COEXISTENCE OF INSULATING AND METALLIC ELECTRONIC PHASES IN DOPED SEMICONDUCTORS AT LOW TEMPERATURE : ELEMENTS OF A CONCENTRATION TEMPERATURE DIAGRAM

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COEXISTENCE OF INSULATING AND METALLIC

ELECTRONIC PHASES IN DOPED

SEMICONDUCTORS AT LOW TEMPERATURE :

ELEMENTS OF A CONCENTRATION

TEMPERATURE DIAGRAM

P. Leroux Hugon, A. Ghazali

To cite this version:

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JOURNAL DE PHYSIQUE _{Colloque C6, supplPment au no}_8, _Tome:_39,_aoiit_1978,_page C6-1074

COEXISTENCE OF INSULATING AND METALLIC ELECTRONIC PHASES IN DOPED SEMICONDUCTORS

AT LOW TEMPERATURE

:

ELEEIENTS OF A CONCENTRATION TENPERATURE DIAGRAF?

P. Leroux Hugon and A. Ghazali

Laboratoire de Physique des SoZides, C. IIT. R. S., BeZZevue,

I, Place Aristide Briand, 92190 Meudon, France.

RBsumB.- Nous pfdsentons une analyse delatransition isolant-mBtal dans les semiconducteurs dop6s 2 basse tempgrature (transition de Ilott) basde sur le formalisme de la fonctionnelle de la densite

(Kohn-Sham) dans l'approximation cellulaire (Wiper-Seitz). Nous montrons que dans un domaine Btroit de concentration en impuretBs, coexistent au niveau de Fermi des Btats localis6s et des Btats Qten- dus et nous presentons des BlBments d'un diagramme de phase. L1interprBtation de diffBrentes exp6- riences 1 partir de ces rLsultats sera brisvement discutLe.

Abstract.- The metal-insulator transition in doped semiconductors at low temperature (Mott transition) is studied with the help of the density functional (Kohn-Sham) formalism within the cellular (~igner-Seitz) approximation. It is shown that localized states and extended states coexist at the Fermi level in a narrow impurity concentration range. Elements of a phase diagram are presented. The relevance of this result to interpret various experiments is briefly discussed.

1. INTRODUCTION.- The metal-insulator (M-I) transi-

-

V + v )

+

-

'

d r

+

v [n(%l

2 m* + + XC

tion which takes place in doped semiconductor /l/ Ir-r'I

e.g. P-doped silicon, when the donor concentration

_JI:

_(r)+ ₌_E:$, I I I increases beyond a critical value is an example of

where the external potential v(;) is here the Cou- electronic phase transition driven by electron-

lomb potential of the impurity nuclei and where the electron interactions. This transition manifests it-

exchange-correlation potential v depends only on

self by the vanishing of the activation energy of

rc

the local electronic density n(r) which is itsqlf the electrical conductivity /2/ but a number of

given by the selconsistency relation : physical properties, namely specific heat /3,4/, ESR

+

spectrum / S / magnetic suscebtibility and NMR spectrum = i

1

i

1

f(~i)

/ 6 / , Raman scattering 171, dielectric constant 181, _{£(ci) being the Fermi function.}

etc, exhibit quiet different behaviours on either

Neglecting at this stage the effect of disor- side of the transition. The aim of this communication

der which arises from therandomess of impurity dis- is to present a formalism which properly accounts for _{tribution, we assume the existence of a quasi-perio-} the electron-electron interactions and allows one

dic nuclear lattice. We then solve variationally to describe the phase transition. Our major finding _{the wave equations by taking trial wave functions} is that insulating and metallic phases do coexist _{choosen as to satisfy the Wigner-Seitz-Bardeen boun-} in a narrow concentration range ; this confirms the _{dary conditions}_/10/. _{In doing so, we account from} conclusions independently achieved by a number of _{the beginning for the formation of impurity band}

experimentalists 13,s-71. _{and/or conduction band. The variational calculation}

provides us with two distinct kinds of solutions, 2 . LOCAL SPIN DENSITY FUNCTIONAL FORMALISM AND

localized ones and extended ones which we identify, CELLULAR APPROXIMAT1ON'- In Order the

respectively, with the bound and the free states of lomb, exchange and correlation terms of electron-

the electron. In order to satisfy the requirement electron interactions whatever be the ground state

that the.8ounds states carry a definite magnetic of the system, insulating or metallic, we use the

moment, we use the local spin-density functional general formalism of the density functional / g / . In

extension

1 1 1 1

of the Kohn-Sham formalism, in which this method, one has to solve a set of one-electron

the exchange correlation potential vxc depends on

wave equations : _{the local electron density n(r) and on the local}+

+

spin polarization p(r), both of them being calcula-

(3)

ted selfconsistently. A more thorough discussion of our method will be given elsewhere.

3. RESULTS.- Within our method, the localized states and the extended states which appear as solutions of the wave equations build up respectively the impurity band and the conduction band of the system. At zero

temperature and for a given impurity concentration (which defines the radius rs of the Wigner-Seitz sphere), we have studied the filling up of the bands as a function of the quasi-wave vector k , the requirement of selfconsistency being fulfilled. It turns out that at low impurity concentration, the filled impurity band is separated by an energy gap from the empty conduction band, while at higher concentration only the conduction band states are occupied up to the Fermi level. In between, there exist a narrow concentration range (4.6 5 r 3 3.6) for which the two bands overlap at the Fermi level (Fig. I(a)).

Fig. 1 : Electronic density of states D(€) for an intermediate impuritjr concentration (re=4.10) at T=O showing the overlap of impur't and conduction bands at the Fermi level. rs = ( ;a: N)-'l3, where a, is the effective Bohr radius and N the donor concentration. Energies E are expressed .in effective Rydberg units and D(&) with the concentration unit

T - 3

No = -(4ao)

.

The inset shows, for the same rs, the variazion of the reduced electron density n(p)=N(p)/N and the spin polarization p(p) inside the Wigner- Seitz sphere (p = r/a,rs).

This implies that localized spin-p.olarized states coexist withextended states, as in the Friedel- Anderson scheme. An example of the resultant spatial distribution of both electron density and spin polarization is given in figure I (b).

This calculation is readily extended to the finite temperature situation. We will focus our attention on the intermediate impurity concentration range. Figure 2 shows that the ratio of the metallic phase concentration to the total one essentially

V I I

0 0.1 0.2 0

Fig. 2 : Variation of the ratio of the metallic phase concentration to the impurity concentration nf/ nt as a function of the reduced temperature 8 =kBT/R for various r, insidethe range of phase coexistence. The lower and upper curves correspond respectively to the insulating and metallic bounds.

increases with increasing temperature. Note that in the figures, energies and temperatures scale with the effective Rydberg R = m*e4

m*

where K is the

static dielectric constant and m the effective mass, and lengths and concentrations with the effective

KP

Bohr radius a. = 7

.

At this stage, it is worth m*e

pointing out that as a consequence of electron- electron interactions the relative position of the band edges on the band filling andhenceon temperature. In other words, this is not a rigid band scheme as in the independent electron picture.

4. COMPARISON WITH EXPERIMENTS.- We have first to point out that our model has been worked out in the simple case of a single isotropic band within the effective mass approximation and for a hydrogenic donor. This applies to e.g. n-CdS / 7 / butthe conclusions remain essentially valid for the more studied materials Silicon and Germanium

Let us emphasize that most of theexperiments, which have been performed as a function of impurity concentration on both sides of the transition,point to the coexistence in a narrow concentration range of metallic and insulating phases. For instance, in doped semiconductors, the spin susceptibility

X

may be interpreted, depending wheter the material is insulating or metallic, in terms of Curie-FJeiss susceptibility

X

or Pauli susceptibility

X

.

cw P

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concentration is slightly higher than the concentration N = 3.2 X 10" cm-3 161 at which the conductivity activation energy vanishes. Specific heat /3/ and transport /5/ measurements performed on the same sample lead to similar conclusion. These experimen- tal evidences for the coexistence of metallic and insulating phases have been interpreted by some authors /6/ as a consequence of disorder which leads to metallic clusters embedded in an insulating ma-

trix. Instead, our model predicts an equilibrium between two strongly interacting homogeneous electronic phases in a semimetallic configuration, even in the absence of disorder. Some support to this picture is supplied by Spin-flip-Raman-scattering experiments. Geschwind et a1 171 suggest also the coexistence, in some impurity concentration range, of two electronic phases, but as they argue, this arises from a dynamic equilibrium between electrons localized on impurity sites and the electron sea.

References

/ l / For a review see the Proceedings of the Confe- rence on Metal-non-Metal Transitions, Autrans,

1976, J. Physique

37

(1976) C4

/2/ See e.g. Yamanouchi, D., Piizuguchi, K. and Sasaki, W., J.Phys.Soc.Japan

22

(1967) 859 / 3 / Marko, J. Harrison, J. and Quirt, J., Phys.Rev.

B - 10 (1974) 2448

/4/ Kobayashi, N. Ikehata, S., Kobayashi, S. and Sasaki, W., Solid State Commun.

2

(1977) 67 /5/ Quirt, J., Marko, J., Phys.Rev. B

1

(1973) 3842

/ 6 / Sasaki, W., in Reference /l/, p. C4-307 /7/ Geschwind, S., Romestain, R. and Devlin, G.,

in Reference / l / , p. C4-313

/8/ Castner, T., Lee, N., Cielaszyk, G. and Salinger

G., Phys.Rev. Lett.

24

(1975) 1627

/g/ Kohn, W. and Sham, L. Phys.Rev.

140

(1965)A1133 /l01 See Anderson, P., Concepts in Solids, (W. Ben-

jamin) 1963, p. 43