matrix in the kondo problem

16
188 FERROMAGNETIC HEAT CAPACITY NEAR CRITICAL POINT 947 However, /dM\ (dH,/dT)u (dH/dT)M (_) = = _, (A10) \dT/ He (dH e /dM) T (6H/dM) T +D where the second equality utilizes (2.6). Finally, the Maxwell relation 4 (6S/dM) T =- (dH/dT) M (All) completes the derivation. I. INTRODUCTION I N this paper, we study numerically the low-temper- ature anomalies of dilute magnetic alloys. The s-d exchange model (or Kondo Hamiltonian) is widely ac- cepted as a reasonable description of the interaction between conduction electrons and the localized mag- netic moments of the impurity ions. This model Hamiltonian consists of a simple contact interaction between the impurity spin S/ n,p and the electron spin density at the position of the impurity: / H. d = H kin L S/^-s^Ry). (1.1) N i Here H kin is the kinetic energy of the noninteracting * Research sponsored by research grants from the National Research Council of Canada. f Present address: Department of Physics, Simon Fraser Uni- versity, Burnaby, B. C, Canada. % Present address: Institut Max von Laue-Paul Langevin, Aussenstejle Garching, 8046 Garching, Germany. In computing C e it is convenient, given H ei to first choose M, then evaluate X = TM~ 2 by combining (2.6) and (Al), and express the right-hand side of (A7) in terms of x and M. If D = D/V is set equal to zero in (A7), C e is identical to C//. It is clear from (A4) that T~ l C M satisfies a "scaling" relation. The same is true of T~ l C H . It can, in fact, easily be shown that T -i CH = Co(T)+f(r\H\-^)-\\n\H\ (A12) for an appropriate function f(x) and constant A. electron system, N is the number of atomic cells, and J is the coupling constant. We shall consider only the case of antiferromagnetic coupling (/>0). In the course of explaining the resistance minimum of dilute magnetic alloys on the basis of the s-d exchange Hamiltonian, Kondo 1 discovered that in perturbation theory the one-electron scattering amplitude has a logarithmic singularity at small temperatures and ener- gies. Various nonperturbational methods 2-7 have since been developed to explain quantitatively the anomalous behavior of the physical properties of dilute magnetic alloys. Of particular interest to us are the methods of 1 J. Kondo, Progr. Theoret. Phys. (Kyoto) 32, 37 (1964). 2 H. Suhl, Phys. Rev. 138, A515 (1965); 141, 483 (1966); Physics 2, 39 (1965). 3 A. A. Abrikosov, Physics 2, 5 (1965). 4 Y. Nagaoka, Phys. Rev. 138, All 12 (1965); Progr. Theoret. Phys. (Kyoto) 36, 875 (1966). 5 J. Appelbaum and J. Kondo, Phys. Rev. Letters 19, 906 (1967); Phys. Rev. 170, 542 (1968). 6 K. Yosida, Phys. Rev. 147, 223 (1966); Progr. Theoret. Phys. (Kyoto) 36, 875 (1966). 7 A. Okiji, Progr. Theoret. Phys. (Kyoto) 36, 712 (1966). PHYSICAL REVIEW VOLUME 188, NUMBER 2 10 DECEMBER 1969 Numerical Study of the t Matrix in the Kondo Problem* R. K. M. CHOwf AND H. U. EVERTS! Department of Physics, University of Toronto, Toronto, Canada (Received 7 July 1969) A numerical study of the Kondo problem is presented. The calculations are based on the Suhl-Abrikosov- Nagaoka integral equation for the scattering amplitude t(a},T) of the s-d exchange Hamiltonian. Use is made of the exact analytic solution first given in detail by Zittartz and Muller-Hartmann. It is shown that, because of the resonance in Imt(oj,T) which occurs at the Fermi energy (w = 0) at low temperatures, the tunneling density of states of a dilute paramagnetic alloy is very slightly reduced at zero bias voltage. There is, however, a possibility of detecting this change by studying the derivative of the conductance. The details of the Zittartz-Miiller-Hartmann expression are found to be unimportant for the low-tem- perature behavior of the transport coefficients with the exception of the thermoelectric power. A rein- vestigation of the thermoelectric power shows that some care is necessary in the evaluation of the integrals K n —S-^dc0(*) n no(a))T(c0,T)df/dco, because of the strong w dependence of the electronic lifetime T(O),T) a [Imt(co,T)~]~ 1 at low temperatures. While the transport coefficients reflect the behavior of the scattering amplitude in a small energy interval about the Fermi energy, the specific-heat anomaly is found to be related to the temperature derivative of t(o),T) at large values of the energy variable w comparable to the bandwidth. We also point out that the quasiparticle approximation is not valid for electrons interacting with impurity spins due to the rapid variation of t(w,T) near the Fermi energy.

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Page 1: Matrix in the Kondo Problem

188 F E R R O M A G N E T I C H E A T C A P A C I T Y N E A R C R I T I C A L P O I N T 947

However,

/dM\ (dH,/dT)u (dH/dT)M ( _ ) = = _ , (A10) \dT/He (dHe/dM)T (6H/dM)T+D

where the second equality utilizes (2.6). Finally, the Maxwell relation4

(6S/dM)T=- (dH/dT)M (All)

completes the derivation.

I. INTRODUCTION

IN this paper, we study numerically the low-temper­ature anomalies of dilute magnetic alloys. The s-d

exchange model (or Kondo Hamiltonian) is widely ac­cepted as a reasonable description of the interaction between conduction electrons and the localized mag­netic moments of the impurity ions. This model Hamiltonian consists of a simple contact interaction between the impurity spin S/n , p and the electron spin density at the position of the impurity:

/ H.d = Hkin L S / ^ - s ^ R y ) . (1.1)

N i

H e r e Hkin is the k ine t ic energy of the nonin te rac t ing

* Research sponsored by research grants from the National Research Council of Canada.

f Present address: Department of Physics, Simon Fraser Uni­versity, Burnaby, B. C , Canada.

% Present address: Institut Max von Laue-Paul Langevin, Aussenstejle Garching, 8046 Garching, Germany.

In computing Ce it is convenient, given Hei to first choose M, then evaluate X = TM~2 by combining (2.6) and (Al), and express the right-hand side of (A7) in terms of x and M. If D = D/V is set equal to zero in (A7), Ce is identical to C//.

It is clear from (A4) that T~lCM satisfies a "scaling" relation. The same is true of T~lCH. I t can, in fact, easily be shown that

T-iCH = Co(T)+f(r\H\-^)-\\n\H\ (A12)

for an appropriate function f(x) and constant A.

electron system, N is the number of atomic cells, and J is the coupling constant. We shall consider only the case of antiferromagnetic coupling ( / > 0 ) .

In the course of explaining the resistance minimum of dilute magnetic alloys on the basis of the s-d exchange Hamiltonian, Kondo1 discovered that in perturbation theory the one-electron scattering amplitude has a logarithmic singularity at small temperatures and ener­gies. Various nonperturbational methods2-7 have since been developed to explain quantitatively the anomalous behavior of the physical properties of dilute magnetic alloys. Of particular interest to us are the methods of

1 J. Kondo, Progr. Theoret. Phys. (Kyoto) 32, 37 (1964). 2 H . Suhl, Phys. Rev. 138, A515 (1965); 141, 483 (1966);

Physics 2, 39 (1965). 3 A. A. Abrikosov, Physics 2, 5 (1965). 4 Y. Nagaoka, Phys. Rev. 138, Al l 12 (1965); Progr. Theoret.

Phys. (Kyoto) 36, 875 (1966). 5 J. Appelbaum and J. Kondo, Phys. Rev. Letters 19, 906

(1967); Phys. Rev. 170, 542 (1968). 6 K. Yosida, Phys. Rev. 147, 223 (1966); Progr. Theoret. Phys.

(Kyoto) 36, 875 (1966). 7 A. Okiji, Progr. Theoret. Phys. (Kyoto) 36, 712 (1966).

P H Y S I C A L R E V I E W V O L U M E 1 8 8 , N U M B E R 2 1 0 D E C E M B E R 1 9 6 9

Numerical Study of the t Matrix in the Kondo Problem*

R. K. M. CHOwf AND H. U. EVERTS!

Department of Physics, University of Toronto, Toronto, Canada (Received 7 July 1969)

A numerical study of the Kondo problem is presented. The calculations are based on the Suhl-Abrikosov-Nagaoka integral equation for the scattering amplitude t(a},T) of the s-d exchange Hamiltonian. Use is made of the exact analytic solution first given in detail by Zittartz and Muller-Hartmann. I t is shown that, because of the resonance in Imt(oj,T) which occurs at the Fermi energy (w = 0) at low temperatures, the tunneling density of states of a dilute paramagnetic alloy is very slightly reduced at zero bias voltage. There is, however, a possibility of detecting this change by studying the derivative of the conductance. The details of the Zittartz-Miiller-Hartmann expression are found to be unimportant for the low-tem­perature behavior of the transport coefficients with the exception of the thermoelectric power. A rein­vestigation of the thermoelectric power shows that some care is necessary in the evaluation of the integrals Kn—S-^dc0(*)nno(a))T(c0,T)df/dco, because of the strong w dependence of the electronic lifetime T(O),T) a [Imt(co,T)~]~1 at low temperatures. While the transport coefficients reflect the behavior of the scattering amplitude in a small energy interval about the Fermi energy, the specific-heat anomaly is found to be related to the temperature derivative of t(o),T) at large values of the energy variable w comparable to the bandwidth. We also point out that the quasiparticle approximation is not valid for electrons interacting with impurity spins due to the rapid variation of t(w,T) near the Fermi energy.

Page 2: Matrix in the Kondo Problem

948 R . K . M . C H O W A N 1) H . U . E V E R T S 188

Suhl,2 Abrikosov,'* and Nagaoka.4 Suhl applied disper­sion relation technqiues to the s-d scattering problem and obtained a set of two coupled nonsingular integral equations for the non-spin-flip and the spin-flip parts of the electron scattering amplitude. Essentially the same equations were obtained by Abrikosov, who used a graphical technique to sum up the most singular terms in the perturbation series for the scattering ampli­tude. The complete equivalence of the Suhl and Abrikosov methods has been proved by several au­thors.8,9 Nagaoka considered the hierarchy of equations of motion starting with the equation of motion for the one-electron Greeks functions. He decoupled the higher-order Green's function so that a closed set of equations for the first two Green's functions was obtained. Later on, Falk and Fowler10 and, independently, Hamann11

showed that Nagoaka's equations could be cast into the form of a single nonlinear singular integral equation for the non-spin-flip scattering amplitude. The solution of this integral equation is due to Hamann,11 Bloomfield and Hamann,12 and Zittartz and Miiller-Hartmann (ZMH).13 Hamann, as well as Hamann and Bloomfield, truncated the integral equation by neglecting the regular contributions to the kernel as small compared to the singular ones. Their solutions for the t matrix are ac­curate for small energies and thus quite sufficient to explain the anomalies in those physical quantities which only depend on the scattering amplitude for small values of the energy. The exact solution of Hamann's integral equation is due to ZMH.

In Sec. I I we discuss the relation between the / matrix and the one-electron spectral density and the density of states. In Sec. I l l , we present the results of a detailed numerical study of the ZMH analytic expres­sion for the scattering amplitude. Section IV is devoted to the question as to how far the details of this expres­sion are of importance in the computation of measur­able physical quantities.

II. SPECTRAL DENSITY AND DENSITY OF STATES

The spectral density of one-electron states in an alloy is given by

Palloy(k,w)

1 - ImS(k ,« ) = . (2.1)

w [^-ek-Re2(k,aj)]2+[ImS(k,c«;)]2

Here ek is the kinetic energy of our electron with mo­mentum k. The effect of interactions on the spectral

8 S. D. Silverstein and C. B. Duke, Phys. Rev. 161, 456 (1967). 9 H. Keiter, Z. Physik 214, 22 (1968). 10 D. S. Falk and M. Fowler, Phys. Rev. 158, 567 (1967). 11 D. R. Hamann, Phys. Rev. 158, 570 (1967). 12 P. E. Bloomfield and D. R. Hamann, Phvs. Rev. 164, 856

(1967). 13 J. Zittartz and E. Miiller-Hartmann, Z. Physik 212, 380

(1968).

density is entirely contained in the self-energy function S(k,w) which, in general, may be split into three contributions:

S(k,a>) = Sel .e l (k,w) + Sel.ph(k,C0) + S e l - i m p ( k , « ) . ( 2 .2 )

We shall neglect the first two terms which result from electron-electron and electron-phonon interactions since they are also present in the pure host material. In so doing, we assume that 2ei-ei and 2ei-Ph are the same for the pure host material and for the alloy. This assump­tion is quite generally made for low concentration of impurities although it is hard to justify rigorously. One may incorporate the real parts of the first two terms on the right-hand side of (2.2) by interpreting ek to be the renormalized electronic energy. Near the Fermi surface, such effects may be simply expressed in terms of an effective mass m*, i.e., ek = &2/2w*.

To lowest order in the impurity concentration, Sei-imp is given by

S e l - imp(k,a>)=Wi/k k(co) , (2 .3 )

where Ui is the impurity concentration and /kk(co) is the one-electron non-spin-flip forward-scattering amplitude. Since the interaction which we are considering is a con­tact interaction, the scattering amplitude and hence the self-energy are independent of the momentum k of the electron

*kk(w) = /(o>), 2ei-imp(k,a;) = 2(a>). (2.4)

This fact makes it especially easy to compute the density of states of the alloy. In terms of the spectral density, the density of states can be written as

1 W(co) = W0(w)H X) Palloy(k,C0)

X k

= »o(co)+ / dek?io(ek)paiioy(k,a>), (2.5)

where w0(ek) is the density of states of the host metal. We choose n0(ek) to be a Lorentzian, namely,

no(ek) = noD2/(ek* + D2). (2.6)

The reason for this choice is that the ZMH solution for the / matrix, which we shall use in our later work, is also based on this bare density-of-states function. no=no(ek=0) is the density of states at the Fermi energy (all energies are measured relative to the Fermi energy €*•). From the exact sum rule,

/ W 0 ( € k ) d € k = l , J —oo

we see that »o=lAZ> (2.7)

in the case of the Lorentzian band model.14

14 «0 is the density of states per unit cell. To obtain the total density of states, one has to multiply by the number of atoms in the system.

Page 3: Matrix in the Kondo Problem

188 / M A T R I X I N T H E K O N D O P R O B L E M 949

Using (2.3), (2.4), and (2.6) in the evaluation of (2.5), one finds that the change in the density of states is given by

An(<t),T) = n((i),T)— n0(oo)

rw

= Wi«02(<o) 2—Re/ret(aj,r)

L D

+(i—^Jim/ret(«,r) , (2.8)

where tret(a),T) is the retarded forward-scattering ampli­tude. For energies well within the band (|co|<3CD), this result simplifies to

An(co) = fiitio2 Imtret(o),T). (2.9)

I t is well known that the imaginary part of the scatter­ing amplitude is bounded by the unitarity limit

- Im/ r e t (« , r )^ l / i rWo, (2.10)

and hence we see that15

An(w,T)

n0

(2.11)

The relative change in the density of states is less than the concentration of impurities. For values of the con­centration of the order of 10~3 which are common for dilute magnetic alloys, it would therefore be difficult to observe An(oo)/n0 in a tunneling experiment. However, one might try to do experiments with magnetic impurity concentration of order 10~2 as long as one has a sufficient concentration of nonmagnetic impurities which would destroy the long-range Ruderman-Kittel-Kasuya-Yosida (RKKY) polarization effects that induce im­purity spin-impurity spin coupling. Moreover, there might be a chance of observing the derivative (d/doo) An(ooyT), which is large because of the rapid variation of A»(co) with energy.16 We shall come back to this point in Sec. IV A.

III. NUMERICAL RESULTS FOR NON-SPIN-FLIP t MATRIX

In this section we shall discuss the results for the scattering amplitude tvct(o),T) which we have obtained by evaluating numerically the analytic expression for t(co,T) as given by ZMH.17 I t is evident from Sec. I I that this provides all the information needed to deter­mine the one-particle excitation spectrum as well as the density-of-states function.

18 A. Griffiin, in Lectures Notes of McGill Summer School on Superconductivity, edited by P. R. Wallace (Gordon and Breach, Science Publishers, Inc., New York, to be published).

16 This was suggested by A. Griffin. 17 A similar analysis of Bloomfield-Hamann solution for t(o),T)

is contained in a recent paper by B. N. Ganguly and C. S. Shastry, Phys. Rev. (to be published).

ZMH were able to obtain an exact solution of the approximate Suhl-Abrikosov-Nagaoka integral equa­tion for t(co,T) under the assumption that the density-of-states function of the pure metal is a Lorentzian [see Eq. (2.6)]. For convenience, we state their solution for / re t(w,r) here18

1 / X(co,T) \ tret(a>,T) = ( 1 —«"<«.*•>). (3.1)

2xi»0(«)\ [_K{o>,T)J^ /

The functions X(u,T) and K(co,T) are defined as follows:

X(co,r)^l-(7r7)2Po(co)[i5(5+l)+^(T)]+TPo(co)

x W H - W ( f - i - j l , (3.2a)

p0(o,)=Z)2/(a,2+Z)2), (3.2b)

K(»,T)= 1 X(uyT) | 2 + [ 7 T 7 P O ( " ) ] 2 S ( S + 1 ) . (3.3)

Here \f/(z) denotes the digamma function and y=Jno is a dimensionless coupling constant which is assumed to be much smaller than unity. The function 0(o>,r), which we will henceforth refer to as the phase, is given by a principal-value integral

2TT

lnK(o,T) -dJ. (3.4)

The function A(T) which enters K(oo,T) via X(o),T) has to be determined self-consistently from

1 r00 u-2iD A(T)= / dw(tanh£#o) /ret(-o>). (3.5a)

2iiry J-x (u—iD)2

For convenience, we shall use the alternative condition obtained by ZMH to determine A(T),

M0(A,T)-= \nK(a,T)du=0. (3.5b)

The physical meaning of A (T) becomes apparent from the exact result

A(T)= (2TD\y\)-i(AEkin-Eint), (3.6)

which relates it to the difference between the incre­mental kinetic energy of the electrons,

A F i • = Fi • a l l°y__ 77, . host metal

and to the interaction energy. The following general features of / r e t(0,r) follow

immediately from (3.1):

18 /ret as given here differs from the corresponding quantity in the paper of ZMH by a factor of 7, i.e., *ret(co,r) =//re t

Z M H(w,r).

Page 4: Matrix in the Kondo Problem

950 R. K. M . C H O W A N D H . U . E V E R T S 188

(a) If the density-of-state function of the host material w0(o>) is an even function of co, then

Im/ r e t («,r) = I m / r e t ( - « , r ) , (3.7a)

Re/ret(a>,F) = - R e / r e t ( - ^ r ) . (3.7b)

This applies, of course, to our case since w0(co) is a Lorentzian.

(b) Im/ r e t («,r) ^ 0 for all co and T. (c) Making use of the asymptotic form of the func­

tion X(co,T) for r = 0 , namely,

lim X(a,T) = 1 - (7r 7 ) 2 Po(a J ) [ i5(5+l)+ .4]

+7Po(co)(In| D/co| +i\ir sgnw)

and of the fact that

6(u> = 0,T) = 0, (3.8)

one can easily prove the inequality

- T m / r e t ( ^ r ) ^ - T m / r e t ( 0 , 0 ) = Tr-Kunitarity limit). (3.9)

Furthermore, for arbitrary temperature, X(o>=0, T) takes the simple form

x(«=o , r ) = - 7 i n ( r / r K ) , (3.10)

where the Kondo temperature TK is defined by

TK=(D/kB)exp((l/y){l-(*ymS(S+l)+Al}), (3.11)

with D= (2/w)D exp(7BuI„) = 1.13D.

Here, .4(2") is treated as a temperature-independent constant for reasons which will become clear later on. Because of (3.8), using (3.10), we find that

/ret(w=0, T)

1 / \n(T/TK) \ = ( 1 J. (3.12)

2i7rn0\ {[}n(T/TK)J+T'S(S+l)y^/

The remarkable feature of this simple formula is that the right-hand side depends on the coupling constant only through the ratio T^T/TK- Thus TK completely determines the temperature scale on which /ret(a>=0, T) varies.

(d) A more general statement can be made in the weak coupling limit ( |T|—>0). It has been shown by Mliller-Hartmann19 (MH) that, in this limit, the scat­tering amplitude approaches a function which is in­dependent of the coupling constant y. This obtains for fixed values of the dimensionless temperature r and the dimensionless energy variable e=cx)/kBTK. Moreover, this function does not depend on the specific form of the

19 E. Muller-Hartmann, Z. Physik. 223, 267 (1969).

density-of-states function of the host metal w0(w). In mathematical terms, if we define

F ( 6 , r ) ^ l i m [ - ( l / 7 ) X ( c o , r ) ]

= l n r + ^ ( i - i e / 2 i r r ) - ^ ( i ) , (3.13)

then tret(e,r) is given by

/ret ( ( ) )(wJr)=lim/ ret(aJ,r) 7 - *0

1 / Y(e,r) \ = ( i g»M«.r) , (3.14)

2iVn0\ [ ^ ( e , r ) ] 1 / 2 / where

Ky(e,r)= | F(e , r ) | 2 +7r 2 S(S+l) (3.15) and

1 r" \nKu(t',r) 0K(e,T)=—P dt' ; . (3.16)

The right-hand side of (3.14) is evidently independent of the coupling constant y and of the density-of-state function of the host metal n0(<*>). The important question of how well this simple limiting expression for /,et(e,r) approximates the scattering amplitude for finite but small values of y will be answered by our numerical results.

The analytic expression for the scattering amplitude is complicated a great deal by the presence of the uni-modular factor eid in the expression for tTet(o)yT) and by the fact that the function X(ca,T) depends on the func­tion A(T) which has to be determined self-consistently from Eq. (3.5). Neither 0(co,T) nor ^4(7") can be ex­pressed by simple analytic functions.

In deriving (3.11), we have treated -4(7") as a con­stant. This is justifiable since, as was pointed out by ZMH, the over-all magnitude of A(T) is small, being of the order of the coupling constant y. Thus for small enough 7, A can be neglected relative to the term J 5 ( 5 + l ) in (3.11). For small 7, one can obtain an asymptotic expansion for A by expanding the integrand in the Eq. (3.5b) in powers of 7. This yields

A(T)= - | 7 5 ( 5 + l ) [ l + 0 ( r / P ) ] + 0 ( 7 2 ) . (3.17)

For r = 0 , this expansion has been derived by ZMH. As one can see, the temperature dependence of the lowest-order contribution for A(T) is negligible for all physically reasonable temperatures.

The results of the numerical calculation of Eq. (3.5b) agree qualitatively with the expansion (3.17). To obtain the values of A{T) listed in Table I, we computed the integral

M0(A,T)= [ dalnKfaT) (3.18) J —00

for specific values of A. To find A, we then determined graphically the zero of M0(A,T) as a function of A.

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188 / M A T R I X I N T H E K O X D O P R O B L E M 951

TABLE I. The values of A obtained for two different values of the reduced coupling constant 7. See Eqs. (3.17) and (3.18) in text.

-0 .08 0.031 -0 .11 0.050

Within the numerical precision of this procedure, we could not detect any temperature variation in A over a temperature interval of 100°K.

Let us note here, that, as far as the scattering ampli­tude is concerned, A(T) is of no importance. In con­trast, however, it turns out that A(T) is crucial for the calculation of the specific heat of dilute magnetic alloys.

To compute 0(co,T) numerically, we use the expression

1 /•« co /K(u\T)\ 0 ( w , D = - P / du' ln( J, (3.19)

T J-x o/2-C02 \/C(«, T)f

which is identical to (3.4) since

/•- co P I </«' = 0

and K(»',T) = K(-o',T).

I t is obvious that the integral in (3.19) is easier to handle in a numerical integration than the integral in (3.4). The convergence of the latter depends on the delicate compensation of the contributions from large positive and large negative values of the integration variable a/.

FIG. 1. sin0 is plotted as a function of lne = ln(w/7V) for four values of the dimensionless temperature r = T/TK. See Eq. (3.4) in the text. We have taken 7=—0.11 which corresponds to TK = 9.84:°K. Subsequent figures involve the same dimensionless variables and the same value of 7.

FIG. 2. The real part of the scattering amplitude is shown as a function of Inc. The unimodular factor ei$ is included in the computation of these curves. See Eq. (3.20).

In Fig. 1 we have plotted sin0(e,r) as a function of e for four different values of the dimensionless tempera­ture r. The dimensionless coupling constant 7 was chosen to be 7 = —0.11. The curves for finite values of r and the one for r = 0 differ qualitatively. For the values of T7*0 for which we have computed 0(e,r), it approaches zero smoothly from above as e goes to zero. In contrast to that, it appears that 0(e, r = 0 ) goes to zero with a large negative slope if e goes to zero. We have not been able to do the numerical integration of (3.19) for values of e which are small enough so that the increase of 0(e, r = 0 ) becomes visible. The evaluation of (3.19) for small values of co at T= 0 is difficult since K(ai',T= 0) has a logarithmic singularity at a/ = 0.

I t is easy to see that 6y(e,T) defined in (3.16) is ob­tained from (3.19) if the Lorentzian po(co) which occurs in the function K(oo,T) [see (3.2) and (3.3)"] is replaced by unity. We have done this and found that the values of 6y(e,T) so obtained do not deviate significantly from those of 0(e,r). We conclude that 0y(e,T) is a very good approximation to 0(e,r) for values of | Y | as large as 0.11.

Figures 2 and 3 show the results of the numerical evaluation of

1 Re/ret(co,r) = #-1/2(a>,r)[ReX(a>,r) sin0(co,r)

27TWo(to)

+i7T7Po(w)tanhi/5co cos0(a>,r)] (3.20) and

1 Im/ r e t(«,r) = {l-K-vtfaT)

2irno(co)

X [ReX(o>, r)sin0(co, T) +i7T7Po(co)

Xtanh£/&> sin0(co,r)]}. (3.21)

[To obtain these expressions for Re, Im/ re t, we have used (3.2a) and the mathematical identity Im\p(%—ix)

Page 6: Matrix in the Kondo Problem

952 R . K . M . C H O W A N D H . U . E V E R T S 188

= — \tc tanh7nxf|. Re/ret and Im/ret are plotted as func­tions of the dimensionless energy variable e for various values of the dimensionless temperature r. The results are independent of the coupling constant | y | for values of | 7 | as large as 0.11. The resonance peak in the Im/ret(e,r) gets sharper with decreasing r, and its maxi­mum value approaches the unitarity limit [lm/ret(0,0) = — 1/w] as r goes to zero. At T= 0, the resonance peak (which has a horizontal tangent at € = 0 for all finite temperatures) degenerates into a cusp with an unde­fined derivative at e = 0:

lim - Im/ret(e, T=0) = =fcoo . (3.22) «-*±o de

P * 0.041-

This behavior can be derived from the following asymp­totic expression for /ret(€, r = 0 ) , given by M H :

Im/ret(e, 2irnQL

w2S(S+l) 2 + +0

2 In2 Mn'lel/J

e - ^ 0 . (3.23)

The singularity of Im/ret(e, r = 0 ) at e = 0 is reflected in Re/ret(€, T = 0 ) by an infinite derivative at € = 0.

To give a quantitative picture of the influence of the unimodular factor eie on Re/ret and Im/ret, we compare these functions in Figs. 4 and 5 to the real and ima­ginary parts of the function

! t^°(co,r)^[27rm0(co)]-J

x[i- -#-i/2(co,r)X(o>,r)], (3.24)

which is obtained from (3.1) if the unimodular factor is replaced by unity. We should emphasize that, at r = 0 , the imaginary part of the function defined in (3.24) has the same asymptotic behavior for small e as the imagi­nary part of the exact ZMH solution (3.23). As the graphs show, the imaginary parts of tTet

0~0 and of the exact solution (3.1) differ considerably only for e > l .

0 3

0.2

0.1

T = 0.25

-T a |

T=!0

1 1 -4

\ N

1 -2

\ \

1 1 0

l 2

1 1 4

FIG. 3. The imaginary part of the scattering amplitude is shown as a function of Inc. The unimodular factor eie is included in the computation of these curves. See Eq. (3.21).

FIG. 4. Comparison of Re/ret(e,r) with Re/retfl~0(e>r) for r = l. The dashed curve represents Re/ret

e=0(e,T). See Eq. (2.34).

For T— 1 the real parts of these two functions are also in fairly good agreement as long as e< 1. In view of the behavior of 0(e, r = 0 ) it is not surprising that, for r = 0 , the real parts of tre%

b=Q and of the exact solution differ both for e < l and for e > l . Nevertheless, the approxi­mation (3.24) which has a relatively simple analytic structure may be useful for the computation of low-temperature transport properties which only depend on the value of Im/ret(*,r) at small values of e.

IV. PHYSICAL QUANTITIES DIRECTLY RELATED TO THE SCATTERING AMPLITUDE

Having worked out quantitatively the details of ZMH's expression for the non-spin-flip scattering ampli­tude in Kondo's model, we now turn to the question of how much of the detailed structure is reflected in measurable physical quantities.

A. Tunneling Density of States

Since there is no way of measuring the spectral func­tion p&lloy(a),k) directly, the most detailed information about the scattering amplitude that one could hope to obtain would come from a tunneling experiment. The quantity which one probes in a tunneling experiment is the density-of-state function »(co). Let us, for simplicity, assume that the tunneling diode has the following structure: a pure metal on one side of an oxide layer and a dilute paramagnetic alloy of the same metal on the other side of the oxide layer. No paramagnetic ions will be assumed to be in the oxide layer itself. The conductance of such a diode can be expressed as

G(V)=dI(V)/dV

d r00

= 47r^( |T | 2)— / 6?cowmetal(w)walloy(co+eF) dV J-v

X[/ (a>) - / ( co+eF) ] . (4.1)

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188 / M A T R I X I N T H E K O N D O P R O B L E M 953

Here 47r( | r | 2 ) is the average tunneling probability and V is the applied voltage C/(w) is the Fermi func­tion]. wmetal(co) and walloy(co) are the density-of-state functions of the pure metal and of the alloy, respec­tively. For small voltages (eF«eF), wmetal(o>) can be treated as a constant. Using the expression (2.8) for the difference between the density of states of the alloy and that of the pure metal, one obtains for the relative change of conductance A G ( F ) = [ G ( F ) - G ( 7 = 0 ) ] / G(V=0) at zero temperature,

AG(V) = Trn0n£lmtTet(eV) —lm/ re t(0)]

l+irnofii lmtret(0)

=sraoM*[Im/ret(e7) — lm/ r e t (0) ] ,

for m«l (4.2)

where n0 denotes the density of states of the pure metal. Since the impurity concentration in dilute magnetic alloys is usually small, nf^\Q~z or less, AG(V) is a small quantity. However, for small temperatures it varies very rapidly with V. For example, at zero temperature it follows from (3.25) that

d wS(S+l) • I m / l e t ( w ) - <o->0. (4.3)

Thus at zero temperature the derivative of AG(V) with respect to V becomes infinite as V approaches zero. For finite temperatures (but T<£TR), the derivative re­mains finite even for V=0. The maximum value of d[_AG{V)~]/dV occurs at V^TK and is of the order ni{T\n.T/TK)~l, so that one has to go to very low temperatures if one wants to make d\_AG{V)~]/dV large. Nevertheless, it might be possible to detect the rapid variation of AG(V) experimentally. This would, at least, give some information about the width of the resonance in the scattering amplitude.

FIG. 5. Comparison of ImtTet(e,T) with Im/ret(,~0(f,T) for r— 1. The

dashed curve represents Im/ret'=0(e,T). See Eq. (3.24).

FIG. 6. Resistivity ratio R(T)/RQ (where Ro^—3Jni/n0e2VF2)

is plotted as a function of T. The dashed curve represents the result of the approximation (4.6) in which R(T)/RQ— — lm/ret(e = 0, T). The solid curve represents R/RQ= —JnoVF2/2tiiKo, where KG is computed numerically according to Eq. (4.4).

The large anomalies in the conductance of tunneling diodes containing magnetic impurities20,21 certainly can­not be explained by the change in the bulk density of states of the alloy. Other mechanisms such as scatter­ing from impurities located inside the oxide layer are probably responsible for these effects.22

B. Transport Properties

While a tunneling experiment, in principle, enables one to measure the density of states and hence the scattering amplitude at arbitrarily high energies, the transport coefficients depend on Im/ret(co,r) for small values of the energy only. In general, the transport co­efficients can be expressed in terms of the following integrals23:

Kn = — I do) v2(o))cx)nn(o)}T)—»o(co), y_oo do)

(4.4)

where w0(co) denotes the density of states of the pure host metal as before and v(co) is the speed of an electron with energy co.

The electronic lifetime due to impurity scattering is given by the imaginary part of the scattering amplitude,

r r 1 ( « , r ) = -2n<Im/ r e t ( « , r ) . (4.5)

Since df(e)/de is sharply peaked at e = 0, the main con­tribution to the integral (4.4) comes from a narrow region around e = 0, This suggests the following well-known approximation for these integrals

Kno^~ n0 VF'

2tii l m / r e t ( 0 , r ) / do) o)n—,

7-oo da) (4.6)

20 A. F. G. Wyatt, Phys. Rev. Letters 13, 401 (1964). 21 J. M. Rowell and L. Y. L. Shen, Phys. Rev. Letters 17, 15

(1966). 22 J. Solyom and A. Zawadowski, Phys. Status Solidi 25, 473

(1968).

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954 R . K . M . C H O W A N D H . U . E V E R T S 188

where vF is the Fermi velocity. This approximation is not too good, however, since at low temperature Im/ ret(w,r) is itself a rapidly varying function of c.24-25

We have illustrated this in Fig. 6 using the example of the electrical resistance which is proportional to i£0. For other transport coefficients which involve integrals Kn with n>0, such as the thermal conductivity, the approximation (4.6) is even worse. On the other hand, computation of the resistivity shows that, to a very good approximation, Im/ret(co,r) may be replaced by Im/ r e t^°(w,r) [see (3.27)]. We infer from this that no conclusion about the detailed structure of tTet(oo,T) can be easily drawn from a comparison of the experimental data for the transport coefficients and the theoretical expressions.

So far, we have completely neglected the normal po­tential scattering arising from the impurity ions. If one intends to compare the theoretical expressions for the transport coefficients with experimental results, this has certainly to be taken into account. As we shall show in some detail, the anomalous low-temperature maxi­mum in the thermoelectric power cannot be understood even qualitatively if the scattering of the electrons from the electrostatic impurity potential is neglected.26-27

In terms of the integrals Kny the thermoelectric power Q(T) can be expressed as

Q(T)=(l/eT)Kl/K0 (4.7)

For a band which is described by a Lorentzian density -of-state function of width 2D, we find the following dispersion of the electrons for energies near the Fermi energy (oi<^D):

v{u) = vF{l+m/D). (4.8)

Here v is a constant of the order of unity. As a direct consequence of the symmetry (3.7a) of Im/ iet(a>,r),

it follows that at low temperatures

K^OiT/D).

Thus Q(T) becomes small of the order 1/D and inde­pendent of the temperature in contradiction to the experimental observations.28 However, the symmetry (3.7a) is removed if the ordinary potential scattering is taken into account in addition to the exchange scatter­ing.

Assuming that the electrostatic potential V(r) of the impurity can be approximated by a contact potential Vd(r), one can write the scattering amplitude with the inclusion of potential scattering in the form29,30

where

tj+v{u,T) = tv+e™ytj{^T),

/F=(2ixw 0)- 1( l~^ 2 i 6 F)

(4.9)

(4.10)

is the scattering amplitude of the electrostatic potential alone, 8v being the phase shift associated with it. J is an effective coupling constant given by

J= J cos25F. (4.11)

The expression (4.9) was derived from a more general expression for tJ+v developed by Schotte29 under the assumption that a>«Z) so that the band structure of the host metal is unimportant. For the computation of the integrals K\ and K0i one needs the electronic lifetime

r t(co,r)= (2^-)-1[Im/ J + F(co,r)]-1

= - [2w i]-1[(2xw0)-1(cos26F - 1 ) + cos28v Xlmtj(ccyT) + 2 sin25F Re/j(co,r)]~1. (4.12)

For the range of energy jco) <kBT for which we need r»(co), the phase shift of the electrostatic potential can be treated as a constant. Then tj(w,T) has the same sym­metry properties as /ret(co,r). T,-(a>) can be written as the sum of an even and an odd function

Ti(u,T)= T; ,even(c0,r)+7\- ,odd(a>,r) ,

where

Tz,even(ct>) = '

T,\odd(co) = •

1 (27rn0)~1(cos25F — l)+cos2<5F Imtj

2fa [(27rn0)~1(cos25F--l)+cos25F Im /7 ] 2 - [ s in25 F Retjj'

1 -sin25FRe/^(co,r)

2m [(27T^0)-1(cos25F-l)+cos25F I m / j ] 2 - [ s i n 2 5 F Re/j]2*

(4.13)

(4.14)

23 See, e.g., J. M. Ziman, Electrons and Phonons (Oxford University Press, London, 1960), p. 384. 24 I t should be emphasized that, while the approximation (4.6) for Kn is inaccurate because of the rapid variation of Im/ret(co T)

the usual expressions for the transport coefficients involving the integrals Kn (see Ref. 23) are still exact in spite of the strong J dependence of ImtTet(a),T).

25 K. K. Murata and J. W. Wilkins, in Proceedings of the Eleventh International Conference on Low Temperature Phxsics St. Andrews 1968, edited by J. F. Allan, G. M. Finlayson and D. M. McCall. (University of St. Andrews Printing Department 'St Andrews' Scotland, 1969), p. 1242. ' '

26 J. Kondo, Progr. Theoret. Phys. (Kyoto) 34, 372 (1965). 27 K. Fischer, Phys. Rev. 158, 613 (1967). 28 D. K. C. MacDonald, W. B. Pearson, and I. M. Templeton, Proc. Roy. Soc. (London) A266 161 (1962) 29 K. D. Schotte, Z. Physik 212, 467 (1968). / > v ;. ;}0 Y. Nagaoka, Progr. Theoret. Phys. (Kyoto) 39, 533 (1968).

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188 M A T R I X I X T H E K O X D O P R O B L E M 955

We note that rev:n enters in K0j while r0<i<i determines Kx. We also recall that the integral Ko gives the con­ductivity. In the approximation (4.6), we obtain

n0vF2/

Ifii V

Using (3.12), we can write this as

1

2irn0

-(COS25F — 1 ) + C O S 2 5 F Im/; 7(co = 0 ) )

AV /To2t>2/ lnr X"1

I1 -COS25 , ' I . n i \ [ ln2r+irtf(S+l)]1 / 2 /

(4.15)

(4.16)

In the same approximation, K\ can be reduced to

irno2vF2 / lnr \ 2 C do) / w \

# i ~ — s in2M 1 -cos2o F J [ l n 2 r + 7 r 2 ^ + l ) ] - ~ 1 / 2 / co cosh"-2! ) m \ [ l n 2 r + x 2 5 ( 5 + l ) ] 1 / 2 / J 4kBT \2kBT/

X(l/\y\)lReX(^T)sine((afT)+JmX(u9T)cose(o)9T)2. (4.17)

The integral still can not be evaluated exactly and so, as a further approximation, we set

by Fischer27 and Kondo.81 By expanding

7

0(«,r)=o, (4.18) ( l n r ) - 1 - -

which means that we adopt the approximation (3.24) for the real part of tTet(oj,T). We are aware of the fact that this might not be a good approximation for very low temperature, where, as shown in Fig. 5, 0(<a,T) becomes important in determining Re/ ret(w,r). As we shall see, the approximation (4.18) becomes also invalid at high temperatures, where perturbation theory is ap­plicable. For temperatures of the order of the Kondo temperature, however, the approximation should not be too bad. Using (4.18), we find from (4.17) that

1 - 7 ln(77Z>)

in powers of 7, we find, in this limit,

Q^rt(r)=(7rykB/2e) s i n 2 5 , [ l + 7 ln(r /Z>)] . (4.23)

In contrast to this, Fischer and Kondo found

C )pert(r)cC7 , ' ?-

This failure of (4.22) to reproduce the correct perturba-tive result for Q(r) is due to the approximation (4.19).

11 i sin 2 5 v A'!~ kBT-

2r? o V [ln2r+7r25*C5,+ l)]1 ' '2 (4.19)

and hence

where

Q(T) = - (wkB/2e)sm28v q{r), (4.20)

^ ( r )={[ ln 2 T+7r 2 5(5+l ) ] 1 / 2 ~cos25 F ln r} - 1 . (4.21)

q(r) is plotted in Fig. 7 as a function of r. From (4.19), one can see that the sign and the magnitude of the thermoelectric power depend on the sign and the mag­nitude of the phase shift 8v through the factor sin25F-The same dependence of Q(T) on 8v was found by Fischer,27 who calculated the thermoelectric power in perturbation theory. In the high- and low-temperature limits (|lnr|2$>l), Q(r) reduces to

/ \ ( d )

1 \ sinSy =0 451

/ yC - " "^s<^\

if \ V \

\ 1 \ 1

\°X^^ ^^ - " "s in Sv= 0 08

sin 0\i =0 08

^ " \ (c) s

^ \ S t n O v =0 08

-1 1

Q{r)^-{irkB/2e) sin2<5F (lnr)~ (4.22)

In the high-temperature limit, (4.22) should agree with the perturbative expression for Q(r) obtained

FIG. 7. Thermoelectric power ratio Q(r)/Q0 [where Qo = — (irkB/2e) m\2bv~] is shown as a function of r. Curves (a) and (b) corresponds to the approximation in Eq. (4.20) and Maki's result given by Eq. (4.26), respectively. Curves (c) and (d) are obtained by computing K0 and K} numerically.

:J1 J. Kondo, Progr. Theoret. Phys. (Kyoto) 40, 695 (1968).

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956 R . K . M . C H O W A N D H . U. E V E R T S 188

In the limits lnr —> ± <*>, the integrals KQ and K\ can be evaluated exactly. The results are given in Appendix A [(A 14) and (A15)]. Using these expressions, one ob­tains the following exact asymptotic expressions for the thermoelectric power:

Q(r)=-irks sm25^ 7r25(5+l)

2e l+cos2<H I lri7

X 1 + cos25F w2S(S+l)

L 1+COS25F 2 ln2r \ ln 3 r / J

and

* 0

(4.24)

Q{r)=-irkn sin25F T T ^ S + I )

2e 1—cos25F

cos28v X 1-

| l n r | 3

7T26(5+1)

1—cos25F 2 ln2r -0

\ln3r/J

Since (4.25)

\]nr\-*= \y\*{l+0[y WT/D)]},

the high-temperature expansion of Q(T) clearly agrees with perturbation theory. We note that the asymptotic expansion (4.25) is meaningful only if

cos2S> TT 2 S(S+1)

1 — cos2e>v - « 1 .

For small values of the phase shift 5v, this requires very large values of r.

In a very recent publication, using the solution of Hamann and Bloomfield for the scattering amplitude and making the same approximation that led us to (4.20), Maki32 obtained the following expression for the thermoelectric power33 :

(?MakiM =—— ^7-7 sin2(5r

2e [ ln 2 r+7r 2 5(S+l ) ] 1 / 2 -cos25 F ln r

w2S(S+l) X-

ln2r+7r25(5+l) (4.26)

In the two limits lnr —» ± °°, this expression agrees with the exact expansions (4.24) and (4.25). For temperatures such that

ln2r<7r25(5+l), the factor

TT2S(S+ l)/[ln2r+7r25(5+1)]

by which Maki's expression differs from our approxi­mate result (4*20) is nearly unity. In particular, at the

32 K. Maki, Progr. Theoret. Phys. (Kyoto) 41, 586 (1969). 33 The negative sign in Maki's expression is erroneous (private

communication).

Kondo temperature the expressions (4.20) and (4.26) are identical. We have plotted the functions (4.20) and (4.26) in Fig. 7. Experimentally, a fairly sharp extremum of the thermoelectric power is observed near the Kondo temperature.28,34 I t is evident that the func­tions (4.20) and (4.26) do not exhibit this extremum. In order to avoid the approximation (4.6), we have evaluated numerically the integrals K0 and K\ in (4.7), using the complete expressions tj for two values of the phase shift due to normal potential scattering, sin^F = 0.08 and 0.45. These results are included in Fig. 7. I t can be seen that Q{r) is very sensitive to the phase shift considered. | (?(r) | attains a maximum near TK and decreases at higher temperatures. These features are in agreement with experiment and with earlier numerical calculations by Suhl and Wong.35

We should note that the thermoelectric power we have computed is independent of the impurity con­centration. The impurity concentration is a multiplica­tive factor in both K$ and Kh and consequently cancels out. However, at temperatures higher than the Kondo temperature, the scattering of electrons from phonons becomes comparable to, and possibly more important than scattering from impurities. Consequently, the integrals K\ and K0 in (4.7) have to be computed with the total electronic lifetime

1/not = 1 Arf-lAph (4.27)

rather than T,- from (4.12) alone. Since Tph(r?) is in­dependent (dependent) on the impurity concentration, the resultant thermoelectric power will depend on the impurity concentration in general. In the limits, however,

l / r t » l / r P h (4.28a) and

l / r P h » l / r t - . (4.28b)

Q(T) will become concentration independent. The former situation is realized at temperatures r < 1. The latter condition holds true already for temperatures much less than the Debye temperature and, in this case, the thermoelectric power approaches that of a pure metal, Qmetau Qmetai is usually at least an order of magnitude smaller than that of the alloy near the Kondo temperature.

In summarizing the discussion of the transport coefficients, one can say that they do not provide in­formation about the detailed structure of the Kondo resonance as it was worked out in Sec. I I I . Just as in the case of the density of states, the best one can hope for is some information about the width of the reso­nance. In addition, we have seen that measurements of the thermoelectric power allow one to indirectly draw some conclusions about the electrostatic potential associated with a paramagnetic impurity in a metal.

34 M. D. Daybell and W. A. Steyert, Rev. Mod. Phys. 40, 380 (1968).

35 H. Suhl and D. Wong, Physics 3, 17 (1967).

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188 t M A T R I X I N T H E K O X D O P R O B L E M 957

C. de Haas-van Alphen Effect

Several authors36-37 have interpreted the observed field and temperature dependence of the damping of the de Haas-van Alphen oscillations in dilute magnetic alloys in terms of the electronic lifetime T(O),T) [see (4.5)]. Their approach was to take Dingle's38 expression for the periodic part of the free energy of the alloy in an external magnetic field and to replace the electronic lifetime r, which in Dingle's paper is treated as a constant, by the energy- and temperature-dependent lifetime that results from the s-d exchange interaction. The effect of the magnetic field on the s-d scattering amplitude is neglected, however.

We feel that such a procedure is not firmly based on theoretical arguments. A rederivation of Dingle's ex­pression for the periodic part of the free energy by Brailsford39 shows that it is not necessary to assume that the electronic lifetime is a constant. The electronic ex­citation spectrum however, must have the characteristic features of a quasiparticle spectrum, i.e., near the Fermi energy, the electronic self-energy 2 (a>) must be a slowly varying function of o>. As we have shown in Sees. I I and I I I of this paper, this is not the case for the Kondo model. 2 (co) is a rapidly varying function of oo near a> — 0 and the quasiparticle picture does not apply. Further­more, it is questionable whether it is a good approxima­tion to neglect the influence of the magnetic field on the s-d scattering mechanism. The magnetic fields applied in the experiments which are reported in Refs. 36 and 37 range 2-5 kOe and the temperatures at which the measurements were carried out range 1-5 °K, so that for the extreme values T= 1°K and H=S kOe we have

Thus, in this case, the magnetic energy is of the same order of magnitude as the thermal energy and certainly not negligible.

I t appears that a more careful theoretical considera­tion of these points is necessary before conclusions about the scattering mechanism can be drawn from the ob­served field- and temperature-dependent damping of the de Haas-van Alphen oscillations in dilute magnetic alloys.

D. Specific Heat

I t has been shown by several authors1213-40 that the anomaly in the electronic specific heat which is ob­served in dilute magnetic alloys can be attributed to the resonant scattering of the electrons from magnetic impurities. In terms of the change in the density of states An(ai) discussed in Sec. I I , the incremental

36 E. B. Paton and W. B. Muir, Phys. Rev. Letters 20, 732 (1968).

37 H. Nagasawa, Solid State Commun. 7, 259 (1969). 38 R. B. Dingle, Proc. Roy. Soc. (London) A211, 517 (1952). 39 A. D. Brailsford, Phys. Rev. 149, 456 (1966). 40 W. Brenig and W. Gotze, Z. Physik. 217, 188 (1968).

specific heat can be expressed by the well-known exact formula

d r™ AC(T)=— / An((a,T)uf(w)d(a, (4.29)

dTj-„

where the integral represents the incremental internal energy of the system due to the interaction with im­purities. I t is suggestive to split AC(T) in the following two contributions:

ACn(T)-- do) An(cx)yT)o)- (4.30)

ACa(T)= f dJ—A«(co,r)Jco/(co). (4.31)

Henceforth we shall call ACn and ACa the normal and the anomalous contribution to the specific heat. We recall that for an electron gas interacting with normal nonmagnetic impurities the change in the density of states An(cx),T) is temperature-independent, so that there is no contribution to the specific heat from the derivative of An(a>,T) with respect to the temperature.

In what follows we shall examine ACn(T) and ACa(T)9

given by (4.30) and (4.31), using the expression (2.8) for An(u,T). ZMH and Brenig and Gotze40 developed an elegant way of singling out the important contribu­tion to the specific heat. They were able to relate the internal energy of the system to the coefficients of the asymptotic expansion of /ret(cu,r) for large o). Neverthe­less, we find it worthwhile to discuss the integrals (4.30) and (4.31) directly. We will show that, in contrast to all other quantities which we have discussed so far in this section, the specific heat depends crucially on the presence of the unimodular factor eie and on the tem­perature dependence of the function A(T) in the ZMH expression for the scattering amplitude /ret(co,r).

Let us first consider the normal contribution to the specific heat. Because of the factor df/dT in the in­tegrand of (4.30), An(c*)7T) is only needed in the energy interval \(a\<ksT. Thus it is sufficient to work with the approximate expression (2.9) for An(u),T), with the result

' / • ACn(T)=nin0

2 / daw Im/ret(co) dT

(4.32)

Introducing the dimensionless variables e and r again, we can write this as

ACn(T)

where

= nino2kBTK / * / ( € , T )

de e Im/ret(e,r) , (4.33)

/ ( e , r ; s < e x p ( € / r ) + l ] - i .

With (2.10), we then can give the following estimate for ACn(r)

ACn{T)<\ninJzB2TKT

<nin^kB2T. (4.34)

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958 R . K. M . C H O W A N D H . U . E V E R T S 188

Thus ACn(r) goes to zero at least as fast as T. Compared to the specific heat of the free-electron gas

ACn is small because of the concentration factor w,\ I t is possible to separate AC« from the measured total specific heat of the alloy but this is not very interesting.

At this point, it is necessary to define more precisely what type of contribution to the specfic heat we con­sider to be typical for the Kondo effect. Looking at the first line of (4.34), we see that, for fixed r, ACn vanishes in the limit y —> 0. In contrast to this, there are quanti­ties which, in the same limiting process, remain finite. An example of this is the thermoelectric power as given by Eq. (4.18). As was pointed out by MH, the latter type of behavior is typical for physical quantities which show Kondo anomalies. I t may, in fact, be used to characterize a Kondo anomaly in any physical quantity.

Following ZMH, we shall now search for a contribu­tion to ACa which remains finite in the limiting process described above and disregard all contributions which vanish like ACn in the limit y —> 0. Using the expression (2.8) for A»(co,r), we have

ACa(T) = nino' r2w/ d

dec cop2(co) —I Re/ret(< ID -AdT

«,r))

+(1-—Y—Im/ret(co

= m(IHe+IUn).

*)] / (« ) ,

(4.35)

As we show in Appendix B, the contribution from the second term in the integrand vanishes as y goes to zero. The important anomalous contribution comes from the first term. Since the first term contains a factor (cc/D), this means that the anomalous contribution arises from the behavior of Re/ret(co,r) at energies of the order of D and is not directly related to the reso­nance structure in /ret(w,jH). The latter is confined to the energy region | « | <KBTK* As the calculations in Appendix B show, the part of ACa which remains finite (in the limit y —> 0) is proportional to the derivative of A(T) with respect to T

ACa(T) = niTry2D(dA/dT). (4.36)

This clearly indicates the importance of the function A(T) in the ZMH forward-scattering amplitude. The fact that we did not find any temperature variation in A (T) in our numerical calculation of A (T) does not mean that ACa(T) as given by (4.36) is small. The large factor D makes up for the smallness of SA/dT which, as Eq. (B20) shows, is of the order Et~l. The derivation of (4.36) in Appendix B also shows clearly that, as was first pointed out by Hamann and Bloomfield, the presence of the unimodular factor in the exact solution

is closely related to the occurrence of an anomalous contribution to the specific heat.

From the expression (B20) for (iryYDdA/dT, Eq. (4.36) may be written in the form

ACa(r)--nikE

2TT2

(a/ar)]F(e,r)l2

F(e,r)l2+7r\9(S+l)' (4.37)

This agrees with the results found by ZMH as well as by Brenig and Gotze. AC«(r) is evidently independent of the coupling constant.

Integrating ACa(r) with respect to temperature, we can find that portion of the total internal energy the temperature variation of which is responsible for the occurrence of the anomalous contribution to the specific heat

Ea(T0)-- i ACa{r)dT

= TK\ ACa{r)dr. (4.38)

The integral in (4.30) is a dimensionless quantity of the order of unity. Thus the magnitude of E&n0m(To) is governed by kBTK = Delly. On the other hand, there are certainly perturbative contributions to the total inter­nal energy which are of the order y2D. For small 7, these are much larger than Eanom(To). Any attempt to first compute the total internal energy of the system and to obtain the specific heat by differentiating the result with respect to temperature appears to be hopeless.

ACKNOWLEDGMENTS

We wish to express our sincere gratitude to Dr. A. Griffin for suggesting this problem, for discussions, and for helpful comments on the manuscript. One of the authors (H.U.E.) gratefully acknowledges a travel grant from the Deutsche Forschungsgemeinschaft.

APPENDIX A

Introducing the variable # = J/3o>, we can write the integrals Kn in the form

Kn=%vF2n0[ 0"L d% XnTi(xyT) COSh~2X. ( A l )

Here we have assumed that v(w) and w0(w) can be treated as constants and have replaced them by their values at the Fermi energy. Because of the factor cosrr t ; in the integrand of (Al), the electron lifetime n is important in the interval \x\<l only. To obtain the asymptotic behavior of K0 and Kt in the limits lnr —> ± oc y it is therefore sufficient to derive an as}Tnptotic expansion for n which holds for all x such

Page 13: Matrix in the Kondo Problem

188 t M A T R I X I N T H E

that | # | < 1 . The most convenient starting point for such an expansion is the following expression for the scattering amplitude which can be derived from (3.1) by a mathematical transformation (see MH)

where

S(X}T) =

tret(x,r)= (2iTrn0) *[1—S(X,T)] .

I"(€,r)|

(A2)

x — ixo

[\Y(eJT)\*+T2S(S+l)~]1'2 x+ixo

dx* / \Y(xfyr)\2

K O N DO P R O B L E M 959

Y(X,T) x—ixo

[| Y(X,T) 12+ir2S(S+l)']1/2 x+iXQ

r T T 2 5 ( 5 + 1 ) / 1 \ n

= I 1 l + O ) [ l + 2 i x r - 0 ( « 2 T 2 ) ] >

L 2 ln2r \ l n 3 r / J

r - * 0 (A7)

Y(X,T) x—ixo

r i r* dxf / \Y{X',T) z \ - | X exp —P / ln( ,

L27T J^x-x' \ F ( ^ r ) | 2 + 7 r 2 5 ( 5 + l ) / J

[ | Y(x,T)\2+ir2S(S+l)J12 x+ixQ

T2S(S + 1)

in which

Y(x,T) = \iiT+g(x)9

(A3)

(A4)

= 1-2 1n27 - < • ( - ) •

\ln3T/

(A8)

The principal-value integral in the argument of the exponential in (A3) can be expanded as follows:

is the dimensionless function defined in (3.12). Here ixo is the solution of the equation

•fA Y(X\T)\2

F ( * , T ) = 0 , T < 1 (A5)

Y{x\r) | 2 + T T 2 S ( S + 1 ) / x - x '

w2S(S+l) : — ^ 2 _

In3

and it is zero otherwise. I t is easy to verify that in the limit of r going to zero

\ dx'

/ x—x'

x+o(—V

l n r - » ± ° o . (A9)

* « ( T ) = 1 / T + 0 ( T ) . (A6j

The following two asymptotic expressions are im­mediate consequences of (A4) and (A6)

To obtain this expansion, we have made use of the analyticity of g(x) in the upper half of the complex x plane. Combining the results (A7)-(A9), we find the following asymptotic expansions for Re^et and Im/ r et:

Re/ret(#,r) = 1 T T 2 S ( S + 1 )

27rn0 ! lnr | 3 1 —

i r 2 5(5+l )

2ln :

M / 1 V —+0[ r \ l n 3 r / .

i7rtanhx, l n r — > ± ^

Im/ re t(#,r) = — 2wn(]

L 2 ln2r \ l n 3 r / J

| | -7r25(5+l) / J \ n

IL 2 ln2r \ l n 3 r / J '

T - » 0

(A10)

(All)

r — > Q C

Inserting these expansions into (4.13) and (4.14), we obtain asymptotic expansions for the even and odd parts of the electronic lifetime

irrio

Hi

cosldv w2S(S+l S+l) / 1 V

n2r \ l n 3 r / . l+cos2<5FL

I 1 r cos25F T T 2 S ( S + 1 ) / 1 \ — 1 +0[ )

U-cos25 F L l - c o s 2 5 F 2 ln2r \ l n 3 r /

l+cos2S F 2 lr

cos25F T T 2 S ( S + 1 ) / 1

, T-+0

t »oc

(A12)

Hi t 7T2S(S+1) Ti,odd = sin25F- Jx tanho; Tj.even, In r—>±^ .

wn0 l^nr]4 (A13)

In (A13) the appropriate asymptotic expressions for r,,eVen given in (A12) have to be inserted. Finally, we obtain

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960

for KQ and K\y

R . K . M . C H O W A N D H . U . E V E R T S

cos25F T T 2 5 ( 5 + 1 )

188

fto

ni

r 1 r cos25F T T 2 5 ( 5 + 1 ) / 1 \ l 1 + + 0 ) , r - ^ 0

l+cos25FL l+cos25 F 2 ln2r V l n W J

1 r cos2<5F T T 2 S ( S + 1 ) / 1 \ 1 1 + 0 ( ) , r->oo

U-cos25 F L l - c o s 2 5 F 2 ln2r \ l n 3 r / J

K^\*kBT-rii

sin25 w2S(S+l)/ 1T2S(S+1)

v 1-i lnr l 3 V In2

) ^o 2 , l n r • . ± 0 0

(A14)

(A15)

APPENDIX B

To evaluate the anomalous part of the specific heat given by (4.35), we must compute two integrals. To begin with, we consider the integral

and n0 /•" / u\lm(Xeie)

I m = / do)up(u)[\ ) 2TJ-X \ D2/ K1'*

r ( *\ = n0

2 / dwtop2(co)( 1 J

X—Im/ r e t (« , r ) / (a>) . (Bl) dT

/dd\ dA dd\ X I — + ) / («) . (B5)

\dT\A dTdA/

We shall show that these integrals vanish in the limit 7 —» 0 at any fixed value of r, which is less than a cer­tain upper limit (see B7). Starting with Ix

lm, we first notice that

d I m X ( « , r ) / d r = 7 ^ ( d / a r ) t a n h i / S M (B6) Taking the temperature derivative of Im/ret(co,i ) as given by (3.4), one gets a variety of terms. I t is con- decays exponentially for | cu |»& B r . Using this, one can venient to regroup them in the following way: easily show that the third term in the integrand of

, Zi Im behaves like ACn as a function of temperature (see J ' m = J ' 1 ' 2 ' " 3 > ^ ' main text). I t is therefore negligible. To estimate the

other two terms, we need upper bounds for the func­tions \(ReXy/K-l\ and \(ReX)(ImX)/K\. We adopt the procedure of ZMH and limit the temperature

where

/ i

(ReX) 2 \ R e X I m X ~]dX

n0 r00 / u \ i m = _ _ / dw»i>{<A\ )K-1

2w ]-„ \ DV

r l m X ReX /

by kBT<D«(kBTKy-« ( 0 < a < i ) . (B7)

Icos0-K

(ImX)2 '

sin0 — JdT\A

J sin0

Then we consider these functions in two intervals, namely, |co| <coc and |co| >coc, where

o>c = D(l-a)(kBTK)«. (B8)

dX) X l m — / ( « ) . (B3)

dT)

Omitting the details, we merely state the upper bounds that we have found.

(i)

The symbol dX/dT\A means that A is considered as a constant in the differentiation,

flo dA hlm= (TTT)2—

2TT dT <Wp2(a>)ir-1/2

x[(-T) ^ ReZ ImX

cos0H

coI <coc:

| ( R e X ) 2 / i : - l | < l ,

\ReXImX/K\<c.

(ii) jco[>toc :

\(ReX)*/K-l\<cy*?(»),

\ReXImX/K\<c\y\p.

sin0 \f(o)) (B4) I n t n e s e estimates, c is a constant of order unity which K A is independent of y. Using these bounds, we have

(B9a)

(B9b)

(BlOa)

(BlOb)

n01 r T / (ReA)2 \ RzXlmX n | / 1 i m | = _ / du><ap(u))(l-u*/D*)\ I 1 )cos6H sin0

2T\J^ LA K ) K J

dX R e —

dT

iff, 2 i rU 0

<c—I / ^cocoliC-^Re- ' 27rU0 l an

r i + / duulK-1

/ 2 Re

A

dX

dT ( 7 P ( « ) + Y V ( « ) ) (BH)

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188

Now

t M A T RI X I X T H E K O N D O P R O B L E M 961

#- 1 / 2 (w)<l /M > *><«c (B12a)

K'll2(<a)<c} CO>COC. (B12b)

Furthermore, using the asymptotic expansion for $(h+z) a n < i d\J/(^+z)2/dz, one finds the following bounds for Re(dX/dT\A) (see Ref. 13, Appendix C).

(i) co<coc:

jdReXl

! dT

1 <: 7~ 1 —

/3w d

2ird((3u/2w) Ret̂ l

( • • 2T/\ 12\pD/ J

(ii) co>coc:

d ReX

dT

i 1T!P(W)/ 2 T \ 2

/3D/

(B13a)

(B13b)

Using (B12a), we obtain

I a ReX rfww IK-1'2

o I dT ^CikB

2TTK

'kBTK\2a r / ^ \ /IZBTKV* -l

4|ln'i+1"fe>hr) + 4 <W4a) and from (B12b) and (B13b),

I dX do>w\K~l^Re—

I dT \yp(w)+y2p2(u)

^ck^TTK\y\^+\y\)- (B14b)

Since no=ir/D, it follows that

| / i I m | = C I * « T - ( | lnr + l n

+c2+;Ti2+|T (B15)

In the limit | *v j —> 0, r fixed, this vanishes obviously. J i I m differs from Zi Im by their term in the integrand of Iilm which we have already shown to be negligible. Thus (B15) completes the proof that 7i I m does not con­tain a Kondo-type anomalous term.

To estimate the integral l2 I m , we make use of the bounds (B9a)-(B10b), (B12a) and (B12b). We obtain

| / 2I m | <c(no/2T)(wy)HdA/dT)

X[coc2/[7| + | 7 l / > 2 ( l + | 7 | ) ] . (B16)

To obtain dA/dT, we differentiate the condition (3.5b) and solve for dA/dT. This yields

dA r" 1 T / dX\ / dX\l / ( X 7 )2—= / do> Re( x— ) + I m X— ) /

dT ;_« i^(co,r)L \ dT/ \ dT/M RtX(coyT)

</cop(a>)—— . (B17) K(^T)

The integrals involved in this expression have been estimated in Ref. 13, Appendix C. Neglecting terms which vanish in the limit 7 —> 0, one finds

( a / d r ) | F ( € , r ) | 2

de , (B18) , ! F ( 6 , r ) i 2 + x 2 5 ( 5 + l )

where y(e,r) is the dimensionless function defined in (3.12). Evidently (try)2 dA/dT remains finite in the limit 7 - > 0 at'fixed r. From (B17) and (B18), it follows that its magnitude is of order \/D. Therefore

+ |7| + |7i2W (B19)

(iry) dA kB f°

dT 2 T T Z > J _ ;

lim 7 2I m< lim c{iry)2[

17I-0 M-o \\y\{pKD)* < —

Using the definition (3.4) for 0, the integral TV11

can be written

no r / u\lm(Xeie) P 3 I m = ~ rfco cop(co)( 1 V /(co)

2 W - „ \ DV K1'" 00 dJ /d\nK(a',T) r00 dec' / .

x / — ; ( J-oo C O — CO \ ar

/2 " 2TT

dA dlnK(w',T)

dT dA • ) •

(B20) With the help of the condition (3.5b), we have

I*lm = n0f ^p 2 (co)M-^JRe/ r e t (co , r )7Xco, r ) , (B21)

where

F(a,T)=— / d<J— ( 2w J-oo co'—co\

co' /d\nK(co'yT)

dT

+ dA d\nK(u'yT)\

. (B22) dA / dT

We obtain the estimate for 73I m

I / 3I m j < n/.nax / du p(co) j Re/rot(co, T) I , (B23)

where F m a x is the maximum value of F(GJ,T) for all co and for kBT<Da(kBTK/D)^-a\ Without going into the details, we state that

r. J<apM\Ret(u,T)\$cikBTK+c*D\y\ , (B24)

where the first term arises from the resonance in /ret(co,r) near co = 0 and the second term is the contribu­tion to the integral from large values of co. Since FmtaL

remains finite as 7 approaches zero, it follows that

lim \l3Im\ = 0. (B25) [ 7 I - 0 1

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962 R . K . M . C H O W A N D H . U . E V E R T S 188

Thus we have shown that the integrals iYm , 72I m , and

73Im , and hence 7 I m , vanish in the limit [7) —> 0, with

r fixed. We now turn to the evaluation of the other integral

in (4.35)

Wo CO

(I02 co—p(co) D

X l m \eJ 1_Y| L Va(irV-0 VK/J

! / (") , (B29)

and / R e = 2 n 0

2 / do> co--p2(co)(— Re/ret(co,r) ) / (« ) . (B26)

I3Re= / do) 00—p(co)

We again regroup the various terms that arise from w J_M D performing the differentiation in the integrand in three integrals just as in the case of 7 I m

7 R e = / i R e + / 2I l e + / 3 R e , (B27)

Re(Xeie)/d6 X ( —

dA 66 \ + ) / («) . (B30)

A dTdA/ where

n0 /•* w 7i R e = / duo:—,

ir J-x D

rdX\ / d x \ X l m — ( )•

LdT\A\dXVK/

-p(co) )

~dX\ / d X / ( « ) , (B28)

The integrals 7iRe and 72Re can be shown to vanish in

the limit 171 —> 0 with r fixed by the same methods that were used to show that 7i I m and 72

Tm vanish in this limit.

Using (3.1), we can write 73Re as

(-B31)

where

and

73 iR e= / do—p(co)[l+2ir«o(co)Im/ret(c n ' D

,T)lf(»)pj co' Re[X(co',r)(<3/ dT)X{J,T)~\

dj— (B32)

/ 3 2R e =

K(a',T)

dA ReX(o>',T) fio fx co P r* co' R e = ~ / Jco-p(co)[l+27rp(co)Im/ret(co,r)]/(co)- / dJ (TT7)2—p(co')

7T ./_„ D w J - * co'-co $T K(a',T) (B33)

The integral 73iRe can again be shown to vanish in the limit 7 —» 0 with r fixed. The important contribution to 73

Re, and hence to 7Re, comes from 732Re which can be written

no dA / 3 2 R e = - ( 7 T 7 ) 2 —

T 6T\

' r co P r* co' z-00 co P r « ' /ReXfa'T) \ / da-f(a)— / </co'~-~p(co')+ / cico-p(co)/(co)— / </«' p(co')( 1)

. ;_« D T J^ co'-co ; _ « D T J-x co'-co \ K(u',T) / ) -

7T J

P r co P r™ co' ReX(w',r)- | +27r / ^/co—p2(co)Im/ret(co,r)/(co)— / </«' . (B34)

J-00 D w i-oc co'-co K(co',T) J

The first integral in the square brackets can be calculated by elementary methods

r co p r* <*' r /r2\-/ dor-p(co)/(co)— / </co -p(co') = 7rZ)2 l+<9( — 1

J-oo D W J^ CO'-CO L \D2/. (B35)

The second and third integral can again be shown to vanish in the limit \y\—» 0 so that for | 7 J « 1 and T<KD

IRe^l32Re^noD2(wyy2dA/dT. (B36)

The significance of this expression is discussed in the main text.