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Discussion of “some questions about electron localization in disordered systems” by F. Brouers
JOURNALOF NoN-CRYSTALLINESOLIDS4 (1970) 433--435 © North-Holland Publishing Co., Amsterdam
D I S C U S S I O N OF " S O M E Q U E S T I O N S ABOUT E L E C T R O N L O C A L I Z A T I O N IN D I S O R D E R E D S Y S T E M S " BY F. B R O U E R S * P. W. ANDERSON Cavendish Laboratory, University of Cambridge, Cambridge, England
A number of papers have appeared in the past few years contradicting the result of ref. 2 of Brouers' article. That of Bonch-Bruevich 5) is the most comprehensive, but the logical fallacy in all, including the present one, is identical. Since our present models both of impurity band conduction and of amorphous semiconductors are to a great extent based on the central concept of localisability introduced in my paper, it seems essential to make a public answer to this criticism. After eq. (7) for the self-energy A~ of the state at site I Brouers makes the statement: " T h e quantity Aa has to be averaged over all the possible configurations of the disordered system". Two paragraphs later, it is stated "Anderson discussed ... (A~) ..." ( ( ) meaning average). Actually, this is precisely the point at which a configuration average may not be taken. It is perhaps as useful as anything to reiterate the language of my original paper. (Omissions are not substantive but omit remarks irrelevant to the present discussion; the notation has been changed somewhat.) "We find that the (self-energy A~) must be studied as a probability variable: that is, we pick a starting atom ! and an energy E+is and study the probability distribution of A~... (we show that) there is a region in which, with probability unity, I m A ~ 0 as s ~ 0 .... The probability distribution of X = I m A t / s . . . falls off as 1/(X) ~ for large X. This shows that the mean value of X is infinite. Clearly this is the result of a very few large values. It can be shown that even these few large values are illusory .... In ref. 2 [A. M. Portis, Phys. Rev. 104 (1956) 584, where the method of the article under discussion is first given] the transition probability is calculated by taking the mean of X .... The resulting finite transition probability is therefore meaningless." As can be seen, my paper was a rebuttal of an earlier one in which A. M. Portis had made the obvious, sensible, but unfortunately incorrect * See preceding paper. 433
434
DISCUSSION
assumption that averaging the self-energy is permissible. Thus it seems that Brouers, Bonch-Bruevich e.a. have simply reiterated the 13 year old Portis argument. The reason that A~ must be studied as a probability variable is obvious: if localizability is to occur, we cannot expect the physical sample to exhibit homogeneity in any sense: every atom is really different from every other. Thus no mean value of any property is really representative, no law of large numbers operates, and the only way in which one may reassert statistical determinacy is by studying distributions not averages. To lay the argument to rest permanently, let us remark two false conclusions to which the averaging theory leads: (1) it does not give localization in one dimension, where entirely independent arguments for it exist; (2) it has no dependence on sample size so it leads to the manifest absurdity that two or three atom wave-functions cannot be local at any energy. (This was pointed out to me by Professor Mott.) Two further remarks. Ziman has pointed out privately to me a much more serious query of the original result: is it stable against small perturbations from an external reservoir? The kind of question here is: The argument is valid at absolute zero or in the absence of phonons, but if the states acquire a breadth through contact with phonons will the original averaging procedure not be correct? Here the basic answer has been given by Mort, from whose work one can assert that it can be shown that even these large values in the above quotation are illusory. Mott's considerations show essentially that there is almost certainly a "hole" in the distribution of poles of A~ centered about the perturbed energy E t of the localized state at l, caused by mutual repulsion of energy levels. This effect probably does indeed cut off the tail of the X distribution and cause stability of the localization effect. But no really rigorous theory of this effect has yet been published. Finally, we should remark that there is completely overwhelming experimental evidence in favour of localizability. The conductivity deduced from the diffusion coefficient of Brouers would have entirely over-ridden the Miller-Abrahams-Pollak-Geballe conductivity in the multitude of cases in which that has been observed. Answer by F. Brouwers to comments of P. W. Anderson
Anderson's theorem (see ref. 2 of my paper) on the absence of diffusion in a random lattice is only valid at low densities of impurity and it applies to the situation described by Miller and Abrahams [Phys. Rev. 120 (1960) 745], where the concentration is so low that "banding does not occur and conduction takes place by hopping". In this case there is no continuous band
DISCUSSION
435
and obviously I do not deal with conductivity in this situation. In the model considered in my paper, where there is a random potential on each lattice site (cellular disorder), it is not surprising that one does not find strictly localized states if the band is supposed to be continuous. The way of averaging the density of states and the self-energy critisized by Anderson has been used extensively by a number of people and led to sensible physical results. This method gives the correct density of states of the disordered system considered here, which agrees with the exact result of Lloyd for a Lorentzian distribution of potentials. The relevant question is to know whether or not to take account of the singularities of the locators when d is averaged. By neglecting the contribution of singularities one would obtain a non-continuous density of states. This is probably true for sufficiently low concentration when the density of states is a weighted sum of 6 functions. [n the model of cellular disorder, to obtain a continuous density of states one must take account of the contribution of these singularities and the states are not localized. The conclusion of my paper is simply that Anderson's theorem on impurity conduction does not prove that in a continuous disordered tightly bound band, states at the centre of the band can become localized, a statement which is often found in the literature. In my opinion, if some states are localized at the edge of an amorphous band, they belong to a discontinuous part of the energy spectrum and, I agree, cannot easily be obtained by the usual average procedure. If these states are "quasi-localized" (this is the case in my model), the transition from band to hopping conduction occurs when 7 is comparable to phonon frequencies. The conclusions of ref. 2 applied to the model of cellular disorder, however, would predict that all the states of the band become localized. There is no evidence of such a sudden transition of the whole spectrum in this case.
D I S C U S S I O N OF " S O M E Q U E S T I O N S ABOUT E L E C T R O N L O C A L I Z A T I O N IN D I S O R D E R E D S Y S T E M S " BY F. B R O U E R S * P. W. ANDERSON Cavendish Laboratory, University of Cambridge, Cambridge, England
A number of papers have appeared in the past few years contradicting the result of ref. 2 of Brouers' article. That of Bonch-Bruevich 5) is the most comprehensive, but the logical fallacy in all, including the present one, is identical. Since our present models both of impurity band conduction and of amorphous semiconductors are to a great extent based on the central concept of localisability introduced in my paper, it seems essential to make a public answer to this criticism. After eq. (7) for the self-energy A~ of the state at site I Brouers makes the statement: " T h e quantity Aa has to be averaged over all the possible configurations of the disordered system". Two paragraphs later, it is stated "Anderson discussed ... (A~) ..." ( ( ) meaning average). Actually, this is precisely the point at which a configuration average may not be taken. It is perhaps as useful as anything to reiterate the language of my original paper. (Omissions are not substantive but omit remarks irrelevant to the present discussion; the notation has been changed somewhat.) "We find that the (self-energy A~) must be studied as a probability variable: that is, we pick a starting atom ! and an energy E+is and study the probability distribution of A~... (we show that) there is a region in which, with probability unity, I m A ~ 0 as s ~ 0 .... The probability distribution of X = I m A t / s . . . falls off as 1/(X) ~ for large X. This shows that the mean value of X is infinite. Clearly this is the result of a very few large values. It can be shown that even these few large values are illusory .... In ref. 2 [A. M. Portis, Phys. Rev. 104 (1956) 584, where the method of the article under discussion is first given] the transition probability is calculated by taking the mean of X .... The resulting finite transition probability is therefore meaningless." As can be seen, my paper was a rebuttal of an earlier one in which A. M. Portis had made the obvious, sensible, but unfortunately incorrect * See preceding paper. 433
434
DISCUSSION
assumption that averaging the self-energy is permissible. Thus it seems that Brouers, Bonch-Bruevich e.a. have simply reiterated the 13 year old Portis argument. The reason that A~ must be studied as a probability variable is obvious: if localizability is to occur, we cannot expect the physical sample to exhibit homogeneity in any sense: every atom is really different from every other. Thus no mean value of any property is really representative, no law of large numbers operates, and the only way in which one may reassert statistical determinacy is by studying distributions not averages. To lay the argument to rest permanently, let us remark two false conclusions to which the averaging theory leads: (1) it does not give localization in one dimension, where entirely independent arguments for it exist; (2) it has no dependence on sample size so it leads to the manifest absurdity that two or three atom wave-functions cannot be local at any energy. (This was pointed out to me by Professor Mott.) Two further remarks. Ziman has pointed out privately to me a much more serious query of the original result: is it stable against small perturbations from an external reservoir? The kind of question here is: The argument is valid at absolute zero or in the absence of phonons, but if the states acquire a breadth through contact with phonons will the original averaging procedure not be correct? Here the basic answer has been given by Mort, from whose work one can assert that it can be shown that even these large values in the above quotation are illusory. Mott's considerations show essentially that there is almost certainly a "hole" in the distribution of poles of A~ centered about the perturbed energy E t of the localized state at l, caused by mutual repulsion of energy levels. This effect probably does indeed cut off the tail of the X distribution and cause stability of the localization effect. But no really rigorous theory of this effect has yet been published. Finally, we should remark that there is completely overwhelming experimental evidence in favour of localizability. The conductivity deduced from the diffusion coefficient of Brouers would have entirely over-ridden the Miller-Abrahams-Pollak-Geballe conductivity in the multitude of cases in which that has been observed. Answer by F. Brouwers to comments of P. W. Anderson
Anderson's theorem (see ref. 2 of my paper) on the absence of diffusion in a random lattice is only valid at low densities of impurity and it applies to the situation described by Miller and Abrahams [Phys. Rev. 120 (1960) 745], where the concentration is so low that "banding does not occur and conduction takes place by hopping". In this case there is no continuous band
DISCUSSION
435
and obviously I do not deal with conductivity in this situation. In the model considered in my paper, where there is a random potential on each lattice site (cellular disorder), it is not surprising that one does not find strictly localized states if the band is supposed to be continuous. The way of averaging the density of states and the self-energy critisized by Anderson has been used extensively by a number of people and led to sensible physical results. This method gives the correct density of states of the disordered system considered here, which agrees with the exact result of Lloyd for a Lorentzian distribution of potentials. The relevant question is to know whether or not to take account of the singularities of the locators when d is averaged. By neglecting the contribution of singularities one would obtain a non-continuous density of states. This is probably true for sufficiently low concentration when the density of states is a weighted sum of 6 functions. [n the model of cellular disorder, to obtain a continuous density of states one must take account of the contribution of these singularities and the states are not localized. The conclusion of my paper is simply that Anderson's theorem on impurity conduction does not prove that in a continuous disordered tightly bound band, states at the centre of the band can become localized, a statement which is often found in the literature. In my opinion, if some states are localized at the edge of an amorphous band, they belong to a discontinuous part of the energy spectrum and, I agree, cannot easily be obtained by the usual average procedure. If these states are "quasi-localized" (this is the case in my model), the transition from band to hopping conduction occurs when 7 is comparable to phonon frequencies. The conclusions of ref. 2 applied to the model of cellular disorder, however, would predict that all the states of the band become localized. There is no evidence of such a sudden transition of the whole spectrum in this case.