HAL Id: jpa-00216549
https://hal.archives-ouvertes.fr/jpa-00216549
Submitted on 1 Jan 1976HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
ELECTRON CORRELATION EFFECT AT FINITE
TEMPERATURE
K. Chao, K.-F. Berggren
To cite this version:
JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 10, Tome 37, Octobve 1976, page C4-213
ELECTRON CORRELATION EFFECT AT FINITE TEMPERATURE
K. A. CHAO and K.-F. BERGGREN Department of Physics and Measurement Technology
University of Linkoping, Linkoping, Sweden
R&umC. - Le schema variationnel de Gutzwiller est Ctendu a temperature h i e en construisant un ensemble orthogonal des fonctions d'essai pour les electrons correles et en calculant l'entropie des equations de la thermodynamique. L'accroissement de la masse effective de I'electron, la suscep- tibilite de spin de Pauli et le dkplacement de Knight sont obtenus pour tout le domaine de temp& rature et des forces de corr6lation. Cependant, la chaleur spkifique est renforck seulement pour
k-s T/A
2
O,11 ( A = largeur de bande). La theorie est appliqu6e afin d'expliquer les resultats exp6- rimentaux sur Si : P.Abstract.
-
The Gutzwiller variational scheme is extended to finite temperature by constructing an orthogonal set of trial functions for the correlated electrons and by deriving the entropy from the thermodynamic equations. The enhancements of the electron effective mass, the Pauli spin suscepti- bility and the Knight shift are obtained for the entire range of temperature and correlation strength. However, the electronic specific heat is enhanced only for k~ T/A2
0.11 (A = bandwith). The theory is used to explain the experimental results on phosphorus-doped silicon.1. Introduction.
-
The ground state properties of the S-band Hubbard model [l]have been extensively studied by many authors using either the Green-function decoupling scheme [2] or the Gutzwiller's variational method [3] to investigate the magnetic ordering and the metal-non metal (MNM) transitions 241 in strongly correlated electron systems. On the other hand, the finite-temperature regime of the Hubbard model has not been thoroughly examined. The first T # 0 study by des Cloizeaux [5] derived from the Hartree-Fock approximation have been extended in recent years [6j. At large U / A ( A is the bare
bandwidth) Kimball and Schrieffer [7] and Plischke [8]
found a Ntel temperature kTN
-
A2/U. Nevertheless,none of these T # 0 results describes the important features of the enhancements of the susceptibility, the specific heat and the electronic mass, as well as the effect of electron correlation on the Knight shift in highly correlated metals.
Using the variational method at T = 0 Brinkman and Rice [9] obtained a simultaneous enhancements of the Pauli susceptibility and the electronic effective mass by the same order of magnitude as the MNM transition is approached from the paramagnetic metal- lic side. Furthermore, they suggested that the electro- nic specific heat is enhanced in a similar manner. It is therefore the purpose of the present work to generalize the Gutzwiller's approach to finite temperatures as well as to extend the Brinkman-Rice result to this case.
The essential points of the Gutzwiller's variational
method are as follows. The ground state of the Hub- bard Hamiltonian is determined by a detail balance between the electron hopping tg,, and the intra-atomic Coulomb repulsion energy U. The hopping probability varies with the atomic configurations of the initial atom from which the electron hops out and of the final atom to which the electron hops in. Hence both the electron hopping and the irrtra-atomic correlation depend crucially on the atomic configurations. For the simple Hamiltonian eq. (1) without orbital degeneracy, the correlation effect can be measured through the total number of atoms v which are doubly occupied by two antiparallel spin electrons. In pure band model
v = 114 as its band limit. With increasing electron correlation, v decreases continuously to zero where the MNM transition occurs [9]. In Gutzwiller's variational scheme v is treated as a variational parameter to minimize the total energy.
For details of the variational approach, the reader is referred to the original work. Here we only briefly summarize the main results for T = 0. Using the quasi- chemical approximation [l01 for a half-filled band, one found the optimum number of double occupancies in a paramagnetic state as
where the critical value U, =
-
8 E is expressed in terms of the average band energy per electron 5. The energy is so normalized that for a full band Z = 0. Hence U,, is always positive. The electron hopping probability is reduced (or the bare bandwith is nar- rowed) by a factorC4-2 14 K. A. CHAO AND K. F. BERGGREN and the total energy per electron of a correlated system
becomes
E, = (1 - U/U0)' E . (4) Based on these results Brinkman and Rice [ g ] pre- dicted that the susceptibility, the electronic effective mass and the electronic specific heat are enhanced roughly by a factor l/D, the inverse of the band narrow- ing. Furthermore, they found a MNM transition in nonmagnetic phase at U = U,. However, recently Florencio and Chao [l l ] incorporated the virtual elec- tron hopping to the variational method to discover that an antiferromagnetic (AFM) ground state sets in when U
2
0.75 U,. In a model calculation using aparabolic density of states and bare bandwidth A , the Brinkman-Rice result eq. (1) is indicated in figure 1 along the T = 0 axis together with the AFM ground state marked as the darked area.
FIG. l.
-
The optimum number of doubly occupied atoms given by the solid curves.2. Finite temperature variational scheme. - The complete treatment for a generalization of the varia- tional method to finite temperature will be given elsewhere. In this report we only outline the key points and discuss them in physical terms. For T f 0 one should minimize the total free energy with respect to v(T) which is now temperature dependent. We will restrict ourselves to the strongly correlated metallic regime where the system does not show AFM ordering. Then we can consider a microcanonical ensemble with energy sharply peaked at a specific value E,. The free energy of the correlated system can consequently be expressed as
The entropy S, and the energy E, must satisfy the ther-
modynamic equation
which has a special solution as
E, {T, = 2 D(v(T))
C f
{T,49)
~ ( k )+
U,
k
(7)
and
where S ,
{
v(T)) depends on Timplicitly through v( T), andHere we have set the chemical potential to zero since that is true for a symmetric density of state which we will consider later.
When eqs. (7)-(10) are substituted into eq.
(9,
the free energy satisfies exactly the conditionsfor all k. We should point out that the above solution reduces to the correct zero temperature limit. It also approaches to the correct band limit if we require
S ,
{
v(T)}
approaching to zero with sufficiently small U or large T.Gutzwiller [3] has derived the occupation probability for electrons in the reciprocal space at T = 0. His result can be readily extended to the finite temperature as
n(T, k) then consists of two parts : a scaled Fermi distribution function D
{
v(T)}
f {T, v(k))
correspond- ing to the correlation-reduced electron hopping, and a constant ( l-
D ( v(T) })/2 throughout the entire band manifesting again the localized properties of electron due to the correlation. The entropy in a correlated system should be determined by the dual localized- itinerant characteristics. However, because of using the quasi-chemical approximation, they contribute sepa- rately to the entropy eq. (8).In order to construct the correct S ,
{
v(T)3
to give the right entropy for all the possible phases in the U/A-
T/A phase diagram, one must know exactly how the many-electron eigenfunctions evolve from the atomic limit to the band limit. Lacking such informa- tions, however, we found that it is sufficient to use the simple metallic solutiorz S1{
v(T))
= 0 and determine the conditions for its validity. This is not unreasonable because our main interest is the correlated metallic properties which happens to be not depending crucially on the exact form of S ,{
v(T)).
In fact, the following numerical results indicate that for the physical quan- tities we are interested in, this simple metallic solution is valid even for very strong electron correlation.ELECTRON CORRELATION EFFECT AT FlNITE TEMPERATURE quantities per electron. A minimization of free energy
with respect to v(T) yields
where F(T) is the free energy per electron of an uncor- related system and Uo(T) =
-
8 F ( T ) 2 0.To determine the validity of the approximated
metallic solution we have performed a model calcula-
tion using a parabolic density of states of width A
centered at E = 0. In figure 1 we plot v(T) in solid curves for various values of A/U. The dashed curves A and B correspond to S,(T) = 1.6 X In 2 and Sc(T) =In 2
respectively, and the darked area indicates the AFM
phase derived from ref. [ll]. The region above curve A is in deep metallic phase where our solutions are certainly excellent. Since the curves in this region must join smoothly to their T = 0 limits, namely to the Brinkman-Rice solutions [g], our results should be very good estimates in the area between the curve A, the vertical axis and the A/U = 0.61 contour. On the other hand, in the extremely strongly correlated region
A/U
<
0.5, the part of curve B corresponding to the paramagnetic insulating state should be much closer to the horizontal axis. This is the regime where the metallic solution breaks down. However, in this paper we areonly interested in the paramagnetic metallic region with A/U > 0.6 excluding the AFM phase.
3. Results and applications.
-
We will use the above theory to investigate the effect of electron corre- lation on the magnetic susceptibility, the Knight shift and the electronic specific heat, and then apply them to the phosphorus-doped silicon. The impurity band of Si : P can be treated with the S-band Hubbard Hamiltonian through a temperature range from1 Kto 77 K. Its susceptibility, specific heat and knight shift have been measured by Quirt and Marko [12], Sasaki
et al. [l31 and Holcomb et al. 1141, showing very strong
electron correlation effect.
We will not give the detailed mathematical derivation in this paper. The magnetic susceptibility is obtained by minimizing the free energy at the presence of a weak applied field. If x,(T) is the Pauli spin susceptibility per atom at U = 0, then the susceptibility in a correlat- ed system is
where the enhancement factor is
The enhancement factor is shown in figure 2 as the solid curves. For strong correlation and low tempera- ture, the susceptibility is strongly enhanced. This is due
FZc. 2. -Enhancement factors for the spin susceptibility qx
and for the Knight shift VK.
to the increase of unpaired spins (small v) which give large Curie-type contribution. When applied to Si : P,
this model gives excellent agreement with experi- ment [l 51.
The Knight shift can be derived from the spatial dis- tribution of the electronic spins in a correlated system as
where Ko(T) is the Knight shift in an uncorrelated system. In figure 2 we have plotted the qK(T) as the dashed curves. Note that q, is less than q, by
a
factor D(T). The reason is as follows : As the unpaired spins are increased in number by the correlation, they also become more localized. Therefore the magnetic field associated to the conduction electron begins to pile upl%. 3.
-
Electronic specific heat in the metallic phase where the metallic solution is valid.(3-216 K. A. CHAO AND K. F. BERGGREN around the P-impurities. This explains why the 31P line
shift increases sharply while the line shift drops to zero as the MNM transition is approached [13].
The electronic specific heat can be obtained from eq. (6) as
C,(T) = D(T) Co(T)
+
16 U2 S ( T ) ~ T / u ~ ( T ) ~ , (18)where Co(T) is the specific heat from the band model. Since dE,/dT contains a term dv/dT, the slope of the
v(T) vs T curve for constant A/U plays an important role. In figure 1 the correct v(T) in the area between
curve B and the verticsll axis may be as indicated by the dotted curves. The small difference between the dotted curves and the solid curves gives a negligible effect for the susceptibility and the Knight shift which depend on the value of v(T). However, it leads to a large change of the slope and therefore change the specific heat drastically. If we delete this uncertain region, the
specific heat in figure 3 shows a general enhanced characteristics, in agreement with the observed beha- viour [12].
We should make a final remark on the validity of the Gutzwiller's approach as applied to the real systems. Though our nondegenerate band model treatment rea- sonably explains the various experimental results of doped semiconductors, it is indequate to predict quan- titatively the behaviour of transition metal oxides due to the orbital degeneracy. However, the general line of approach to such complicated systems is clear in view of the extensions of the Gutzwiller's original treatment to include the orbital degeneracy [3] and the anti- ferromagnetic ordering [l l], provided one can calculate the Hubbard parameters accurately. Nevertheless, the lack of accurate values for the Hubbard parameters is the fundamental problem for the Hubbard model, not only for the variational approach.
References [l] HUBBARD, J., PYOC. R. SOC. A 276 (1963) 238 ; A 281 (1964)
401.
[2] NAGAOKA, Y., Phys. Rev. 147 (1966) 392 ; H A R F ~ , A. B. and LANGE, R. V., Phys. Rev. 157 (1967) 295 ; ROTH, L. M., J. Phys. Chem. Solids 28 (1967) 1549 ; EDWARDS, D. M. and HEWSON, A. C., Rev. Mod. Phys. 40 (1968) 810 ; ESTERLING, D. and LANGE, R. V., Phys. Rev. B 1 (1970) 2231.
[3] GUTZWILLER, M. C., Phys. Rev. Lett. 10 (1963) 159 ; Phys. Rev. 137 (1965) A 1726 ; CHAO, K. A., Phys. Rev. B 4
(1971)-4034; B 8 (1973) 1088 ; J. Phys. C 7 (1974) 2269 ; OGAWA, F., KANDA, K. and MATSUBARA, T., Prog. Theor. Phys. 53 (1975) 614.
[4] For a general survey, see MOTT, N. F., Metal-lnsulatov Transitions (Taylor & Francis Ltd., London) 1974. [S] DES CLOIZEAUX, J. J., Phys. Radium 20 (1959) 606.
[6] LANGER, W., PLISCHKE, M. and MATTIS, D., Phys. Rev. Lett. 23 (1969) 1448 ; KAPLAN, T. A. and BARI, R. A. J. Appl. Phys. 41 (1970) 875 ; BLACKMAN, 3. A. and ESTER- LING, D. M., J. Phys. C 4 (1971) L 238 ; BARI, R. A. and KAPLAN, T. A., Ph.ys. Rev. B 6 (1972) 4623.
[7] -BALL, J. C. and SCHRIEFFER, J. R., Magnetism in Alloys, Ed. Beck P. A. and Waber J. T. (A. I. M. M. P. E.) 1972, 21.
[S] PLISCHKE, M., Solid State Commun. 13 (1973) 393. [9] BRINKMAN, W. F. and RICE, T. M,, Phys. Rev. B 2 (1970)
4302.
[l01 For a detailed description of QCA, see CHAO, K. A. J., Phys.
C 7 (1974) 127 ; Solid State Commun. 14 (1974) 525. [Ill FLORENCIO, J. Jr. and CHAO, K. A., Phys. Rev. Lett. 35
(1975) 741 ; Phys. Rev. B(to be published).
1121 QUIRT, J. D. and MARKO, J. R., Phys. Rev. B 5 (1972) 716 ;
B 7 (1973) 3842.
[l31 SASAKI, W., KTNOSHITA, J. and KOBAYASHI, S., Phys. Lett.
42A (1973) 429 ; J. Phys. Soc. Japan 38 (1974) 1377 ;
IKEHATA, S., SASAKI, W. and KOBAYASHI, S., J. Phys. Soc. Japan 39 (1975) 1492.
[l41 SANDFORS, R. K. and HOLCOMB, D. F., Phys. Rev. 136 (1964) A 810 ; BROWN, G. C. and HOLCOMB, D. F., Phys. Rev. B 10 (1974) 3394.
