Solid State Communications, Vol. 78, No. 2, pp. 133-135, 1991. Printed in Great Britain.
0038-1098/91 $3.00 + .00 Pergamon Press plc
THERMAL EXPANSION OF MIXED VALENCE COMPOUNDS G. Gangadhar Reddy and A. Ramakanth Department of Physics, Kakatiya University, Warangal 506 009, India
(Received 10 September 1990; in revised form 30 November 1990 by P. Burlet) The anamolous thermal expansion of mixed valent systems is studied using the periodic Anderson model (PAM). The coefficient of thermal expansion is related to the temperature derivative of the 4f-level occupancy. The 4f-level occupancy and its temperature derivative have been evaluated self-consistently using the PAM in the mean-field approximation for a wide range of temperatures. Thereby, the interpolation, the earlier authors employed to obtain a peak from the low and high temperature results, is avoided. The order of magnitude and the qualitative behaviour obtained here are in agreement with earlier results. IT IS NOW well established that the mixed valence (MV) state of the 4f-ion in the rare-earth compounds or alloys is due to the quasidegeneracy of two electronic configurations namely 4f"(5d6s) m and 4f"-l (5d6s)m+ where n and m are respectively the electron number in the 4f-shell and the 5d6s-conduction band. Systems with MV state show marked anomalies [1-3] in practically all the experimentally measured properties such as in lattice properties (anomalously high compressibility and thermal expansion), in specific heat (anomalously large linear specific heat at low temperatures), in magnetic susceptibility (a pronounced peak in the low temperature region), in transport properties (a resistivity anomaly at low temperature), etc. Recent investigations both theoretical [4, 5] and experimental [6, 7] have shown that the MV systems under suitable conditions show spinglass like and re-entrant magnetic type behaviour too. Theoretically this has been explained as due to the iincrease of 4f-level occupation n: with increasing temperature. It is well known that the metallic radius of the rare-earth atom with n electrons in the 4f-shell is greater than with n - 1 electrons in the 4f-shell (Lanthanide contraction). Therefore, the rate at which the occupancy of the 4f-level changes with temperature may be a good candidate to explain the above said marked anomalies. A few theoretical works have tried to explain the experimentally observed [8] anomalous thermal expansion in YbCuA1 and CeSn3. Based on the periodic Anderson model (PAM), using a perturbation expansion method, Muller-Hartmann [9] has shown that the thermal expansion goes as l I T at high temperatures. Then using plausibility arguments, he conjuctured a peak like behaviour in the low temperature region. Recently, Sanjeev Kumar et al.
[10], using a Greens function decoupling method in a single impurity model, have shown that the thermal expansion in the low temperature limit is proportional to T and in the high temperature limit it goes a s ! / T 2. Eventhough both the works lead to a peak like behaviour of thermal expansion as a function of temperature, they could not obtain the results for the entire range of temperature. In this communication, we present results for the entire range of temperature based on PAM in the mean-field approximation. The coefficient of thermal expansion is defined as 0In V 0In VOn/ ctOT On/ OT (1) where V is the volume of the system and T the temperature. The increase in the 4f-shell occupation enhances the screening of the nuclear charge. This results in not only the radial expansion of the valence orbitals of rare-earth ion, but also in a corresponding expansion of the lattice of the rare-earth compound. The size of this effect is typically of the order of 15% volume increases per f-electron [9], i.e. 01n V ~- 0.15. On:
(2)
The combination of this effect with the temperature dependence of n: may lead to an anomalous thermal expansion of mixed valence compounds. The temperature dependence of n:is calculated in the following, using the PAM in the mean-field approximation. The Hamiltonian of PAM can be written as
133
It,o
+
", Vki
(")
e - flit - R i
+ h.c.)
(3)
134
T H E R M A L EXPANSION OF M I X E D V A L E N C E C O M P O U N D S
The notation is conventional. In the meanfield approximation, we obtained the following expressions (in the paramagnetic situation) for the total number of electrons per lattice site nc and for the 4f-level occupancy nj-:
n,, -
2
U ~ f ( E ( k ' t))
Vol. 78, No. 2
12
-7
I0 8
(4) x
2 Fir
z
~
~3E(k, t) ~ E t ~ f ( E ( k , t))
6
(5t
where
E(k, t = +_) = ½ [ L + e(k) __+ x / ( L # i -
e(k)) 2 + 4V2];
0
0.02
Ej- = Ey + Uns/2, f ( E ( k , t)) = and
fl = 1/KBT. From the above equations we have obtained the following expression for the temperature derivative of the 4f-level occupancy:
~3nt -
2KBfl2
(6)
where
' g'= N~L4\ ~3/~t 2cosh 2{~(E(k,t)- 4 02E(k, t) t))] t ~ f(E(k,
L- 1t -
-fl Z ctE(k, 4Nk., ~3E,-t) cosh 2 IL,-fl (E(k' t) - #)} fl ~ c o s h 4 N k,,
/4
4N ~
0.06
KBT/W
[exp {fl(E(k, t) - /~)} + l] '
14 - I2I~~ -i7,/
0.04
2{~(E(k,t)-I~)}(E(k,t))-#)
t3E/
x cosh 2 { f l ( E ( k , t ) - # ) } ( E ( k , t ) - # ) . After solving the equations (4) and (5) selfconsistently for the determination of the chemical potential # and 4J:level occupancy nr, Onfl~T has been evaluated. Then the coefficient of thermal expansion is obtained from equation (1). The dependence of the
Fig. 1. The coefficient of thermal expansion as a function of temperature for different 4f-level positions. (a) Er/W = 0.4, (b) 0.3, (c) 0.18 and (d) 0.1: V/W = 0.25 and n,, = 1. coefficient of thermal expansion on temperature for different values of the 4f-level position is shown in figure. All the model parameters have been normalized with respect to the conduction band width W. It can be seen that ~ increases with temperature and reaches a maximum (~ma*) at some characteristic temperature (Tm~x) and then it decreases for further increase of the temperature. It is found that both ~max and Tmax increase with the shifting of the 4f-level position Es towards the conduction band. In the low temperature region i.e. T < Tmax, ~ is proportional to T. In the high temperature region T > Tmax, ~ is not varying either as 1/T or 1/T 2 as was obtained by MullerHartmann [9] or by Sanjeev Kumar et al. [10]. However, the order of magnitude of the maximum of ~ [e.g. ~max = 12.2 X 10 6 K l for Es/W = 0.1, V/W = 0.1, nc = 1 and U/W = 0.25] obtained in the present calculation is consistent with the results of Pott et al. [8] and Muller-Hartmann [9]. The anomalous behaviour of the thermal expansion in the low temperature region is due to the atomic like character of the f-electrons and their associated capability to screen the nuclear charge and therefore is purely electronic in nature. We have thus, unlike the earlier workers obtained the dependence of ~ on T, though numerically for the entire temperature range and observe a peak like behaviour without using any interpolation. It may be tempting to reduce analytically the present results in the high- or low-temperature limits and compare with earlier works [9, 10]. It should be pointed out that, within a certain domain of parameter
Vol. 78, No. 2
THERMAL EXPANSION OF MIXED VALENCE COMPOUNDS
space, the occupance of 4f-level is very sensitive to the change in T and this makes many physical quantities anamolous. For example at a certain 4f-level position, a slight change in Ei/W say by 10-3, the system undergoes transition from ferro- to para-magnetic phase or from para- to ferro- to para-magnetic phase as a function of temperature [5] or displays spin-glass-like behaviour [4, 5]. Therefore, it is difficult to obtain an approximate expression for ~ with the required accuracy. These various features are not reflected in the behaviour of ~ as a function of T in the present calculation because, a paramagnetic phase has been assumed at the beginning itself. In this communication, we have analysed the thermal expansion of mixed valence system based on the periodic Anderson model in the mean-field approximation. It has been found that anomalous thermal expansion is due to the increase of the 4f-level occupancy and is purely electronic in nature. The results obtained are in good agreement with the results obtained earlier both experimentally and theoretically. It can be concluded that the anomalous behaviour observed in several physical properties of the mixed valence systems may be due to the change of the 4f-level occupation with temperature.
135
Acknowledgements - One of the authors (GGR) acknowledges Council of Scientific and Industrial Research, Government of India for the financial support in the form of a Research Associateship.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
J.M. Lawrence, P.S. Riseborough & R.D. Parks, Rep. Prog. Phys. 44, 1 (1981). D.M. Newns & N. Read, Adv. Phys. 36, 799 (1987). P. Schlottmann, Phys. Reports 181, 1 (1989). G. Gangadhar Reddy, A. Ramakanth & S.K. Ghatak, Solid State Commun. 71, 395 (1989). G. Gangadhar Reddy, A. Ramakanth & S.K. Ghatak, to be published in J. Phys.: Condensed Matter. U. Rauschschwalbe, U. Gottwick, U. Ahleim, H.M. Mayer & F. Steglich, J. Less Common Met. 111, 265 (1985). S.K. Dhar, Jr., K.A. Gschneidner, C.D. Bredl & F. Steglich, Phys. Rev. B39, 2439 (1989). R. Pott, R. Schefzyk & D. Wohlleben, Z. Phys. B - Condensed Matter 44, 17 (1981). E. Muller-Hartmann, Solid State Commun. 31, 113 (1979). Sanjeev Kumar, P.R. Ahluwalia & K.C. Sharma, Solid State Commun. 73, 65 (1990).