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Fermi-liquid theory of the electron-electron interaction effects in disordered metals
Solid State Communications, Vol.46,No.6, pp.429-435, ] 9 8 3 . Printed in Great Britain.
0038-]098/83/]80429-07503.00/0 Pergamon Press Ltd.
FB~R~I-LIQUID THEORY 0F THE ELECTHON-ELECTRON INTERACTION ~ P E C T S DISORDERED B.L.Altshuler
IN
M~.TALS
and
A.G.Aronov
Leningrad Nuclear Physics Institute Ga tchina, Leningrad 188350, USeR (Received
15 January 1983
by
E.A.Kaner
)
Influence of the arbitrary strong interaction between the electrons on the thermodynamic and transport properties of disordered metals is considered. The contributions of the interaction in the diffusion channel to all of the quantities are shown to depend on only one Fermi-liquid interaction constant (to take interaction in the Cooper channel into account one has to include one constant more). It is found very convenient to divide all of the diffusion corrections on the parts which correspond to different values of the total spin of the electron and hole j:j = 0 and j = I. It is the interaction w i t h j=1 that leads to magnetic field dependence of the diffusion contributions. In the presence of spin-dependent scattering of the electron only the contribution related to the interaction w i t h j=O are important. These contributions to all of the quantities are universal, i.e. they ~o not depend on an~ interaction constant.
this perturbation theory is 2 ~ / p F the ratio of the electron wave-length 2~/PF (p~ is the Fermi momentum) and the screenLug length ~'-~ .
Si~aificant progress in the understanding of the electron properties of disordered metallic systems has been achieved in the last few years./1-~/ In particular, the importance of the role, which the interaction between the electrons plays has became clear. The value of the interaction contribution to the frequency and temperature dependence of the conductivity /5,4/ is not less than the value of the weak localization contribution /1-3/ which is not caused by the interaction. ~oreover, taking electron-electron interaction into account leads to significant corrections to the one-particle density of states / 5 , 4 / ~ , heat capacity and spin susceptibility ~ /6,7/. But for the interaction these quantities would not be essentially modified even by the metal-insulator transition. It has been ascertained also that the decay time ~-ee of the one-particle excitations grows w i t h decrease of the temperature T or quasiparticle energy mmch slower in disordered metals than in the ideal Fermi-liquid /8,9/. The corrections to all quantities mentioned above have the universal T and dependence. On the other hand, these corrections depend on the interaction constants which are quite different for various quantities. Formerly these constants have been calculated for the Coulomb interaction between the electrons in the first order of perturbation theory. The parameter of
The aim of the present w o r k is to connect the effective interaction constants related to various physical properties with each other in the general case of arbitrary strong interaction. If the interaction effects in the Cooper channel (when the momenta sum of the two interacting particles is small) are not taken into account the values of all the interaction effects are determined by only one Fermi-liquid constant. This constant is to be determined from the experiment. let us,for example, consider the correction to the conductivity due to the diffusion channel interaction (the momenta difference of the two interacting electrons is small). As it has been shown in ref. /4/, in two dimensions when the screened Coulomb interaction is taken into account in the first order of perturbation theory this correction is ee~2(T)
-- 4,_¢2----~
(2-2:F)ln(T-g/'~). (I)
Here G"2 R ~ is the conductivity per square, -~" - the electronic m e a n free p a t h time determined by the elastic scattering on the random potential. The two terms in brackets in eq.(1) are the contributions of the exchange and di, 429
THE ELECTRON-ELECTRON INTERACTION EFFECTS IN DISORDERED METALS
430
rect interactions. Coulomb potential averaged over the Fermi-surface F can be expressed as a function of the ratio /4, lO/fJ' 4,~ 1 when ~<< pF). Pinkelshtein/11 / have pointed out that eq.(1) is not correct even if understood as the first two terms of the expansion over F (i.e. over ~ / p F ). To obtain the correct expression for o¢G'2('/") at ~ I it is sufficient to substitute the factor (2 - 3~/2) for
~l~/pw
(2-2F) in eq.(1).
To understand the cause of this substitution we must recollect: wh~ the contribution of the exchange interaction is independent of ~R/p . The bars Coulomb potential~in coordinate representation is e=//~ . Its form in m o m e n t u m representation depends on the effective dimensionality of the sample d /4/
~4~re 2
-e21n
(d =3
)
(d=l),
q2a2
where a is the transversal size of the wire at d=1 (or film thickness at d--2). The condition for d = I ,2 case to take place is 131 :
a ~ ~
,
@* being the characteristic energy scale. Debye screening leads to the effective potential which depends both on the energy transfer Ud(°) (q)
where
/3,4/ D q2
(q,
) = "$d - i ~ + D q
2
'
(4)
~d is the d-dimensional density of states (e.g. in two dimensions "O2 is the number of states per unit energy interval and unit area). If U~°J(q) -~oo as q -~0, we can neglect unity in the denominator of eq.(3). Hence
i_
U(q,~-,) = ~)d
-i~J+ Dq 2
Oq 2
(5)
Vol. 46, No. 6
~/PF (i.e. of the effective electron" charge). The exchange contribution to the conductivity at T ~<~/Z" is (T)
i
--
,'rd
d¢~
¢dcth ¢ ~ )
U (q, c,2) D q2
(6)
where (dq) = ddq/(2~') d, ~ is the wire conductivity per unit l~ngth, ~ - the usual three-dimensional conduc%ivity. In eq.(6) small values of q and are e s s e n t i a l C O ~ T , q ~ . So we can make use of eq.(5). Substituting eq.(5) in eq.(6) we obtain the first universal term in eq. (I). It should be noted now that exact Coulomb form of the potential Ud~°~q) is not necessary for eq.(5) to be valid. The only condition is U d t ~ q ) > > I. But it is impossible to break this condition by short-range c o r r e c t i o ~ to the Coulomb potential (if q is sufficiently small). Therefore, if the electron w~s spinless, the correction to the conductivity would depend neither on ~/P~(i.e. on F) nor on any other coupling constant (e.g. due to the exchange by virtual phonon ). Since the electron has a spin, it is sensible to classify various contributions to the conductivity according to the st~mary spin of the interacting electron and hole. Being averaged over the random potential the two-particle Green function becomes invariant under the rotation in the spin space even in the presence of the interaction or spin-dependent scattering. This fact makes total spin j and its projection good quantum numbers to characterize the ~veraged two particle Green function, Just as the invariance of this propagator under the real space shift makes the summary momentum to be the same il. each intermediate state. Note that the sum of the electron and hole momenta is equivalent to the m o m e n t u m transfer q in eqs.(2)-(6). The contributions of the interaction with different j and ~ are additive. There is one point in which j=S and j=O contributions are different: at j=1 there can be no intermediate state consisting only of a single Coulomb propagator. Thus, the effective interaction amplitude at j=1 is not universal. Consider first the exact static amplitude u( j=1) (q ,CO=O) . There are no reasons for this quantity to depez~ on q significantly when q ~ 0 . Let us denote
Thus, the effective interaction potential U ( q , ~ ) for small enough q and ~J is universal-independent of the
X
F u(J=l)(q,c,3= O) =
•
2"~ d
(7)
Vol. 46, No. 6
THE ELECTRON-ELECTRON
At small ~ / ~ Coulomb interaction leads to 2 ~ d u(j=I)(~- O) equal to F calculated in refs./4, I0/. Later on we shall understand F in ~he sense of eq. (7). If c~ is not exactly zero but small enouy~h, the significant dependence of u(J=I) on q and ~ appears due to the diffusion pole in the intermediate state (see ref./9/). The ~eneral relation between U(j--O)(q,O) an~ UU:°)($~}cau be obtained from eqs. (3) and (4) by excluding Ud¢~q) dThe~ relation between u(j=1)(q,~) ~ is of the same
type
Thus, the j= I contribution absolute value can be three times larger than the one related to j=O (we can see from eqs. (5), (8) that u(J = O ) = U ( j =I) if F= 2) The relative value of the diffusion correction to any thermodynamic or transport quantity can be written as
(8)
F
=
U (j=1)(q,a,)
43]
INTERACTION EFFECTS IN DISORDERED METALS
- i ~ +Dq 2 =
~[,
--n
*I d/2 - I
(d =I ,3 )
c'r
(d =2 )
,
(11)
where The validity of eq.(8) is quite general and independent of the nature of the interaction or its strength. It is important to note that up to now we have not taken the Fermi-liquid modification of the one-particle electron propagator into account. This modification can be shown (see e.g. ref./11/) to lead only to the diffusion coefficient D renormalization. It is renormalized value of D that is connected with the conductivity by the Einstein relation
e2 D
~/4
where /4 and Nd
are the electron
chemical potential and d-dimensional density (e.g. N 2- the number of the conduction electrons per unit area). Eqs. (4)-(6) and (8) are valid only provided this renormalization is taken into account.
~jl~,
~P~)a
e# C
1
E* is either T or ~-quasiparticle e n e r ~ reckoned from the Fermi level. C t°i is the d-dimensional heat capacity
c(O )
~r 2
=
~d-- T 3
Ta~lel~ gives the values of ~(j=O) and ,Itj= for arbitrary F. Therefore, actually only one unknown constant F determines the diffUsion corrections to all quantities under consideration. Numerical factors 4 d) are listed -in Table 2. To obtain the values of ~(J) and J) we have to substitute the exact litudes eq.(5) or eq.(8) into the familiar expressions for the density of states correction /5,4/
Substituting eqs. (8),(5) into eq. (6) we obtain the expression for the correction to the conductivity which is valid for arbitrary F : (9) =
,,[(8/d
+
e2
T
) (d-2)/2
(12) "
and for the energy relaxation time
~-4 -9 1 /
(d=l) (d =2 )
(.0.915
(d=3)'
~J
where
r .32 ~(j
=1 )
F
(d=2) (lO)
1-411+,~,.E in (1
(13) Re-i
o;Dq
"
(1_~)d/2
=
;'~
a
T e e (~)
•
d
1 -~-
d(7-2) .....
.
F/2)]
(d=2)
In two dimensions this result had been obtained in r e ~ / ~ ' / . The factor 3 in eq.(9) is due to the multiplicity of the j=1 state.
TO determine ~(J ) we have to calculate the correction to the thermodynamic potential of the sample~.The renormalization of the potential is not very easy in this case : when one calculates ~ the additional factor I/n appears in the n-th order of the perturbation theory /12/. We can avoid this difficulty making use of the equality
432
THE ELECTRON-ELECTRON INTERACTION EFFECTS IN DISORDERED METALS Table 1.
Values of the constants
~(j=O)
and
Vol. 46, No. 6
~ x (j=l)iF) (j =0 )
~x 32
IlFD-" 114 F - (I - F / 2 ) d'2]/'1
d=l ,3
d (d-2)
%
,,,
s
(j=1) = -26_
T
-4 -
8(1 - F / 2 ) l n
4
-I( 1 -
d - 2
2 ].n (1
(1 - F / 2 ) (d-2 )/2
F/2)
(2)
d=2 "I
-
11
d=l
-F/2)
,3
4 =
(2)
d=3 ( v d j=O) - ~~(j= I) (2,,U
d=2
d=1,2
4/d
o"c/c ~" m
[(I
-F/2) d/2
- I] ,
4~
4F
£~e
(~
Table 2 ~X
£
~-d
-
(1-
-
712) d/2]
- coefficients
a(d)
d=2
4. 91 8yT2
d=3
0.~.,91..._!5 8JT2
I
~
Fig.
8~)d
1
,~'C
-o.o
GZ'~I('@ )
8JT
1
1
I
0.018
1 62)-
16 2 " 2
-&£,
Nd =
d4-= J 7 : 1 7 ( 2 )
d (4.-F)
d=l
~Sd
[I
J
I/Vd ~/v
'
(14)
where v~ is the volume of the sample, v 2 - th~ film area, v I - the length of the wire. Unlike the thermodynamic potential, the concentration N d can be calculated making use of the usual diagram technique /12/. Typical graphs for Nd are given in fig. I. Tile insertion of the vertex leads to the derivation of the corresponding electron propagator o v e r / W The sum of all the possible insertions into diagrams for F reduces to t h e ~ - d e r i v a t i v e of this value :
I.
Typical diagrams for the elect~on density coz~cection. Solid lines electron propagators. Dashed lines denote the averagis4; over random po tent ia 1.
= ~/2 9 - - 7 " " Summing the diagrams of the fig. f type in the Matsubara technique /12/ and then substituting the real frequencies for the imaginary ones we obtain
o" (j=1
= -3
i
cth . . . .
2T
-i~(1
- F/2)
(dq)ln
+
The substituion (15) gives the ~(j=l)~
+ Oq 2
-
(15) "
of eq.(14) into eq. correction to ~ : 3
=
Vd
9/
~
- --
9~
d~
I
(dq) •
exp~7/T - I 0
(16) •
rc tg
q2
D
a
rc tg
-- --
Dq
.
Vol. 46, No. 6
THE ELECTRON-ELECTRON INTERACTION EFFECTS IN DISORDERED METALS
Using eq. (16) ~ne can calculate any thermodynamical quantity. In particular, the heat ca pa city is C
=
T
(17) T~
vd
"
lead to the expression Eqs. (16), (17) lis%ed in Table 1. for @C (J=I)(F) ~j--O) can be calculated by the same me thod. To calculate the spin susceptibility we have to examine the magnetic field H influence on the thermodynamic potential. At j=O the diffusion pole is modified neither by the magnetic field nor by the spin scattering of the electrons. This fact is due to the conservation of the number of electrons w i t h a given energy. This is not the case at j=l. The general expression for the diffusion pole with fixed J and M is /13, 6/ : (18) D ( J,
(q ,~) = [-i~+Dq 2+ i~_ _ i IvlO~s]'£ t s
where~
s = g~sH
is
the
Zeeman split-
ting. t~ -' the total electron spin relaxat$o~ Sims /13/. Thus, to calculate ~ L D = t ) ~ at ts ÷ ~ w e must substitute (c~+~)for ~ in the arguments of arctg and include the summation over ~i instead of the factor 3 in eq. (16). After this we obtain (see refs. /6,7/). Strong magnetic field ( C O s ~ T , ~ / ~ g ) suppresses the ~i = ~I contributions to ~ , C and Y-e~ ~ d , ,hence, ~ s u l t s in s u b ~ i t u t i n ~ (~J---U) + ~ k J = ~ ) ) f o r ( ~ao=O> + 3 ~(~=1)) in eq. (11). The corresponding contribution to magnetoresistance was considered in ref. /14/ (in the first order in F). We see that the effect is determined by ~(.j=17 for arbitrary ~. The density of states ~ energy dependence (and, hence, the tunnel conductivity dependence on the V /5/)is much more i~teresting /13/. If the magnetic field is strong enoug~ two new singularities a t e = + g O (eV = _+ ~Js) appear i n @ ( ~ ) in addition t o ~ = 0 (V = 07 one. ~br O < F < 2 there is a minimum at @ = 0 which becomes deeper as H increasing, w~ile at ~c = -+O0s . (~) has maxima. The ~alue and the s h a p e o f ~ h e s e maxima are dete~nined by ~!4 =' (i.e. F) a~Id t s. But the energ~ (voltage) distance between them is equal to 2 ~ s = 2g~4gH and depends on the electron g-factor only. Note that the condition c ~ ~
433
these two electrons is n e a r the Fermi level ~ 0 while for the other 6 c ~ _~-2~ s ( ~ = -+I/2 denotes its spin projection on the H direction). Under these conditions the value of these two electrons momenta are almost equal. Consider now the role of the spin (spin-spin or spin-orbit) scattering. It can be seen from eq.(18) that this scattering if strong enough (~q/ts~'> t~, ~S. ) suppresses all j=l contribuions (in particular, Zeeman singularities i n ~ ( ~ ) function). Thus we have to substitute R (j=O) for ( ~ ( j = O ) + 3 R ( J = l ! ) ^ i n eq. (11). Nearly all constants ~ ( j = u J are P-i~de~endent. The only exception is ~3= ) at d E 2. Since ~ ( j = I ) ( F ) - ~ -~o~wnen F -~ 2 at d g 2 ( s e e T a b l e I), we can not neglect u~ity^in the eq.(3) j - u) denominator while ~.~Jcalculating. Thus ~(J=O)is different in various cases /4,15/ and much more than unity, Therefore all the diffusion channel contributions are universal (except ~(~) at d ~ 2 ) provided t s is small
enough,
T~is fact simplifies essentially the scaling theory of the metal-insulator transition (IVET) in the interacting disordered electron system. If we assume a scaling theory exists, at the strong spin scattering we must include one scaliz4~ variable - the d~menslone ~ s sample conductance G = ~ ~/e2R. The scaling equation can be written in the same form as the one in the pure localization theory /I/: ~ln
G
f~ (G).
(197
in L
L being the sample size. The localization of the electrons means that ]9 (G)~ -~ln G - ~ - ~ at G ~ 0 . If there are localized spins in the system, the corresponding electron scattering suppresses all the Cooper channel effects (in particular weak localization corrections) /16/ as well as j=1 diffusion contributions. Thus the total correction to the classical conductance G O is determined by the interaction in the diffusion~channel at j=O. Substituting ~ D / L ~ for E* in eq.(11) we see that at G > ~ 1 @ (G) = d-2- I/G. Thus, only insulator phase takes place at d=1,2 ( B < O at any G). The localization length ~ is just the same as the one -in the pure localizati0n theory without spin scattering. E.g., at d=2 ~"
1o " 1 exp ( ~ 2
(1 is the m e a n free path). Note that spin-spin scattering leads to ~o ~" ~-1 exp (G O ) ~ > ~ o if the interaction
434
THE ELECTRON-ELECTRON
INTERACTION EFFECTS IN DISORDERED METALS
is absent /17/. Therefore, interaction results in the enormous d e c r e a s i n K o f the localization length at PF2 l a > > ~ 2 . In three dimensions there is unstable fixed point (since ~ > 0 at G > ~ I and ~ ~ 0 for small G). So ~IT takes place without an~ conductivity jump. These features are the same as in the pure localization theory w i t h only the potential electron scattering. Nevertheless, there are no reasons for the correlation length in these two cases to be equal. At finite T scaling h~pothesis means that in the metal near the ~IT conductivity can be written as (~ = yr2 &
• f
(
),
(21)
where L T = D~/T. f ( x ) = 1 + Ax a t x<~ 1 (A~I). Thns, if T < ~ D / ~ z the t e ~ e r a t u r e dependent part df the conductivity is proportional to In the critical region the main term in ~ 5 should be ~ - i n d e p e n d e n t ; e2 (B ~ 1) (22)
Using the ~ n s t e i n
%-D=
e2
Now let the magnetic impurities to be absent while the spin-orbit scattering is strong. Such a situation takes place e.g. in heavy metals or in p-type cubic semiconductors. The interaction effects in the Cooper channel neglected, there are two contributions to the conductivity: the one due to the diffusion channel interaction at j=O and the weak localization one. The latter ~ s the additional factor (-I/2) as compared with the potentials scattering c~#_#___, /19/. Hence, at L T ~ L ~ I ~ o ---~D-Cso and G~ I
#=
d-2-
1/2Q,
(while without interaction ~ = d-2+ + I/2G). In fig. 2 the solid line is the graph of ~ (G~ function (dashed line is the same function without inte ract ion).
I
"~
d=3
relation we obtain
(.2
1 TI/3 (B
( 23 ~ /N~.... )2/3
(24)
The same index 1/3 for the frequency dependence of the conductivity can be obtained in the same way. Note that for the noninteracting electzons the same frequency dependence was obtained earlier /18/. As to the one-particle excitations decay t i ~ , substituting eq~ (24) ~into eq.(11) gives at ~ * = T ~ D/~: ~ - - 1 (T) ee
Vol. 46, No. 6
= bT
(b ~-1)
(25)
Thus, the condition T T ^ ^ ~ becomes invalid and the P e r m i - l ~ u i d quasiparticle description fails in the ..... critical region. IT and Lin =~fD--A-ee become of the same order. Therefore, the only temperature-dependent length scale in the critical region is Note that due to the Hubbard repulsion and non-homogeneous impurity distribution in semiconductors there mast appear the states near the mobility edge which are occupied by a single e l e c t r o n - i.e. localized spins. Thus, in semiconductors the presented theory must describe the general case.
Fig.
2.
Gell-Kvann - ~ o w function ~ (G) in the spin-orbit case. D a s h e @ l i n e s - pure localization theory. Solid lines correspond to the interacting el ec tr ons.
We see that in two dimensions only the i n ~ l a t o r phase takes place(unlike the noninteracting electrons). The localization length is larger than defined by eq. (20) : ~ .~ ~oL/1. At d=3 the conductivity in the critical region is determined by eq. (23) as before. In su~mnar2 we have shown that either thermodynamic or transport properties of the disordered metalic systems are determined by only one Fermi-liquid constant F, which can be obtained independently from a variety of experiments Cooper channel interaction if taken into account leads to one constant more. Therefore, there are no
Vol. 46, No. 6
THE ELECTRON-ELECTRON INTERACTION EFFECTS IN DISORDERED METALS
four independent constants, as it was argued in ref. /20/ but only two of them. If spin scattering is strong enough, all the diffusion corrections become universal. This fact have been used to construct the one-parameter scaling theory of metal-insulator transition. The simplification of the metal-insulator transition problem in the strong spin scattering case is evidently due to the possibility to Refer 1. 2. 3.
4. 5.
6.
E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys.Rev.Lett., 42, 673 (1979). P.W. Anderson, E . ~ r a h a m s and T.V. Ramakrishnan, Phys.Rev.Lett. , 43, 718 (1979). L.P. Gor' kov, A.I.Larkin and D.E. Khmelnitskii, JETP Letters, 30, 328 (1979) (Pis'ma Zh. Eksper.Teor.Fiz., , 248 (1979)). L.Altshuler, A.G.Aronov and P.A. Lee, Phys. Rev. Lett., ~ , 1288(1980) B.L. Altshuler and A.G. Aronov, Solid State Comn., 36, 115 (1979) and Zh. Eksp.Teor. Fiz., 77, 2028 (1979)(Sov. Phys. JETP 50, 9 ~ (1979)). B.L. Altshuler, A.G.Aronov and A. Yu.Zuzin, Pis'ma Zh. Eksp.Teor. Fi z.('935') 1 5S2) 1 (1982) (JLTP Lett.,35, and Zh. Eksp.Teor. Fiz., 16 ' No4 (1983) (Sov. Phys.JETP (1983) be published). H" ' J~uku~ama 07 (31 98 1' ) J" . 4 Phys "S°c " J a p " __50,
~.
~o
7. 8. 9.
A. Schmid, Z. physik, 271, 251 (1974). B.L. Altshuler and A.G. Aronov, Pis'ma Zh.Eksper. Teor. Fiz., 30, 514(1979) (J~.TP Letters, /Q, 482(1979)) and Solid St .Comm. ,--~8, 11 (1981). 10. T.F. Rosenbaum, K~'Andres, G.A.Thomas and P.A. Lee, Phys.Rev.Lett. ,46, 568 (1981).
435
separate the metal-insulator transition from the attendamt magnetic transitions in this case. Acknowledgement - We would like to thank Drs. A,M.Finkelshtein and A. Yu.Zuzin for numerous and helpful discussions. We are also grateful to E.M. Pavlenko and G.V.Stepanova for the helping in the preparing the manusc ri pt. ences 11. A.~.Finkelshtein. Zh.Eksp. Teor°Fiz., ' No2 (1983) (Sov. Phys.J•TP (1983) be published). 12. A.A. Abrikosov, L.P.Gor'kov and I.E. Dzyaloshinskii, ~ t h o d s of Quantum Field Theory in Statistical Physics, Pergamon, N.Y. (1975). 13. B.L. Altshuler and A.G. Aronov, Pis'ma Zh. Eksp.Teor.Fiz., ~ ~o2(1983). 14. P.A. Lee and T.V. Ramakrlshnau, Phys. Rev. B 26, 4009 (1982) 15. A. Y u . Z ~ - ~ , Pis'ma Zh.~sp.Teor. Fiz. , 377 (1981)(J~TP Letters, 33,360, 981) ). 16. P.A. Lee, J.Non.Crys. Sol.,35, 21
~o
~
--(1980).
17. K.B. Efetov, Zh.Eksp. Teor. Fiz., 83, 833 (1982). 18. F. Wegner, Z. Physik, B_~, 207 (1979); W. Gotze, Philos.~ag., 43, 219(1981) B. Shapiro and E. Abrahams, Phys.Rev.,
B24, 4889 ( 1981).
19. S..Hikami, A.I. Iarkin and Y.Nagaokm, Progress The or.Phy s. Le tt .63,707 (1980). 20. H. Fukuyama, Surface Sci.,1~, 489
--(1982).
0038-]098/83/]80429-07503.00/0 Pergamon Press Ltd.
FB~R~I-LIQUID THEORY 0F THE ELECTHON-ELECTRON INTERACTION ~ P E C T S DISORDERED B.L.Altshuler
IN
M~.TALS
and
A.G.Aronov
Leningrad Nuclear Physics Institute Ga tchina, Leningrad 188350, USeR (Received
15 January 1983
by
E.A.Kaner
)
Influence of the arbitrary strong interaction between the electrons on the thermodynamic and transport properties of disordered metals is considered. The contributions of the interaction in the diffusion channel to all of the quantities are shown to depend on only one Fermi-liquid interaction constant (to take interaction in the Cooper channel into account one has to include one constant more). It is found very convenient to divide all of the diffusion corrections on the parts which correspond to different values of the total spin of the electron and hole j:j = 0 and j = I. It is the interaction w i t h j=1 that leads to magnetic field dependence of the diffusion contributions. In the presence of spin-dependent scattering of the electron only the contribution related to the interaction w i t h j=O are important. These contributions to all of the quantities are universal, i.e. they ~o not depend on an~ interaction constant.
this perturbation theory is 2 ~ / p F the ratio of the electron wave-length 2~/PF (p~ is the Fermi momentum) and the screenLug length ~'-~ .
Si~aificant progress in the understanding of the electron properties of disordered metallic systems has been achieved in the last few years./1-~/ In particular, the importance of the role, which the interaction between the electrons plays has became clear. The value of the interaction contribution to the frequency and temperature dependence of the conductivity /5,4/ is not less than the value of the weak localization contribution /1-3/ which is not caused by the interaction. ~oreover, taking electron-electron interaction into account leads to significant corrections to the one-particle density of states / 5 , 4 / ~ , heat capacity and spin susceptibility ~ /6,7/. But for the interaction these quantities would not be essentially modified even by the metal-insulator transition. It has been ascertained also that the decay time ~-ee of the one-particle excitations grows w i t h decrease of the temperature T or quasiparticle energy mmch slower in disordered metals than in the ideal Fermi-liquid /8,9/. The corrections to all quantities mentioned above have the universal T and dependence. On the other hand, these corrections depend on the interaction constants which are quite different for various quantities. Formerly these constants have been calculated for the Coulomb interaction between the electrons in the first order of perturbation theory. The parameter of
The aim of the present w o r k is to connect the effective interaction constants related to various physical properties with each other in the general case of arbitrary strong interaction. If the interaction effects in the Cooper channel (when the momenta sum of the two interacting particles is small) are not taken into account the values of all the interaction effects are determined by only one Fermi-liquid constant. This constant is to be determined from the experiment. let us,for example, consider the correction to the conductivity due to the diffusion channel interaction (the momenta difference of the two interacting electrons is small). As it has been shown in ref. /4/, in two dimensions when the screened Coulomb interaction is taken into account in the first order of perturbation theory this correction is ee~2(T)
-- 4,_¢2----~
(2-2:F)ln(T-g/'~). (I)
Here G"2 R ~ is the conductivity per square, -~" - the electronic m e a n free p a t h time determined by the elastic scattering on the random potential. The two terms in brackets in eq.(1) are the contributions of the exchange and di, 429
THE ELECTRON-ELECTRON INTERACTION EFFECTS IN DISORDERED METALS
430
rect interactions. Coulomb potential averaged over the Fermi-surface F can be expressed as a function of the ratio /4, lO/fJ' 4,~ 1 when ~<< pF). Pinkelshtein/11 / have pointed out that eq.(1) is not correct even if understood as the first two terms of the expansion over F (i.e. over ~ / p F ). To obtain the correct expression for o¢G'2('/") at ~ I it is sufficient to substitute the factor (2 - 3~/2) for
~l~/pw
(2-2F) in eq.(1).
To understand the cause of this substitution we must recollect: wh~ the contribution of the exchange interaction is independent of ~R/p . The bars Coulomb potential~in coordinate representation is e=//~ . Its form in m o m e n t u m representation depends on the effective dimensionality of the sample d /4/
~4~re 2
-e21n
(d =3
)
(d=l),
q2a2
where a is the transversal size of the wire at d=1 (or film thickness at d--2). The condition for d = I ,2 case to take place is 131 :
a ~ ~
,
@* being the characteristic energy scale. Debye screening leads to the effective potential which depends both on the energy transfer Ud(°) (q)
where
/3,4/ D q2
(q,
) = "$d - i ~ + D q
2
'
(4)
~d is the d-dimensional density of states (e.g. in two dimensions "O2 is the number of states per unit energy interval and unit area). If U~°J(q) -~oo as q -~0, we can neglect unity in the denominator of eq.(3). Hence
i_
U(q,~-,) = ~)d
-i~J+ Dq 2
Oq 2
(5)
Vol. 46, No. 6
~/PF (i.e. of the effective electron" charge). The exchange contribution to the conductivity at T ~<~/Z" is (T)
i
--
,'rd
d¢~
¢dcth ¢ ~ )
U (q, c,2) D q2
(6)
where (dq) = ddq/(2~') d, ~ is the wire conductivity per unit l~ngth, ~ - the usual three-dimensional conduc%ivity. In eq.(6) small values of q and are e s s e n t i a l C O ~ T , q ~ . So we can make use of eq.(5). Substituting eq.(5) in eq.(6) we obtain the first universal term in eq. (I). It should be noted now that exact Coulomb form of the potential Ud~°~q) is not necessary for eq.(5) to be valid. The only condition is U d t ~ q ) > > I. But it is impossible to break this condition by short-range c o r r e c t i o ~ to the Coulomb potential (if q is sufficiently small). Therefore, if the electron w~s spinless, the correction to the conductivity would depend neither on ~/P~(i.e. on F) nor on any other coupling constant (e.g. due to the exchange by virtual phonon ). Since the electron has a spin, it is sensible to classify various contributions to the conductivity according to the st~mary spin of the interacting electron and hole. Being averaged over the random potential the two-particle Green function becomes invariant under the rotation in the spin space even in the presence of the interaction or spin-dependent scattering. This fact makes total spin j and its projection good quantum numbers to characterize the ~veraged two particle Green function, Just as the invariance of this propagator under the real space shift makes the summary momentum to be the same il. each intermediate state. Note that the sum of the electron and hole momenta is equivalent to the m o m e n t u m transfer q in eqs.(2)-(6). The contributions of the interaction with different j and ~ are additive. There is one point in which j=S and j=O contributions are different: at j=1 there can be no intermediate state consisting only of a single Coulomb propagator. Thus, the effective interaction amplitude at j=1 is not universal. Consider first the exact static amplitude u( j=1) (q ,CO=O) . There are no reasons for this quantity to depez~ on q significantly when q ~ 0 . Let us denote
Thus, the effective interaction potential U ( q , ~ ) for small enough q and ~J is universal-independent of the
X
F u(J=l)(q,c,3= O) =
•
2"~ d
(7)
Vol. 46, No. 6
THE ELECTRON-ELECTRON
At small ~ / ~ Coulomb interaction leads to 2 ~ d u(j=I)(~- O) equal to F calculated in refs./4, I0/. Later on we shall understand F in ~he sense of eq. (7). If c~ is not exactly zero but small enouy~h, the significant dependence of u(J=I) on q and ~ appears due to the diffusion pole in the intermediate state (see ref./9/). The ~eneral relation between U(j--O)(q,O) an~ UU:°)($~}cau be obtained from eqs. (3) and (4) by excluding Ud¢~q) dThe~ relation between u(j=1)(q,~) ~ is of the same
type
Thus, the j= I contribution absolute value can be three times larger than the one related to j=O (we can see from eqs. (5), (8) that u(J = O ) = U ( j =I) if F= 2) The relative value of the diffusion correction to any thermodynamic or transport quantity can be written as
(8)
F
=
U (j=1)(q,a,)
43]
INTERACTION EFFECTS IN DISORDERED METALS
- i ~ +Dq 2 =
~[,
--n
*I d/2 - I
(d =I ,3 )
c'r
(d =2 )
,
(11)
where The validity of eq.(8) is quite general and independent of the nature of the interaction or its strength. It is important to note that up to now we have not taken the Fermi-liquid modification of the one-particle electron propagator into account. This modification can be shown (see e.g. ref./11/) to lead only to the diffusion coefficient D renormalization. It is renormalized value of D that is connected with the conductivity by the Einstein relation
e2 D
~/4
where /4 and Nd
are the electron
chemical potential and d-dimensional density (e.g. N 2- the number of the conduction electrons per unit area). Eqs. (4)-(6) and (8) are valid only provided this renormalization is taken into account.
~jl~,
~P~)a
e# C
1
E* is either T or ~-quasiparticle e n e r ~ reckoned from the Fermi level. C t°i is the d-dimensional heat capacity
c(O )
~r 2
=
~d-- T 3
Ta~lel~ gives the values of ~(j=O) and ,Itj= for arbitrary F. Therefore, actually only one unknown constant F determines the diffUsion corrections to all quantities under consideration. Numerical factors 4 d) are listed -in Table 2. To obtain the values of ~(J) and J) we have to substitute the exact litudes eq.(5) or eq.(8) into the familiar expressions for the density of states correction /5,4/
Substituting eqs. (8),(5) into eq. (6) we obtain the expression for the correction to the conductivity which is valid for arbitrary F : (9) =
,,[(8/d
+
e2
T
) (d-2)/2
(12) "
and for the energy relaxation time
~-4 -9 1 /
(d=l) (d =2 )
(.0.915
(d=3)'
~J
where
r .32 ~(j
=1 )
F
(d=2) (lO)
1-411+,~,.E in (1
(13) Re-i
o;Dq
"
(1_~)d/2
=
;'~
a
T e e (~)
•
d
1 -~-
d(7-2) .....
.
F/2)]
(d=2)
In two dimensions this result had been obtained in r e ~ / ~ ' / . The factor 3 in eq.(9) is due to the multiplicity of the j=1 state.
TO determine ~(J ) we have to calculate the correction to the thermodynamic potential of the sample~.The renormalization of the potential is not very easy in this case : when one calculates ~ the additional factor I/n appears in the n-th order of the perturbation theory /12/. We can avoid this difficulty making use of the equality
432
THE ELECTRON-ELECTRON INTERACTION EFFECTS IN DISORDERED METALS Table 1.
Values of the constants
~(j=O)
and
Vol. 46, No. 6
~ x (j=l)iF) (j =0 )
~x 32
IlFD-" 114 F - (I - F / 2 ) d'2]/'1
d=l ,3
d (d-2)
%
,,,
s
(j=1) = -26_
T
-4 -
8(1 - F / 2 ) l n
4
-I( 1 -
d - 2
2 ].n (1
(1 - F / 2 ) (d-2 )/2
F/2)
(2)
d=2 "I
-
11
d=l
-F/2)
,3
4 =
(2)
d=3 ( v d j=O) - ~~(j= I) (2,,U
d=2
d=1,2
4/d
o"c/c ~" m
[(I
-F/2) d/2
- I] ,
4~
4F
£~e
(~
Table 2 ~X
£
~-d
-
(1-
-
712) d/2]
- coefficients
a(d)
d=2
4. 91 8yT2
d=3
0.~.,91..._!5 8JT2
I
~
Fig.
8~)d
1
,~'C
-o.o
GZ'~I('@ )
8JT
1
1
I
0.018
1 62)-
16 2 " 2
-&£,
Nd =
d4-= J 7 : 1 7 ( 2 )
d (4.-F)
d=l
~Sd
[I
J
I/Vd ~/v
'
(14)
where v~ is the volume of the sample, v 2 - th~ film area, v I - the length of the wire. Unlike the thermodynamic potential, the concentration N d can be calculated making use of the usual diagram technique /12/. Typical graphs for Nd are given in fig. I. Tile insertion of the vertex leads to the derivation of the corresponding electron propagator o v e r / W The sum of all the possible insertions into diagrams for F reduces to t h e ~ - d e r i v a t i v e of this value :
I.
Typical diagrams for the elect~on density coz~cection. Solid lines electron propagators. Dashed lines denote the averagis4; over random po tent ia 1.
= ~/2 9 - - 7 " " Summing the diagrams of the fig. f type in the Matsubara technique /12/ and then substituting the real frequencies for the imaginary ones we obtain
o" (j=1
= -3
i
cth . . . .
2T
-i~(1
- F/2)
(dq)ln
+
The substituion (15) gives the ~(j=l)~
+ Oq 2
-
(15) "
of eq.(14) into eq. correction to ~ : 3
=
Vd
9/
~
- --
9~
d~
I
(dq) •
exp~7/T - I 0
(16) •
rc tg
q2
D
a
rc tg
-- --
Dq
.
Vol. 46, No. 6
THE ELECTRON-ELECTRON INTERACTION EFFECTS IN DISORDERED METALS
Using eq. (16) ~ne can calculate any thermodynamical quantity. In particular, the heat ca pa city is C
=
T
(17) T~
vd
"
lead to the expression Eqs. (16), (17) lis%ed in Table 1. for @C (J=I)(F) ~j--O) can be calculated by the same me thod. To calculate the spin susceptibility we have to examine the magnetic field H influence on the thermodynamic potential. At j=O the diffusion pole is modified neither by the magnetic field nor by the spin scattering of the electrons. This fact is due to the conservation of the number of electrons w i t h a given energy. This is not the case at j=l. The general expression for the diffusion pole with fixed J and M is /13, 6/ : (18) D ( J,
(q ,~) = [-i~+Dq 2+ i~_ _ i IvlO~s]'£ t s
where~
s = g~sH
is
the
Zeeman split-
ting. t~ -' the total electron spin relaxat$o~ Sims /13/. Thus, to calculate ~ L D = t ) ~ at ts ÷ ~ w e must substitute (c~+~)for ~ in the arguments of arctg and include the summation over ~i instead of the factor 3 in eq. (16). After this we obtain (see refs. /6,7/). Strong magnetic field ( C O s ~ T , ~ / ~ g ) suppresses the ~i = ~I contributions to ~ , C and Y-e~ ~ d , ,hence, ~ s u l t s in s u b ~ i t u t i n ~ (~J---U) + ~ k J = ~ ) ) f o r ( ~ao=O> + 3 ~(~=1)) in eq. (11). The corresponding contribution to magnetoresistance was considered in ref. /14/ (in the first order in F). We see that the effect is determined by ~(.j=17 for arbitrary ~. The density of states ~ energy dependence (and, hence, the tunnel conductivity dependence on the V /5/)is much more i~teresting /13/. If the magnetic field is strong enoug~ two new singularities a t e = + g O (eV = _+ ~Js) appear i n @ ( ~ ) in addition t o ~ = 0 (V = 07 one. ~br O < F < 2 there is a minimum at @ = 0 which becomes deeper as H increasing, w~ile at ~c = -+O0s . (~) has maxima. The ~alue and the s h a p e o f ~ h e s e maxima are dete~nined by ~!4 =' (i.e. F) a~Id t s. But the energ~ (voltage) distance between them is equal to 2 ~ s = 2g~4gH and depends on the electron g-factor only. Note that the condition c ~ ~
433
these two electrons is n e a r the Fermi level ~ 0 while for the other 6 c ~ _~-2~ s ( ~ = -+I/2 denotes its spin projection on the H direction). Under these conditions the value of these two electrons momenta are almost equal. Consider now the role of the spin (spin-spin or spin-orbit) scattering. It can be seen from eq.(18) that this scattering if strong enough (~q/ts~'> t~, ~S. ) suppresses all j=l contribuions (in particular, Zeeman singularities i n ~ ( ~ ) function). Thus we have to substitute R (j=O) for ( ~ ( j = O ) + 3 R ( J = l ! ) ^ i n eq. (11). Nearly all constants ~ ( j = u J are P-i~de~endent. The only exception is ~3= ) at d E 2. Since ~ ( j = I ) ( F ) - ~ -~o~wnen F -~ 2 at d g 2 ( s e e T a b l e I), we can not neglect u~ity^in the eq.(3) j - u) denominator while ~.~Jcalculating. Thus ~(J=O)is different in various cases /4,15/ and much more than unity, Therefore all the diffusion channel contributions are universal (except ~(~) at d ~ 2 ) provided t s is small
enough,
T~is fact simplifies essentially the scaling theory of the metal-insulator transition (IVET) in the interacting disordered electron system. If we assume a scaling theory exists, at the strong spin scattering we must include one scaliz4~ variable - the d~menslone ~ s sample conductance G = ~ ~/e2R. The scaling equation can be written in the same form as the one in the pure localization theory /I/: ~ln
G
f~ (G).
(197
in L
L being the sample size. The localization of the electrons means that ]9 (G)~ -~ln G - ~ - ~ at G ~ 0 . If there are localized spins in the system, the corresponding electron scattering suppresses all the Cooper channel effects (in particular weak localization corrections) /16/ as well as j=1 diffusion contributions. Thus the total correction to the classical conductance G O is determined by the interaction in the diffusion~channel at j=O. Substituting ~ D / L ~ for E* in eq.(11) we see that at G > ~ 1 @ (G) = d-2- I/G. Thus, only insulator phase takes place at d=1,2 ( B < O at any G). The localization length ~ is just the same as the one -in the pure localizati0n theory without spin scattering. E.g., at d=2 ~"
1o " 1 exp ( ~ 2
(1 is the m e a n free path). Note that spin-spin scattering leads to ~o ~" ~-1 exp (G O ) ~ > ~ o if the interaction
434
THE ELECTRON-ELECTRON
INTERACTION EFFECTS IN DISORDERED METALS
is absent /17/. Therefore, interaction results in the enormous d e c r e a s i n K o f the localization length at PF2 l a > > ~ 2 . In three dimensions there is unstable fixed point (since ~ > 0 at G > ~ I and ~ ~ 0 for small G). So ~IT takes place without an~ conductivity jump. These features are the same as in the pure localization theory w i t h only the potential electron scattering. Nevertheless, there are no reasons for the correlation length in these two cases to be equal. At finite T scaling h~pothesis means that in the metal near the ~IT conductivity can be written as (~ = yr2 &
• f
(
),
(21)
where L T = D~/T. f ( x ) = 1 + Ax a t x<~ 1 (A~I). Thns, if T < ~ D / ~ z the t e ~ e r a t u r e dependent part df the conductivity is proportional to In the critical region the main term in ~ 5 should be ~ - i n d e p e n d e n t ; e2 (B ~ 1) (22)
Using the ~ n s t e i n
%-D=
e2
Now let the magnetic impurities to be absent while the spin-orbit scattering is strong. Such a situation takes place e.g. in heavy metals or in p-type cubic semiconductors. The interaction effects in the Cooper channel neglected, there are two contributions to the conductivity: the one due to the diffusion channel interaction at j=O and the weak localization one. The latter ~ s the additional factor (-I/2) as compared with the potentials scattering c~#_#___, /19/. Hence, at L T ~ L ~ I ~ o ---~D-Cso and G~ I
#=
d-2-
1/2Q,
(while without interaction ~ = d-2+ + I/2G). In fig. 2 the solid line is the graph of ~ (G~ function (dashed line is the same function without inte ract ion).
I
"~
d=3
relation we obtain
(.2
1 TI/3 (B
( 23 ~ /N~.... )2/3
(24)
The same index 1/3 for the frequency dependence of the conductivity can be obtained in the same way. Note that for the noninteracting electzons the same frequency dependence was obtained earlier /18/. As to the one-particle excitations decay t i ~ , substituting eq~ (24) ~into eq.(11) gives at ~ * = T ~ D/~: ~ - - 1 (T) ee
Vol. 46, No. 6
= bT
(b ~-1)
(25)
Thus, the condition T T ^ ^ ~ becomes invalid and the P e r m i - l ~ u i d quasiparticle description fails in the ..... critical region. IT and Lin =~fD--A-ee become of the same order. Therefore, the only temperature-dependent length scale in the critical region is Note that due to the Hubbard repulsion and non-homogeneous impurity distribution in semiconductors there mast appear the states near the mobility edge which are occupied by a single e l e c t r o n - i.e. localized spins. Thus, in semiconductors the presented theory must describe the general case.
Fig.
2.
Gell-Kvann - ~ o w function ~ (G) in the spin-orbit case. D a s h e @ l i n e s - pure localization theory. Solid lines correspond to the interacting el ec tr ons.
We see that in two dimensions only the i n ~ l a t o r phase takes place(unlike the noninteracting electrons). The localization length is larger than defined by eq. (20) : ~ .~ ~oL/1. At d=3 the conductivity in the critical region is determined by eq. (23) as before. In su~mnar2 we have shown that either thermodynamic or transport properties of the disordered metalic systems are determined by only one Fermi-liquid constant F, which can be obtained independently from a variety of experiments Cooper channel interaction if taken into account leads to one constant more. Therefore, there are no
Vol. 46, No. 6
THE ELECTRON-ELECTRON INTERACTION EFFECTS IN DISORDERED METALS
four independent constants, as it was argued in ref. /20/ but only two of them. If spin scattering is strong enough, all the diffusion corrections become universal. This fact have been used to construct the one-parameter scaling theory of metal-insulator transition. The simplification of the metal-insulator transition problem in the strong spin scattering case is evidently due to the possibility to Refer 1. 2. 3.
4. 5.
6.
E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys.Rev.Lett., 42, 673 (1979). P.W. Anderson, E . ~ r a h a m s and T.V. Ramakrishnan, Phys.Rev.Lett. , 43, 718 (1979). L.P. Gor' kov, A.I.Larkin and D.E. Khmelnitskii, JETP Letters, 30, 328 (1979) (Pis'ma Zh. Eksper.Teor.Fiz., , 248 (1979)). L.Altshuler, A.G.Aronov and P.A. Lee, Phys. Rev. Lett., ~ , 1288(1980) B.L. Altshuler and A.G. Aronov, Solid State Comn., 36, 115 (1979) and Zh. Eksp.Teor. Fiz., 77, 2028 (1979)(Sov. Phys. JETP 50, 9 ~ (1979)). B.L. Altshuler, A.G.Aronov and A. Yu.Zuzin, Pis'ma Zh. Eksp.Teor. Fi z.('935') 1 5S2) 1 (1982) (JLTP Lett.,35, and Zh. Eksp.Teor. Fiz., 16 ' No4 (1983) (Sov. Phys.JETP (1983) be published). H" ' J~uku~ama 07 (31 98 1' ) J" . 4 Phys "S°c " J a p " __50,
~.
~o
7. 8. 9.
A. Schmid, Z. physik, 271, 251 (1974). B.L. Altshuler and A.G. Aronov, Pis'ma Zh.Eksper. Teor. Fiz., 30, 514(1979) (J~.TP Letters, /Q, 482(1979)) and Solid St .Comm. ,--~8, 11 (1981). 10. T.F. Rosenbaum, K~'Andres, G.A.Thomas and P.A. Lee, Phys.Rev.Lett. ,46, 568 (1981).
435
separate the metal-insulator transition from the attendamt magnetic transitions in this case. Acknowledgement - We would like to thank Drs. A,M.Finkelshtein and A. Yu.Zuzin for numerous and helpful discussions. We are also grateful to E.M. Pavlenko and G.V.Stepanova for the helping in the preparing the manusc ri pt. ences 11. A.~.Finkelshtein. Zh.Eksp. Teor°Fiz., ' No2 (1983) (Sov. Phys.J•TP (1983) be published). 12. A.A. Abrikosov, L.P.Gor'kov and I.E. Dzyaloshinskii, ~ t h o d s of Quantum Field Theory in Statistical Physics, Pergamon, N.Y. (1975). 13. B.L. Altshuler and A.G. Aronov, Pis'ma Zh. Eksp.Teor.Fiz., ~ ~o2(1983). 14. P.A. Lee and T.V. Ramakrlshnau, Phys. Rev. B 26, 4009 (1982) 15. A. Y u . Z ~ - ~ , Pis'ma Zh.~sp.Teor. Fiz. , 377 (1981)(J~TP Letters, 33,360, 981) ). 16. P.A. Lee, J.Non.Crys. Sol.,35, 21
~o
~
--(1980).
17. K.B. Efetov, Zh.Eksp. Teor. Fiz., 83, 833 (1982). 18. F. Wegner, Z. Physik, B_~, 207 (1979); W. Gotze, Philos.~ag., 43, 219(1981) B. Shapiro and E. Abrahams, Phys.Rev.,
B24, 4889 ( 1981).
19. S..Hikami, A.I. Iarkin and Y.Nagaokm, Progress The or.Phy s. Le tt .63,707 (1980). 20. H. Fukuyama, Surface Sci.,1~, 489
--(1982).