);
f”(c) = [l+e(c-E)‘8]-1
where f is the chemical potential, In the electric field we take s = p+f”r_
(6)
19= kT.
&J+;P
We shal!use quantum numbers n and p, of the operator $&, and, as shown later, in the quasiclassicaf approximation it will be sufficient to regard 4” as a matrix with respect to n and as a number with respect to p,. This is because momentum p, is not quantized. When several electron bands are considered the band number j enters J^ as a parameter. This follows from the fact that in the classical particle motion j remains constant. In the quasiclassical approximation j can change only on collision, and this may be taken into account in the collision integral. The same applies to various spin states s which are taken care of by introducing an additional parameter s in f since the energy levels depend on s :
Ql.
Taking into account the diagonali~ Jb> we have
(9)
of +$$ and
From the equations of motion of the electron in a magnetic field it follows that _[&
(7)
= 40,
i] = t _; Q
(11)
or, from (5) ;(% - Ed) U,,.
= eE GBBs
where c = velocity operator corresponding classical quantity V = &/ap.
tlla) to the
This becomes
According
to (3) +y-ef&
=
(n’-rz)ftw*
Assuming PI
=
-ef&*
(13)
QUANTUM
THEORY
OF
ELECTRICAL
and reducing (13) by w*
[i(n’-
n)+rl*t8,t = Yg7zn~Vnw *
1 Y =-=-; tow*
m*c Ho* Z-------;
Ho* H
et0
(14)
gsw - fnsO-fnO* Es.- En. Assuming
CONDUCTIVITY
OF
13
METALS
This is identical with the classical kinetic equation of the Fourier amplitudes of the distribution function $ = $(E, p,, 9) (9 = angular variable of the particle trajectory in momentum space, proportional to the period of revolution) given in.(l) V, has the meaning of the Fourier amplitudes of the classical velocity vector.
n’ --n = k we obtain fmally (~~~~~Y~
= XkVk
(15)
Here G Wn+k = $k(%a, $4;
Vn,n+k = V;ir(%, Pz)
(20)
(16) gk(h) =
= &,nfk
= &k+k,lt
(17)
f”(%+~fi6J*)--fo(%)
I&o*
*
At K =i 0 the expression gr is undetermined. calculation shows that 7
A simple
go(e) =g which corresponds
(174
to (17a).
For fiw* -+ 0 VO(c) gk(C)+ T
dl = IdPI ;
vL = l/v,2+vy2).
(21)
Returning to the quantum equation (15) we shall replace V, (c,,pJ by their limiting values (21) (this corresponds to the quasiclassical approximation for matrix elements Vnn’). For g, a similar replacement (gk+ ~~*~~~}is not allowed, since f@(6) changes very rapidly near the limiting energy (6-f N 8) and the whole sought-after effect is bound up with this. The solution of the kinetic equation (15) is
(18)
and (15) assumes the form
We shall now return to the general kinetic equation (8) and we shall assume that
(~~)~*n~ = ~kk~~ +k f Frorn$,
=
$k(%
f%);
vk
=
rk(%
where IGkg = operator acting on the variable E_ and p, in the function +k(~nt pz)
?‘z).
+,,&k, = z;swkk,(f,pz;
= PEP we find *
x &(~‘,p’z) Since for any degree m, due to diagonality of $‘0, have
then for an arbitrary function f*(X,)
we obtain
(23)
(E = En;
we
E’,p’z)
d$‘zAE’.
x
(24)
E’ = En’)
(If the transitions between various bands are considered, the kernel W,,. has also the indexes j, j’.) Then, repeating the previous treatment, we transform (8) into a form similar to (15) i~k+~~k~~~
Eauations
=
Ygkvk-
125) and 115) transform
(251
into the
14
E.
M.
LIFSHITS
classical equation (missing in the original) of(l) on replacement of gk(e) by the limiting expression 8f”/& and on transition from suymation in (24) to integration. The quantities W,,. in (25), in agreement with the correspondence principle, represent the Fourier amplitudes of the collision operator in equation (33) of reference (I). Since g,, and consequently $ K change rapidly as a function of E, and slowly 4s a function of p, (for fixed cn = E), it is convenient to integrate with respect to 6 and sum with respect top, in equations (25) and (24), i.e. E and p, are regarded as independent variables. Solving (2) n = n(~, pZ) for p, we find
In the nonelastic scattering by phonons such a replacement is not permissible. As in the classical case, it is not possible to obtain an explicit solution of (25) for arbitrary fields without special assumptions about the collision integral. We shall study therefore the asymptotic behaviour of the electrical conductivity oik in large fields (y < 1) and try to elucidate the connection between quantum and classical expressions for (TV Ir for arbitrary values of y. In the latter case the solution of the classical problem will be regarded as known and the quantum effects described in terms of this solution. 3.
=
I
{ ~~F(GP+Pz~}
pZn = p&z, C);
Ap,,
= g
(26)
de
ehH X3 = I. c I aP2
(The limits of summation in n now depend on Thus for elastic scattering on impurities
t&k& = 2j 1 W,,.PzP’z(,) .S(,-,‘) x[~k(r,~Z)-~~(~‘,p’Z>l
ASYMPTOTIC DUCTIVITY
TREATMENT OF IN STRONG FIELDS
As shown in (I) the asymptotic treatment of the conductivity in strong fields is basically determined by the topology of the Fermi surface. Since quantization of the levels takes place only in the case of closed orbits (finite motion), the whole of our quantum discussion applies only to that case. This means, in particular,
l!)
THE CON(y < 1)
that, from the equation of
mot’on ln a magnetic fie’d [$,&o]
= fVxH
(29)
c
x
WA
which in terms of matrix elements has the form
ikpk = m*Vkxn
(27)
H=Hn
(30)
we find that the diagonal matrix elements of velocity in
= z W,~~~~‘,(~)[~~,(~,pz)-~~(~,p’~)lA~’~.
the plane xy, which is perpendicular to the direction of the magnetic field, are equal to zero
Pi
The quantities +,lc, Vk, g, in (25) and (26) are understood to be functions of a continuous parameter 8 and a discrete parameter p,, = pZ(lz, C)
?&xn
= Voxn
We shall introduce components
=o.
n
(31)
of the operator t&
A
+xn
$k = &d~,Pzn); Vk = Vk(%pm) According
gk = gk(r,Ptn)
=+;
(+,n)
to (13)
,+ p1 = -eeto{(cp, E x n)+
=f”(E+Rftw*)--fo(C). khw*
Taking
(30) into account,
= 9. 1 9(E, n)}.
(32) (33)
we find from (25)
(28) 3f0 go =-z*
In scattering on impurities the parameter Aw*/kT does not occur in the kernel of the collision opeiator (27) and therefore the kernel WkrC,PsP’r(c) may be replaced by its classical value.
ik(pk+$&qti
i%&
= iky---. m*
(34)
Writing down from (34) qr~ in powers of y we shall initially exclude pO. Substituting k = 0 in (34) we obtain ‘PO =
-
eJo-~ti~K,(px,
k’ # 0.
(35)
QUANTUM
THEORY
OF ELECTRICAL
CONDUCTIVITY
OF
METALS
retaining
the terms
15
Therefore
(Pk-YLkk
g*Pt
‘PW
K,k’
= ym,;
#
0. (36)
From
(33) and (38),
-1/H
we
find
and the solution C+XPK may be easily expressed as a series in powers of y glc@ ‘pk
The terms proportional first component.
(l-yf~)kk+~
k’& Y-$+
= Similarly,
=
g.pic.
y&y--+
m*
Since
jxn
..- .
to E, and E, are given by the ri =
-eE,X”,
= ~&J&fQ,
therefore
&I.
(44)
Considering the cur.ent components jz and ju we find, with an accuracy to - l/H%,
for 8~ we have
Hence ak-y.&k+%k,
=
yhk;
and consequently Qk =
yhk+yLktiyhk+...
crqp = ;@&3Qq. From of TX: and 8k, with the exception of a,, begin with the first degree of Y(-J l/N). Also the terms in vk (k $: 0) which are Iinear in y do not depend on the collision integral. The current is given by
(45)
the equality
We see that, as in the classical case, all elements
Thus the terms linear in y-ifH which occur in the components 02x, a~, a~ of the conductivity tensor, are not related to the collision integral and are easily obtained in explicit form. To find the components jx and C we shall calculate jxn. Using (29) and (8)
jxn
and a similar equality
for oyl(i) we fmd
0,x(11 = oyv(ll
= 0.
(46)
Since [js, A] = 1 the transformation of the expression for cr,,@) should be carried out with caution (direct clknge of the order of the m&ipliers in &... is not permitted). Detailed calcuiations show that when all the isoenergetic surfaces (1) in the interval IE-51 N 0 are closed then for each separate band
= esp3ltixn
when &j&z > 0
E.
16
M.
~pJL[3o,x] = &y(f’o-1)= -iv-
LJFSHITS
(47)
where N_+ = number of occupied states with m* > 0 (number of “electrons”) and N_ = number of unoccupied states with mr < 0 (number of “holes”). In this case
+2Re
& V-.&k?‘>
(48)
which is identical with the classical result obtained earlier. For a more complex topology of the Fermi surface, just as in the classical case, equation (48) no longer holds. When the number of the “electrons” and the “holes” is the same o#
= 0.
e2to
2eHh c
For tffrzin (51) we should substitute the solution of the kinetic equation (25). Since (25) and (51) are iinear with respect to (Gg*and gr results of such a substitution may be represented as a “scaler product” ediH 025’ = ---2(x,““‘,g) c
(x”“‘,s> =
Since V# V$J = 0 and g, is present only in the equation for &. . . then none of the components of U~JJ’,except @z, contains g, = 3fO/&.
&CX
=
-2 s
4. RELATIONSHIP BETWEEN QUANTUM AND CLASSICAL EXPRJZSSLONS FOR THE CONDUCTIVITY IN A MAGNETIC FIELD
d=’
z j 1 E,x,““‘gddpz;
x = X,&P&
x
tensor, according
(52)
(49)
We have shown now that those terms of ~~~ (written down as a series in r) which become zero in the classical case need no quantum corrections. This means that in the limit of the quasiclassical approximation special effects may be expected in the temperature dependence of conductivity and in the quantum oscillations. We shall now consider these questions.
The conductivity has the form u++’
(51)
A normalized constant A, can be simply found by comparison with the limiting classical case fiw --t 0 (see p. 75 of reference (1). Since m* Ac, = eHh/c, therefore
A =--G‘--’
G&U = -~(~~+-~~_)
dpz.
m*ap,,
+z zz -2
to (42) and (13)
. * = - eQ-&? Yx $F.
de ;
x,x’ # x,z
sa pm
af” XP~+
+ 2’ x;‘!%] m*Apzs&.
W)
(54)
(55)
k-1
Considering e and & and their diagonality spect to p, we find
with
re-
When % = I, according
to (22) we have
27re2to 2y XX’= ---+--Re,---VkW’@.
‘k
tk+y
(56)
For the diagonal components
(57) The expressions ($6) and (57) are identical with classical equations since, as shown above Va is Fourier amplitude of the classical veIocity vector; find the classical expression for uzz’ it is sufficient
the the to to
QUANTUM
THEORY
OF
ELECTRICAL
CONDUCTIVITY
replace gk by afO/as in (54) and (55) and instead of summing n to integrate with respect to dr. In the general case the quantities xzz’rnay be expressed in terms of a “Green’s function” of the kinetic equation (25) : @Z&
OF %&a,
= $(e--P)--
17
METALS *
kk’
tn
+rwkk’~-
by means of the equality
The classical “Green’s function” equation
is the solution of the
The classical expressions for the conductivity are found by the replacement gk --t af”& and integration
af"
=xX’ = __2 where Wkk, = the classical limit of the collision integral; to find xxx’ mtegration ’ with respect to p, should replace s ummation in (59). For the case when only the elastic scattering on impurities is important, the collision integral does not contain temperature. Consequently, the quantity xx”’ does not contain the parameter HUH,@ = tiu*/@ and the transition to the classical Iimit in x”“’ involves omission of the quantities N hw*/[, compared with Aw/0, which enter the quantum effects through the expressions
gk =
fO(~+k~~*)-fO(e) khw”
-*
Consequently in the above case we can regard x,cxX in (54) and (55) to be equal to their classical analogues.
-
B
~xxx’m*
x,x/,
&= = -2 ss
dpzde
# x,x
8f0 ~(xoze+X”)~*
&de
(6
1)
Since the main part of the quantum corrections is due to the neighbourhood of the points of pronounced instability of the functionfs (i.e. E,, g 6) and the functions WZ*(E,p,) and X(E, p,) are continuously variable, we find, from (54) and (17), that the difference o~~‘-u~~‘~~ (at X, x’ # z, z’) is given with the same accuracy by
af” k aE
xXz’----m*
dcdpz
E. M.
18
LIFSHITS
Summing over k we find that
Z
a(p’@I*) f o -7
0-c
br dp, -
f” ‘Gf
dedpz ) .
n
Integrating obtain
with respect to n, using the equality
aa/& = RwQ = AC and the Poisson
formula
a(‘:: 1,_e(npr,.h~* dp,;
p’(n) = /f”
The expression for the quantum correction obtained here contains both the oscillating part and the part which varies continuousIy with the field. In these corrections the levels with small n (in particular $ (0)) are important. The quasiclassical approximation is accurate for these levels only when the dispersion law is quadratic. In the general case, however, we shall retain only the oscillating part of the conductivity since the continuously variable part represents only a small correction of the fundamental classical value. Thus
LkS”I = 2
ss afo
~x~~*~~~e2ff~kn E
we
(62)
dRd&.
The expressions (63) and (64) are of such form that the oscillating components, contained in the integrals T, and L,, may be easily separated out. A detailed analysis of these components and a discussion of various aspects of the Shubnikov-de Haas effect are given in the following paper.
REFERENCES 1.
M., ASBEL’M. JA. and KAGANOV M. J. Clam. Solids. To be published. D~H& et al. Commun.Whys. Las. Univ. ,Leiden 237 (1935). ACHIESERA. J. Zh. eksp. theor, jiz 9, 726 (1939); DAVIDOVB. J. and POMERANCHVK J. JA. Z?z. eksp. &eor.fiz. 9, 1295 (1939): ZlLBERMhNG. E. Zh. eksp. theor. fiz. 25, 762 (1955). LIFSHITS E. M. and KOSE~ITCHA. M. Zh. eksp. theor. Ji.z. 29, 730 (19551. LIFSHITSE. M. Zh. eksp. tlteor.$z. 30, 817 (1956). IJFSHITS
7. Phvs.
2. 3.
According to (5.5) ~9~ contains an additional term with i?fo,@~ xozzand therefore by a procedure similar to that followed above
4. 5.
E.
Pergamon
QUANTUM
Press 1958. Vol. 4. pp. 11-18.
THEORY
CONDUCTIVITY
OF THE ELECTRICAL
OF METALS IN A MAGNETIC FIELD E. M. LIFSHITS
Physical-Technical
Institute,
Academy of Science of the Ukranian S.S.R., (Received
26 December
Kark’off, U.S.S.R.
19.56)
Abstract-The
paper gives a consistent quantum-mechanical theory of the electrical conductivity of metals in magnetic fields. A kinetic equation is deduced for the statistical operator with general assumptions on the energy spectrum of electrons, and the limit of the transition to a classical case is analysed. Asymptotes of the kinetic coefficients in strong magnetic fields are studied. Starting from the established relationship between the solutions of quantum and classical equations, quantum corrections to the kinetic coefficients, which include resistance and conduction oscillations, are determined.*
1. INTRODUCTION
quantum oscillation effects mentioned above and to find the transition to the classical description. As shown in t4) the energy levels en of a particle, which obeys the law of dispersion (l), in a magnetic field H directed along the z-axis are determined in the quasiclassical approximation by the conduction
IT is well known that the correct interpretation of electrical conduction in metals is basically quantum in character. Nevertheless, since the electron wavelength is small compared with its free path, and, in the presence of the magnetic field, also compared with the radius of curvature of its trajectory, it is usually sufficient to employ only a semiclassical treatment. This means that the electron may be regarded as a classical quasi-particle, which obeys Fermi statistics with a dispersion law E = &z,
P,, Pz)
S(~,pe) =
+(n+y) (n B
1)
(2)
(0 < 61)
where S(E, pB) = area of a section of the surface (1) by a plane p, = const. in quasimomentum space. The separation of the levels, according to (2), is:
(1)
where l r(p) = a periodic function with the period of the reciprocal lattice, j = band number. On this basis a semiclassical theory of galvanomagnetic effects in metals was developed in.(l) In such a treatment, however, certain specifically quantum effects (e.g. resistance oscillations in a magnetic field), connected with the quantization of levels, are lost. These effects were found experimentally@) but their theoretical treatmentt3) does not seem to be satisfactory or consistent. The present treatment aims to develop a consistent quantum theory of metallic conduction in magnetic fields. The theory enables us to study the
Ac, =.s~+~-E~
eH
a* = -;
m*c
=Aw*
r)2* =-.-*
i
2rr
=/.PH
as
ae'
/.L*=- eh
m*c
(3)
m*= m*(e,pZ). (4)
We can expect some intrinsic
effects at Aw* 2
kT. On the other hand to use the quasiclassical approximation and the dispersion law (1) we must have hw < 5 (6 = chemical potential). We shall take the latter condition to be fulfilled in this treatment.
* Some of the results obtained were briefly reported on in reference (5). 11
E.
12
2. KINETIC
EQUATION OF MATRIX
THE
M.
DENSITY
LIFSHITS
es,s =
If the electron is in a uniform magnetic field H and a weak electric field E, we can represent its Hamiltonian as
s=fl;
4pz)fs~oH; eii p-J
=-----.
2mac
(51
Later, unless it will be unavoidable, we shall omit the indices j, s. The kinetic equation for f^’ may be written as
where &‘, is the kinetic energy operator of the electron in a magnetic field. This operator, in the Iimiting case of the quasiclassical approximation, is obtained from c(p) by replacing components of the momentum p by operators of the kinetic momentum 6, and by appropriate symmetrization the exact meaning of which is not important in our case. Since the kinetic momentum operators i,, py, fiz do not commute in a magnetic field H, = H
Here fi/t, denotes a certain linear transformation corresponding to the collision integra1, and te is a relaxation time. To make our discussion clearer we shah start from the simplest case @ = 1. Since clearly [p, &J = 0, and [p, Tj] which is a quadratic function of the electric field, can be neglected, we have
&=
fi=
J&-i-G
--eEr”
[A we shall describe the electron gas state not by a distribution function, but by a density matrix5 In the absence of the eIectric field (7? = 0), the operator f^ = f^” is equal to the equilibrium Fermi function :
fi
=fO~$To);
f”(c) = [l+e(c-E)‘8]-1
where f is the chemical potential, In the electric field we take s = p+f”r_
(6)
19= kT.
&J+;P
We shal!use quantum numbers n and p, of the operator $&, and, as shown later, in the quasiclassicaf approximation it will be sufficient to regard 4” as a matrix with respect to n and as a number with respect to p,. This is because momentum p, is not quantized. When several electron bands are considered the band number j enters J^ as a parameter. This follows from the fact that in the classical particle motion j remains constant. In the quasiclassical approximation j can change only on collision, and this may be taken into account in the collision integral. The same applies to various spin states s which are taken care of by introducing an additional parameter s in f since the energy levels depend on s :
Ql.
Taking into account the diagonali~ Jb> we have
(9)
of +$$ and
From the equations of motion of the electron in a magnetic field it follows that _[&
(7)
= 40,
i] = t _; Q
(11)
or, from (5) ;(% - Ed) U,,.
= eE GBBs
where c = velocity operator corresponding classical quantity V = &/ap.
tlla) to the
This becomes
According
to (3) +y-ef&
=
(n’-rz)ftw*
Assuming PI
=
-ef&*
(13)
QUANTUM
THEORY
OF
ELECTRICAL
and reducing (13) by w*
[i(n’-
n)+rl*t8,t = Yg7zn~Vnw *
1 Y =-=-; tow*
m*c Ho* Z-------;
Ho* H
et0
(14)
gsw - fnsO-fnO* Es.- En. Assuming
CONDUCTIVITY
OF
13
METALS
This is identical with the classical kinetic equation of the Fourier amplitudes of the distribution function $ = $(E, p,, 9) (9 = angular variable of the particle trajectory in momentum space, proportional to the period of revolution) given in.(l) V, has the meaning of the Fourier amplitudes of the classical velocity vector.
n’ --n = k we obtain fmally (~~~~~Y~
= XkVk
(15)
Here G Wn+k = $k(%a, $4;
Vn,n+k = V;ir(%, Pz)
(20)
(16) gk(h) =
= &,nfk
= &k+k,lt
(17)
f”(%+~fi6J*)--fo(%)
I&o*
*
At K =i 0 the expression gr is undetermined. calculation shows that 7
A simple
go(e) =g which corresponds
(174
to (17a).
For fiw* -+ 0 VO(c) gk(C)+ T
dl = IdPI ;
vL = l/v,2+vy2).
(21)
Returning to the quantum equation (15) we shall replace V, (c,,pJ by their limiting values (21) (this corresponds to the quasiclassical approximation for matrix elements Vnn’). For g, a similar replacement (gk+ ~~*~~~}is not allowed, since f@(6) changes very rapidly near the limiting energy (6-f N 8) and the whole sought-after effect is bound up with this. The solution of the kinetic equation (15) is
(18)
and (15) assumes the form
We shall now return to the general kinetic equation (8) and we shall assume that
(~~)~*n~ = ~kk~~ +k f Frorn$,
=
$k(%
f%);
vk
=
rk(%
where IGkg = operator acting on the variable E_ and p, in the function +k(~nt pz)
?‘z).
+,,&k, = z;swkk,(f,pz;
= PEP we find *
x &(~‘,p’z) Since for any degree m, due to diagonality of $‘0, have
then for an arbitrary function f*(X,)
we obtain
(23)
(E = En;
we
E’,p’z)
d$‘zAE’.
x
(24)
E’ = En’)
(If the transitions between various bands are considered, the kernel W,,. has also the indexes j, j’.) Then, repeating the previous treatment, we transform (8) into a form similar to (15) i~k+~~k~~~
Eauations
=
Ygkvk-
125) and 115) transform
(251
into the
14
E.
M.
LIFSHITS
classical equation (missing in the original) of(l) on replacement of gk(e) by the limiting expression 8f”/& and on transition from suymation in (24) to integration. The quantities W,,. in (25), in agreement with the correspondence principle, represent the Fourier amplitudes of the collision operator in equation (33) of reference (I). Since g,, and consequently $ K change rapidly as a function of E, and slowly 4s a function of p, (for fixed cn = E), it is convenient to integrate with respect to 6 and sum with respect top, in equations (25) and (24), i.e. E and p, are regarded as independent variables. Solving (2) n = n(~, pZ) for p, we find
In the nonelastic scattering by phonons such a replacement is not permissible. As in the classical case, it is not possible to obtain an explicit solution of (25) for arbitrary fields without special assumptions about the collision integral. We shall study therefore the asymptotic behaviour of the electrical conductivity oik in large fields (y < 1) and try to elucidate the connection between quantum and classical expressions for (TV Ir for arbitrary values of y. In the latter case the solution of the classical problem will be regarded as known and the quantum effects described in terms of this solution. 3.
=
I
{ ~~F(GP+Pz~}
pZn = p&z, C);
Ap,,
= g
(26)
de
ehH X3 = I. c I aP2
(The limits of summation in n now depend on Thus for elastic scattering on impurities
t&k& = 2j 1 W,,.PzP’z(,) .S(,-,‘) x[~k(r,~Z)-~~(~‘,p’Z>l
ASYMPTOTIC DUCTIVITY
TREATMENT OF IN STRONG FIELDS
As shown in (I) the asymptotic treatment of the conductivity in strong fields is basically determined by the topology of the Fermi surface. Since quantization of the levels takes place only in the case of closed orbits (finite motion), the whole of our quantum discussion applies only to that case. This means, in particular,
l!)
THE CON(y < 1)
that, from the equation of
mot’on ln a magnetic fie’d [$,&o]
= fVxH
(29)
c
x
WA
which in terms of matrix elements has the form
ikpk = m*Vkxn
(27)
H=Hn
(30)
we find that the diagonal matrix elements of velocity in
= z W,~~~~‘,(~)[~~,(~,pz)-~~(~,p’~)lA~’~.
the plane xy, which is perpendicular to the direction of the magnetic field, are equal to zero
Pi
The quantities +,lc, Vk, g, in (25) and (26) are understood to be functions of a continuous parameter 8 and a discrete parameter p,, = pZ(lz, C)
?&xn
= Voxn
We shall introduce components
=o.
n
(31)
of the operator t&
A
+xn
$k = &d~,Pzn); Vk = Vk(%pm) According
gk = gk(r,Ptn)
=+;
(+,n)
to (13)
,+ p1 = -eeto{(cp, E x n)+
=f”(E+Rftw*)--fo(C). khw*
Taking
(30) into account,
= 9. 1 9(E, n)}.
(32) (33)
we find from (25)
(28) 3f0 go =-z*
In scattering on impurities the parameter Aw*/kT does not occur in the kernel of the collision opeiator (27) and therefore the kernel WkrC,PsP’r(c) may be replaced by its classical value.
ik(pk+$&qti
i%&
= iky---. m*
(34)
Writing down from (34) qr~ in powers of y we shall initially exclude pO. Substituting k = 0 in (34) we obtain ‘PO =
-
eJo-~ti~K,(px,
k’ # 0.
(35)
QUANTUM
THEORY
OF ELECTRICAL
CONDUCTIVITY
OF
METALS
retaining
the terms
15
Therefore
(Pk-YLkk
g*Pt
‘PW
K,k’
= ym,;
#
0. (36)
From
(33) and (38),
-1/H
we
find
and the solution C+XPK may be easily expressed as a series in powers of y glc@ ‘pk
The terms proportional first component.
(l-yf~)kk+~
k’& Y-$+
= Similarly,
=
g.pic.
y&y--+
m*
Since
jxn
..- .
to E, and E, are given by the ri =
-eE,X”,
= ~&J&fQ,
therefore
&I.
(44)
Considering the cur.ent components jz and ju we find, with an accuracy to - l/H%,
for 8~ we have
Hence ak-y.&k+%k,
=
yhk;
and consequently Qk =
yhk+yLktiyhk+...
crqp = ;@&3Qq. From of TX: and 8k, with the exception of a,, begin with the first degree of Y(-J l/N). Also the terms in vk (k $: 0) which are Iinear in y do not depend on the collision integral. The current is given by
(45)
the equality
We see that, as in the classical case, all elements
Thus the terms linear in y-ifH which occur in the components 02x, a~, a~ of the conductivity tensor, are not related to the collision integral and are easily obtained in explicit form. To find the components jx and C we shall calculate jxn. Using (29) and (8)
jxn
and a similar equality
for oyl(i) we fmd
0,x(11 = oyv(ll
= 0.
(46)
Since [js, A] = 1 the transformation of the expression for cr,,@) should be carried out with caution (direct clknge of the order of the m&ipliers in &... is not permitted). Detailed calcuiations show that when all the isoenergetic surfaces (1) in the interval IE-51 N 0 are closed then for each separate band
= esp3ltixn
when &j&z > 0
E.
16
M.
~pJL[3o,x] = &y(f’o-1)= -iv-
LJFSHITS
(47)
where N_+ = number of occupied states with m* > 0 (number of “electrons”) and N_ = number of unoccupied states with mr < 0 (number of “holes”). In this case
+2Re
& V-.&k?‘>
(48)
which is identical with the classical result obtained earlier. For a more complex topology of the Fermi surface, just as in the classical case, equation (48) no longer holds. When the number of the “electrons” and the “holes” is the same o#
= 0.
e2to
2eHh c
For tffrzin (51) we should substitute the solution of the kinetic equation (25). Since (25) and (51) are iinear with respect to (Gg*and gr results of such a substitution may be represented as a “scaler product” ediH 025’ = ---2(x,““‘,g) c
(x”“‘,s> =
Since V# V$J = 0 and g, is present only in the equation for &. . . then none of the components of U~JJ’,except @z, contains g, = 3fO/&.
&CX
=
-2 s
4. RELATIONSHIP BETWEEN QUANTUM AND CLASSICAL EXPRJZSSLONS FOR THE CONDUCTIVITY IN A MAGNETIC FIELD
d=’
z j 1 E,x,““‘gddpz;
x = X,&P&
x
tensor, according
(52)
(49)
We have shown now that those terms of ~~~ (written down as a series in r) which become zero in the classical case need no quantum corrections. This means that in the limit of the quasiclassical approximation special effects may be expected in the temperature dependence of conductivity and in the quantum oscillations. We shall now consider these questions.
The conductivity has the form u++’
(51)
A normalized constant A, can be simply found by comparison with the limiting classical case fiw --t 0 (see p. 75 of reference (1). Since m* Ac, = eHh/c, therefore
A =--G‘--’
G&U = -~(~~+-~~_)
dpz.
m*ap,,
+z zz -2
to (42) and (13)
. * = - eQ-&? Yx $F.
de ;
x,x’ # x,z
sa pm
af” XP~+
+ 2’ x;‘!%] m*Apzs&.
W)
(54)
(55)
k-1
Considering e and & and their diagonality spect to p, we find
with
re-
When % = I, according
to (22) we have
27re2to 2y XX’= ---+--Re,---VkW’@.
‘k
tk+y
(56)
For the diagonal components
(57) The expressions ($6) and (57) are identical with classical equations since, as shown above Va is Fourier amplitude of the classical veIocity vector; find the classical expression for uzz’ it is sufficient
the the to to
QUANTUM
THEORY
OF
ELECTRICAL
CONDUCTIVITY
replace gk by afO/as in (54) and (55) and instead of summing n to integrate with respect to dr. In the general case the quantities xzz’rnay be expressed in terms of a “Green’s function” of the kinetic equation (25) : @Z&
OF %&a,
= $(e--P)--
17
METALS *
kk’
tn
+rwkk’~-
by means of the equality
The classical “Green’s function” equation
is the solution of the
The classical expressions for the conductivity are found by the replacement gk --t af”& and integration
af"
=xX’ = __2 where Wkk, = the classical limit of the collision integral; to find xxx’ mtegration ’ with respect to p, should replace s ummation in (59). For the case when only the elastic scattering on impurities is important, the collision integral does not contain temperature. Consequently, the quantity xx”’ does not contain the parameter HUH,@ = tiu*/@ and the transition to the classical Iimit in x”“’ involves omission of the quantities N hw*/[, compared with Aw/0, which enter the quantum effects through the expressions
gk =
fO(~+k~~*)-fO(e) khw”
-*
Consequently in the above case we can regard x,cxX in (54) and (55) to be equal to their classical analogues.
-
B
~xxx’m*
x,x/,
&= = -2 ss
dpzde
# x,x
8f0 ~(xoze+X”)~*
&de
(6
1)
Since the main part of the quantum corrections is due to the neighbourhood of the points of pronounced instability of the functionfs (i.e. E,, g 6) and the functions WZ*(E,p,) and X(E, p,) are continuously variable, we find, from (54) and (17), that the difference o~~‘-u~~‘~~ (at X, x’ # z, z’) is given with the same accuracy by
af” k aE
xXz’----m*
dcdpz
E. M.
18
LIFSHITS
Summing over k we find that
Z
a(p’@I*) f o -7
0-c
br dp, -
f” ‘Gf
dedpz ) .
n
Integrating obtain
with respect to n, using the equality
aa/& = RwQ = AC and the Poisson
formula
a(‘:: 1,_e(npr,.h~* dp,;
p’(n) = /f”
The expression for the quantum correction obtained here contains both the oscillating part and the part which varies continuousIy with the field. In these corrections the levels with small n (in particular $ (0)) are important. The quasiclassical approximation is accurate for these levels only when the dispersion law is quadratic. In the general case, however, we shall retain only the oscillating part of the conductivity since the continuously variable part represents only a small correction of the fundamental classical value. Thus
LkS”I = 2
ss afo
~x~~*~~~e2ff~kn E
we
(62)
dRd&.
The expressions (63) and (64) are of such form that the oscillating components, contained in the integrals T, and L,, may be easily separated out. A detailed analysis of these components and a discussion of various aspects of the Shubnikov-de Haas effect are given in the following paper.
REFERENCES 1.
M., ASBEL’M. JA. and KAGANOV M. J. Clam. Solids. To be published. D~H& et al. Commun.Whys. Las. Univ. ,Leiden 237 (1935). ACHIESERA. J. Zh. eksp. theor, jiz 9, 726 (1939); DAVIDOVB. J. and POMERANCHVK J. JA. Z?z. eksp. &eor.fiz. 9, 1295 (1939): ZlLBERMhNG. E. Zh. eksp. theor. fiz. 25, 762 (1955). LIFSHITS E. M. and KOSE~ITCHA. M. Zh. eksp. theor. Ji.z. 29, 730 (19551. LIFSHITSE. M. Zh. eksp. tlteor.$z. 30, 817 (1956). IJFSHITS
7. Phvs.
2. 3.
According to (5.5) ~9~ contains an additional term with i?fo,@~ xozzand therefore by a procedure similar to that followed above
4. 5.
E.