INTERRELATION OF CORRELATION AND DISORDER. EXPERIMENT AND THEORY.POSSIBLE CONSEQUENCES OF NEGATIVE U CENTERS IN AMORPHOUS MATERIALS

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INTERRELATION OF CORRELATION AND

DISORDER. EXPERIMENT AND

THEORY.POSSIBLE CONSEQUENCES OF

NEGATIVE U CENTERS IN AMORPHOUS

MATERIALS

P. Anderson

To cite this version:

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EXPERIMENT AND THEORY.

POSSIBLE CONSEQUENCES OF NEGATIVE

U

CENTERS

IN AMORPHOUS MATERIALS*

P. W. ANDERSON

Bell Laboratories Murray Hill, New Jersey 07974, U. S. A. and Princeton University

Abstract. - I have proposed, and Mott and others elaborated, a model of amorphous semiconductors in which there is a fairly high density of localized centers near EF which have effectively negative U and can hence accommodate zero or two electrons, in conjunction with a mobility gap which is of order l U I. Two new aspects of these centers will be mentioned here : (1) Varma and Pandey (private communication) have proposed that such centers form at metal-insulator contacts and there may be direct experimental evidence for them. They could be responsible for the well- known Fermi level pinning effect ascribed by Bardeen to surface states (2). Several arguments suggest that the one-electron gap itself will be a function of the pair state occupation. If so, and the relaxation rates of these states are very slow and cover a broad range as expected, they m'ay be seen to lead to : (1) l/f resistance noise ; (2) Long-period photo-electric phenomena ; (3) Switching.

1. Introduction. - Two general approaches. to the problem of understanding the physical properties of amorphous materials can be identified. Our understanding of ordinary solids is very heavily based on proceeding from detailed structural information and deducing spectra from it (band structure, Fermiology, phonon dispersion), proceeding thence t o understanding the macroscopic properties. Structural studies of amorphous materials are thus the natural starting point for many of us, but in the necessary absence of exact information we may, for example, tend to hypo- thecate structures and speculate about the resulting properties without realizing that the electronic and atomic energies determine the structure and not vice versa.

The second possible starting point is to abandon detailed structural ideas at the outset, the structure perhaps to be understood as a consequence rather than as a postulate. In this approach the idea is to search for structure-independent generalizations. Examples are the ideas of localization and the mobility edge which are common to most random structures ; the idea of tunneling centers ; and even the idea of long-wave- length phonons in random structures, which follows from conservation laws and homogeneity.

The latest such structure-independent generaliza- tion is the postulation of negative U or two-electron centers as a general property of amorphous materials [l], especially in equilibrium or quasi-equilibrium

situations such as glasses. This idea follows most naturally from conceiving of the amorphous material a s a random collection of pair-bonds and other possible

(*) Work at Princeton partially supported under NSF grant DMR-76-00886.

electron pair states such as lone pairs and filled ionic rare gas shells. The overwhelming majority of sub- stances have their electrons paired up in such states, as is evinced by the prevalence of diamagnetic materials in nature, and amorphous semiconducting and insulating glasses are no exception; The general reason is that if a site can accept one electron, in general the atoms will move in response to that electron's presence in order to lower its energy and make it favo- rable for a second opposite-spin electron also to occupy the site. Often the binding energy follows from this tendency, as for instance for the H, molecule. It is only relatively rarely that, as in for example donor states in semiconductors or transition metal salts such as MnO, the interelectronic Coulomb repulsion outweighs the coupling of the electrons to atomic motions and repels a second electron with the positive Van Vleck-Hubbard repulsion U, leading to unpaired electron spins.

We imagine; then, the glass as full of electrons in pair bonds, lone pairs, etc. Without prejudice we will often refer to these states as bonds. In a regular material the energies of the possible states are all the same or regularly distributed, so that there can be, and often is, a gap between occupied and empty states even if we ignore atomic motion. In the irregular structure, however, we do not expect an absolute gap, although since the structure will be stabler, the lower the one- electron energies, there will tend to be a relatively low density of states near the Fermi energy, even an order of magnitude or more less than average. But there should always be a finite density of pair states in the sense that there will be a last pair state occupied as we add more electrons : a bond, for instance, which is the weakest and would be the first to break if we removed electrons.

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C4-340 P. W. ANDERSON

The two-electron center model describes this density of randomly situated states by a Hamiltonian

where

Ei

is the one-electron energy in roughly the equilibrium configuration, and the density of states

p(&) may be imagined to look as in figure 1, with low but finite value near the Fermi energy.

SPECTRA OF AMORPHOUS SEMICOND. MODEL

SITE O = E f

RG. 1.

-

Spectra of amorphous semiconductor model : site energies.

To this we add the net effect of the Coulomb repulsion and the coupling of atomic motion to the electron density at site i, which leads by our postulate to a

negative efective U for thermal equilibrium processes :

X2 =

-

C

u i f f

nir nil

.

i

(2) Two things should be said about X , : First, as I have indicated, U:" is not a fixed constant but also a random distribution like Ei and we will see that (1) and (2) do

not necessarily lead to a sharp gap.

Second,

ueff

is the consequence of relatively large amplitude atomic motions, and for rapid motions (2) is not valid. Among other things, there will be a distribution of relaxation times for thermal readjustment of the occupation numbers ni which will probably

extend from very slow to very fast, having a roughly uniform distribution of activation energies Vi

p(Vi) const

.

(1) and (2) imply that at T = 0 the sites i will, to

a

first approximation, be occupied (ni = nit + nil = 2) up to

Ei = U/2 and empty above that, measuring the energy from a Fermi energy E, = 0. This is because the energy necessary to add a pair of electrons is, per electron

The spectra of unperturbed energy Ei and of E z - , are

contrasted in figures 2a and 1.

There are several other spectra which. are important to the problem. As we pointed out before, the spec- trum even of localized one-electron states has a gap if

Ui is finite for all i, even though neither ther2-electron nor the unperturbed one-electron one does. This comes about because it costs at least U / 2 to create an electron

in an empty state or a hole in a filled one, because of the interaction energy-Uin the' latter case. This is shown in figure 2b.

'SLOW" I-ELECTRON ENERGY h - e

X, FIXED

, MOBILITY EDGES \

FIG. 2a. - Spectra of amorphous semiconductor model :

2-electron energy.

RG. 26. - Spectra of amorphous semiconductor model : slow l-electron energy.

FIG. 2c. - Spectra of amorphous semiconductor model : fast site energy E;fP.

Also relevant is the spectrum of mobile or extended one-electron states. To understand this we must recognize again that (2) is not an instantaneous interaction ; U' is the resultant

U'

= U,,,,

-

U,,,, of a true coulomb interaction

u:,,~

and a phonon displacement effect

The latter comes about indirectly from the electron- lattice coupling at the site i :

and when we choose xi by

we get the attractive phonon terms. For mobile elec- trons also we must include a hopping Hamiltonian

C

T i j Cif, Cja

i j a (6) and if T i j is of normal size xi cannot follow the motion of an electron in an extended state and must be taken as fixed.

The net effect of this is that the effective Hamiltonian for mobile electrons becomes

Xmobile =

C

nib

+

C

T j

C: Cja (7)

i where is given by

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that is, the gap in the site energy spectrum for mobile states is even bigger than that for localized ones (see Fig. 2c). But this gap ,gets filled in by the effect of the bandwidth due to

Tij

and the mobility edge may be above or below the gap for one-electron localized states. I argued that in many glasses the experimental evidence suggests it is below that gap, or at least that there are few localized one-electron states. Figure 3 shows the final outcome of all this discussion, suitably smeared for random variations of Ui. Please note that while the gap for localized states need not be sharp, and presumably seldom is, the mobility edge is by defini- tion a sharp phenomenon. This, to my mind, is one of the stronger experimental confirmations of the generai picture given here : that amorphous systems if anyth- ing show a sharper, steeper gap edge than regular ones. The only vertical edges in nature that I know of are the Fermi level and the mobility edge.

FIG. 3. - Spectra of amorphous semiconductor model : final state density with random smearing.

All of the above is exactly what would follow in detail from the rather cryptic discussion of my letter of a year ago. Now I would like to point out some new general directions in which consequences might flow from this. The first general area has to do with properties of bulk amorphous semiconductors, and is work I am doing in collaboration with Ki Ma at Princeton. The crucial observation here is another consequence which follows from the above model : the energy gaps for one-electron states depend on the occupation of the

two-electron states.

That is (see Fig. 4) if we take two electrons from a state with

to another nearby site where

we also modify the basic densities of one-electron states. Specifically, we shift the effective one-electron spectrum as shown in the figure 4 : There is now an empty state

below U/2 by X, and a full one above - U/2 by Y. Thus both the localized and extended one-electron spectrum have their gaps reduced in proportion to the number of such 2-electron excitations and to their energy shifts. This effect is non-negligible both because of the very

EFFECT OF A 2 ELECTRON EXCITATION ON THE ONE ELECTRON SPECTRUM I

X Y _E~ _{Eeff (ONE}_-_ELECTRON ENERGY)

EFFECT OF GENERAL EXCITATION IN THE 2 ELECTRON SPECTRUM

ONE - ELECTRON SPECTRUM

FIG. 4a.

-

Effect of a 2 electron excitation on the one electron spectrum.

FIG. 4b. - Effect of general excitation on the 2 electron spectrum.

high density of such states and because the resistivity depends exponentially on EJT, so that a small shift in E, can have a large effect.

We propose that several well-known effects may follow from this.

(1) Switching : A high field will tend to redistribute carriers among the 2-electron centers essentially ran- domly. The distribution (3) of relaxation times means that slowly relaxing centers will remain out of equilibrium, at least in the presence of a continued current, and as a result the gap will remain smaller. That is, a high field can effectively redistribute the bonding sites, resulting in a decreased gap and much lower resistance. Once the resistance is low the presence of a current will maintain the disequilibrium in the 2-electron states. Experimentally, one might try to measure the gap in switched material to confirm this.

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C4-342 P. W. ANDERSON

(2) Long period photoelectric effects. Again, we rely on the idea that excited carriers will not recombine at first into the most stable possible chemical bonds but can become trapped in less stable ones, resulting in a lowered mobility gap. Thus high photo-excitation can lead to a temporarily unstable state with much smaller gap and higher conductivity.

(3) Resistance noise. In thermal equilibrium the bonds will form and break at a rate given by the spectrum of relaxation times zi of (3). In addition to other electrical effects., there will be a corresponding fluctua- tion in the mobility gap locally, and thus in the number of mobile carriers. This resistance noise is one of the few direct effects which can allow us a spectroscopy of the bond states, their field screening effect being perhaps the major other one. If the distribution of the activation energies

Vi

is uniform in energy, the spectrum will be of the canonical l/f type. Although l/f noise was discovered originally in amorphous carbon, we have been unable to find many definitive measurements of it in characteristic amorphous systems, and we would urge hoise studies as an important approach to the fundamental phenomena in these materials.

As a final topic I would like to move out of the normal range of amorphous physics to the surfaces of semiconductors. Specifically, these are ideas stemming from a suggestion by K. C. Pandey and C. M. Varma [3], who remarked that except under the most rigorous conditions we may surely expect an amorphous, or at least highly disordered, situation at any metal-semiconductor contact. The metal-semiconductor bonds will in general not all have the same character or strength, and again we have every reason to expect that many of them will be weaker than the internal bonds of either component specifically those in the semiconductor. They will therefore lead to 2-electron states in the semiconductor gap, which could have the effect of pinning the Fermi level near the middle of the gap iust as they do in an amorphous semiconductor, and with a high density of such states

-

1015/cm2 or more, there would be no difficulty in explaining the well- known pinning phenomenon ascribed by Bardeen to surface states. D. Haldane, my student at Princeton, has been studying the properties of 2-electron reso- nances with negative U (a negative U anderson model) in contact with the metaIlic Fermi sea and finds that the same n = 0 o n = 2 transition takes place as for localized states, so that Fermi level pinning can take place. Interestingly, recent PS and other data of Rowe

Refere

/l] ANDERSON, P. W., Phys. Rev. Lett. 34 (1975) 953. The main results of that paper are restated and elaborated in a special case by

ADLER, D. and YOFFA, E., Phys. Rev. Lett. 36 (1976) 1197.

[2] STREET, R. A. and MOTT, N. F., P. R. L. 35,1293 have specia- lized the arguments of ref. [l] to a specific electronic model, as have Adler, Kasner and Fritsche in a preprint.

I confess to being somewhat disturbed by these papers'

and Margaritondo reveal very amorphous-semiconductor like sharp edges to the gap as the first few layers of metal are laid down on a semiconductor (see Fig. 5). It is noteworthy that semiconductor surfaces and contacts are well-known to be strong sources of l / f noise.

Si(t1t)

CLEAN 7 x 7

-8 -6 4 -2 0.E:

ENEROY (QV)

FIG. 5. - XPS spectra of Si surface as atomically thin layers of Ga are added. Note disappearance of l-electron spectrum

in gap, appearance of sharper gap edge. After J. Rowe.

In conclusion, the postulate of negative U bond states seems to lead to natural explanations of most of the outstanding mysteries of amorphous semiconductor physics, and promises to extend to surface studies as well via the negative U Anderson model.

Acknowledgements. - I would like to acknowledge the use of unpublished remarks from C. M. Varma, F. D. M. Haldane and Ki Ma as stated in text. Dis- cussions with R. G. Palmer, J. Rowe, C. M. Varma and N. F. Mott have been most useful.

nces

referring to the two electron centers as defects, since one of the strongest features of ref. [l] is that it is consistent with the fact that a glass properly has no defects per se : it has only a statistical distribution of weaker or stronger two- (or sometimes one-) electron bonds. It is, of course, obvious that atoms with lone pair orbitals are more prone to have weakly bound pairs, as both of these papers emphasize.