Perturbative theory of a dilute mixed valence Tm system

0038-1098/87 $3.00 + .00 Pergamon Journals Ltd.

Solid State Communications, Vol. 61, No. 11, pp. 717-720, 1987. Printed in Great Britain.

PERTURBATIVE THEORY OF A DILUTE MIXED VALENCE Tm SYSTEM Tsung-han Lin* International Centre for Theoretical Physics, Trieste, Italy and Chen Liang and Li Yin-Yuan Institute of Physics, Academia Sinica, Beijing, China

(Received 30 September 1986 by E. Tosatti) The perturbative theory for a single mixed valence Tm-impurity system is presented. The ground state and the low temperature properties are studied. 1. INTRODUCTION Tm COMPOUNDS, such as TmSe, are special among the rare earth mixed valence (MV) materials. On the one hand, they share a certain common property of the MV systems; but on the other hand, they show some peculiar behavior which are quite different from the other MV materials. For example, instead of a Pauli-like behavior of susceptibility in low temperature, TmSe undergoes a magnetic ordering at about 3.5 K, from paramagnetic phase to antiferromagnetic phase. Besides, the dynamical susceptibility, the magnetic field dependence of the specific heat, and the electrical resistivity also exhibit peculiarities [ 1 ]. Experimental results clearly show that TmSe is a MV compound with the Tm ions being in two different valence configurations: Tm2÷(4f la) and Tma+(4f 12) [2]. Contrary to all other rare earth MV systems, both valence configurations of the Tm ion involved are magnetic in TmSe. It is important to clarify which of the peculiarities of TmSe stem from this origin. A series of works have been done around this subject. Alascio et al. [1, 3, 4, 5] studied the static and dynamic properties of Tm-system by introducing a simplified model, first for a single impurity then a lattice. Schlottmann and Falicov [6] studied a lattice model. Proetto et al. [7] used the Bethe-Ansatz technique to treat a single impurity with two degenerate configurations (s = ~ and s = 1). Aligia et al. [8] extended [7] to the two degenerate configurations w i t h / l , and ]2 =]1 + ~. In all these works, the degeneracies of the 4 f configurations have been treated in a simplified way. Here we present a perturbative calculation to a single impurity Tm-system in which we consider the degeneracies of 4fconfigurations in details. *Permanent address: Department of Physics, Peking University, Beijing, China.

The rest of this paper is organized as follows: In section 2, we describe the model Hamiltonian and generalize the Keiter-Kimball perturbative theory to this case. The results and discussions are given in section 3. 2. MODEL AND FORMALISM Consider a single Tm impurity embedded in a jellium. The whole system consists of two parts: the local impurity with 4f-shell and the conduction electrons. In strongly local correlation limit, i.e. the f-f Coulomb interaction U ~ ~o, only two 4 f configurations are involved: Tma+(4f 12) and Tm2+(4fla). Neglecting the crystal field splittings, the ground state muttiplets determined by Hund's rule are 3H6(nr = n = 12, S = 1, L = 5 , J = 6, IM[ ~ = I n ; J , M )

(IMI < J), (1)

IM') = I n + 1 ; J ' , M ' )

(IM'I < J').

Hereafter, IM) will be always used to indicate 4 f n states and IM') to indicate 4 f n÷l states. Let XMM' represents Hubbard ionic transfer operator XMM'

-~"

IM)(M'I.

(2)

For the conduction electrons, we take the impurity site as the origin to expand the conduction electron states in partial waves with quantum number (k]m), here k is the wave number, / and m denote the angular momentum and its z-component. Then the degenerate Anderson model can be expressed as the following form

717

A DILUTE MIXTURE VALENCE Tm SYSTEM

718

H=Ho+¢

Zfo = E e-3EjM + E e-~Ej'M', M M'

1210 = E ekjm d;jm dkjm + E /:'aM XMM hjm

M

PM = e- [3EJM/Z[o, and

E {Vkjm(M, M')dkSmXMM' + h.c.}, kjMM'

(3)

where operators d~im, dkj m denote the creation and annihilation operators of the conduction electron with the quantum number (kjm). As in [7], the hybridization has been assumed to be k-independent and rotationally invariant:

Vkjm(M, M') = V(JM/m [J'M'),

(4)

where the Clebsch-Gordon coefficients are non-zero for rn = M ' - - M , representing the conservation of the angular momentum. It has been shown that ] = 7/2 dominates the hybridization effects in heavy rare earth impurities [9]. Therefore, as a good approximation, we can restrict ourselves to the case of ] = 7/2, and simply omit the subscript ] in (3). Then the Hamiltonian be-

conies

PM' = e-3~J'M'/Zto,

are the occupation probabilities for states IM) and IM'}, respectively. E M and E M, are the self-energy functions for the states IM) and [M'). The self-energy functions can be expressed diagrammatically [ 0, 11]. Here we only present the lowest and the next highest order selfenergy functions (see Fig. 1). The analytic expressions of the lowest order self-energy functions are given by

t 2 Z(ffl)(Z) = E ! Vkm (M,M _)[ f(___ek___~_) kM' Z -- (Ej'M' -- EJM) + ekm' and

Z~1),(z) = ~ IVmn(M, M')I2[1 -f(emn)] kM z + (Ea'M' --EaM)

+ 2 Ea' 'XM'M' + E

i ......

(5) IM> i ; ...... ,v. k.ai

The generalization of Keiter-Kimball perturbation theory [10] to a single Tm impurity system is quite straightforward [11]. The grand partition function (G.P.F.), Z = Tr e-OH, can be expressed as a contour integral - - e- t~ Tr 2~i

(10)

ek,,,

I

......

Z =

--

[ M)~i

kMM'

{Vkra(M, M')d[~m XMM' + h.c.}.

(9)

where f(ekm ) is the Fermi distribution function. If we take the non-cross approximation (NCA), i.e. neglect all the crossing diagrams or the vertex correction

~I = ~ ekmd~mdkm + ~., EJMXMM + km M

M'

(s)

is the unperturbed G.P.F. of 4f-shell,

+ E Ea'M'XM'M' M' (I =

Vol. 61, No. 11

I

iM> ," I

(6)

in which the contour circles all singularities of the resolvent. Taking the hybridization V as a perturbation, using the eigenstates of Ho as the basis, one can easily expand the resolvent (z --/7/)-1, and finally obtain

[ M)*

[ M') t

I M> {17

(1) ~M

ZM'

Z/Zo = ~ PM~ dz e-OZ M 2~i Z ---EM(Z)

I 1 M'I)!

1

I

+ E P~'~ dz ~,

e -~z

2 T z -- YMw(O '

[M>,~

(7)

here Zo is the unperturbed G.P.F. of the whole system,

I

Fig. 1. The diagrammatic representations of the lowest order and the next higher order self-energy functions.

Vol. 61, No. 11

A DILUTE MIXTURE VALENCE Tm SYSTEM

[10, 12], the self-energy functions ZM and ZM' satisfy the following coupled integral equations:

719

and F, are introduced, where A =Ej'M'--l£gM is the separation of 4 f configurations before the hybridization

XM(Z) = V IVm~(M,M')12/(ek~) kM' Z -- (Ej' M' --EjM ) + %m -- Y~M'(z -- (EJ'M' --EJM) + ekm) and

~,M'(Z)

kM

Z

IVkm(M, M')I 2 [1 --f(ekm)] + ( E j , M, --EjM ) ekm ~M(Z + ( E j ' M' --EjM)--ekrn)" - -

Since the [z - ZM(z)]-1 and [z -- ZM,(z)]-1 in (7) are analytic everywhere except on the real axis, we shall be only concerned with z = x - - i f with 5 = 0 +. The solutions EjM and Ej'M' of the equations

k?JM = Re

3. RESULTS AND DISCUSSIONS

and

Ej'M' = Re F,M'(Ej'M' -- i6),

(12)

determine the energies of the quasi-particles of the renormalized 4f-holes and 4f-particles. Then the G.P.F. of renormalized 4f-shell can be expressed as

M

e -~(EJ'M'+iJ'M'),

(13)

M'

where EjM +EJM and Ej'M' +Ej'M' are the quasiparticle energies of renormalized 4 f electrons. The ground state energy is the lowest energy among EjM + EjM and Ej'M' + Ej'M'. From (13), one can calculate the thermodynamic potential ~ t , valence v, the static susceptibility Xt and the specific heat C r of 4f-shell g2r = -- kB Tin Z r,

(14)

v = 2 + v,

(15)

P =

(16)

Xr -

(XMM) ,

-~-

in absence of the magnetic field, and P = NFIVI 2 represents the resonant width. Both A and P are measured in the unit of D in the calculation.

~M("EjM--i5),

z r = Y~ e -ateJM+iJM) + ~

R~o'

° 0.005

- 0.015

EGS (18)

The results are obtained by numerical calculation, in which we only consider the lowest order and the next highest order self-energy functions, i.e., we take the approximation ZM "" Z~ ) + Zg ) and ZM' -~ Y-~ ~~-~(1) ! i,~Mt . We choose the Lorentzian as the density-of-states (DOS) of the conduction electrons

- 0.025

- 0.035

- 0,045

- 0,055 - 0,020

' - 0.012

d

' - 0.004

0,004

J 0.012

0.020

A

D2

6(e) = NFe2 + D 2 ,

The ground state which we have obtained is always degenerate. This result is in agreement with the previous theories [1, 3 - 8 ] . Physically, it can be attributed to the symmetry of the hybridization which has been assumed to be rotationally invariant. The dependence of the ground state energy EGs with A for different temperatures and differeht values of F are shown in Fig. 2. With the same F but different T, curves (a) and (b) show a weak temperature dependence. For a fixed A, EGs will be lower as the hybridization getting larger. Figure 3 presents the variation of average valence as a function of A. It shows that the system exhibits a strongly mixed valence state when A is around zero; and tends to be in an almost pure valence state when [A[ is large enough. For a certain value of A, the larger the F is, the stronger the mixed valence effect would be. In

(17)

~2~-~ f

C t = -- T 0T----T .

(11)

- -

(19)

where D is the half width of the conduction band, NF is the DOS at the Fermi level. Only two parameters, A

Fig. 2. The ground state energy EGs as the function of A for different temperatures and different values of F: (a) T = 5 K , F = 0.01; (b) T = 50K, P = 0.01; (c) T = 50 K, F = 0.005.

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A DILUTE MIXTURE VALENCE Tm SYSTEM

720 1.0

Cf

0.3

0.2 I

~

~

(b)

O1 I

0.6

~00

20

40

60

80

100

T

0.4

0.2

0.o[

..... (a) /'=0.005 (b) f =0.001

- 0.036

- 0.018

- 0.000

0.018

0.036

A--0.01

Z~=-0.Ol

Fig. 5. The temperature dependence of specific heat. 0.054

A

Fig. 3. The variation of average valence as a function of A for different values of F. 4.5

(a) £ =0.OO5, A--0.01 (b~ r - 0 . 0 o g A=(~OO 3.6 Zf 2.7

(a~ £=0,01

( b ) £=0.005

Ka')

for T = 1.8 K in antiferromagnetic phase. The value of /ae~f is always less than the values derived from .the Hund's rule ground state multiplets. It seems that this reduction of the moment come from the crystal field effect.

Acknowledgements - One of the authors (THL) is indebted to Su Zhao-Bin and T.K. Lee for many helpful discussions. He would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste, during a period in which an important part of this work was done. This work was supported by the Chinese Science Foundation.

1.8

REFERENCES 1.

0.9

2. 0.0

60

120

180

240

300

T

Fig. 4. The static susceptibility as the function of temperature.

Fig. 4, the static susceptibility Xrvs temperature T is presented. A typical Curie-like behavior in low temperature regime is clearly shown which can be attributed to the magnetic ground state. Figure 5 exhibits the temperature dependence of the specific heat. As a function of temperature, C r has a linear dependence with temperature in low temperature regime, and the linear coefficient is larger when the hybridization is getting smaller. The crystal field splitting which has been neglected in our calculation might play an important role. According to the experiment measurements of TmSe, the effective moment of Tm ion is /~eff = 3-35/aB for 5 K < T < 50K in paramagnetic phase, and/aef ~ "~ 2/aB

3. 4. 5. 6.

7. 8.

9. 10.

11.

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