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High-pressure phenomena in glasses: The role of soft atomic configurations
Journal of Non-Crystalline Solids 232±234 (1998) 257±266
Section 7. Interrelationship between structure and properties
High-pressure phenomena in glasses: The role of soft atomic con®gurations Michael Klinger * Department of Physics, and Jack and Pearl Resnick Institute of Advanced Technology, Bar Ilan University, 52900 Ramat Gan, Tel Aviv, Israel
Abstract The revealed suppression of the anomalous low-energy dynamical, and the related low-temperature, properties of glasses at high pressures is described. Some similarity of this phenomenon to recently observed pressure-induced transitions between `strong' and `fragile' glass-formers (`polyamorphism') is noted. Changes in the electron properties of semiconducting glasses, which are due to delocalization of negative-U centers as basic charge carriers, at high pressures are predicted and analyzed. The phenomena are related to soft atomic con®gurations available in glasses. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Two dierent theoretical approaches are mainly applied for describing properties of amorphous solids. One is based on detailed structural information (see, e.g., [1]). The other proceeds from general concepts that are rather independent of the structure details [2±4]. The purpose of the present paper is to discuss correlations between the variation of some structural features and macroscopic changes in dynamical properties and electronic processes in glasses, actually being 3-dimensional systems. The correlations result from the general concepts of `soft atomic con®gurations' [4] and `negative-U centers' [2±5] for pressure-induced phenomena in glasses as the representative example.
The uni®ed theoretical approach, the `soft con®guration model' (SCM) of glasses, has been developed for the quantitative description of both the low-energy excitations in atomic dynamics of glasses and the electron properties that are related to negative-U centers of non-metallic glasses [4]. The SCM has been derived from the observed similarity between the elastic properties of glasses and their respective crystals, from the macroscopic isotropy of glasses, and from mathematical `catastrophe theory' [6] applied to analyze features of the potential energy of spatially ¯uctuating mediumrange-order (MRO) structures. The basic result was that each soft atomic con®guration, of the MRO length scale LMRO between 1 and 2 nm, actually contains a single soft atomic-motion-mode (x) in which, generally speaking, the potential energy is anharmonic [4] V x V x; g; n A gx2 nx3 x4 :
* Tel.: +972-3 531 8424; fax: +972-3 535 3298; e-mail: [email protected]. 0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 4 0 0 - 1
1
The soft mode as basic random parameters {g > 0 or g < 0} and {n > 0 or n < 0}, with {|g|, n2 } 6
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M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
g* const 1, e.g., g* 0.1, and the resulting spring constant, k k(g,n;A) k0 2A 30±60 eV, the general range for (non-molecular) solids. The modes are localized states of cooperative atomic motion on the length scale LMRO , involving NS 30 (10 6 NS 6 102 ) atoms, at 1 NS 6 NMRO (LMRO /a)3 < 103 with mean atomic spacing a 0.3 nm [4,7,8]. Actually the SCM is the theory of low-energy (E) excitations (in addition to sound waves) in atomic dynamics, their spectra, E(g, n), in the Schroedinger equation with V(x) of Eq. (1), and of the related low-temperature (T) and lowfrequency (x) properties of glasses, at {E;T;hx} hxD , the Debye energy [4]. The new energy scale w 1±2 meV ( 6 0.1 hxD ) obtained in the SCM, is the lowest vibrational energy characteristic of strongly anharmonic soft modes (Eq. (1)), which arises in the quartic anharmonic potential (Ax4 ). Excitations of substantially lower energy (E w) are the well-known `tunneling states' (cf. [9]) while those of higher E, w E hx D , are quasiharmonic soft vibrations. The basic spectral properties, the density-ofstates (DOS), n(E), of the soft-mode excitations with energy spectrum, E(g, n), is related to the probability density distribution, U(g, n). The DOS appears to exhibit similar features, including the main maximum at E EM , for any type of glass. In general, the maximum of n(E) is located either at EM w, the quasiharmonic vibrational DOS nqh (E) growing with E, for w Eb < E hxD at Eb 3w, in a Debye-like manner (Ea , at a 2), or at higher EM , Eb < EM hxD , with nqh (E) increasing, at w 6 E < EM , more rapidly (Ea , at a 3±4) [4,7,8]. Correspondingly, the glassy low-temperature properties are qualitatively similar while quantitatively dierent for different types of glass. Some recent data (see, e.g., [10]) appear to indicate a quantitative dierence in the low-temperature properties of `strong' (e.g., g-SiO2 ) and `fragile' (e.g., Ca0:4 K0:6 (NO3 )1:4 ) glasses, in Angell's classi®cation [11]. Actually, the SCM presents a continuous transition from standard rigid harmonic single-well con®gurations (k k0 ) to quasiharmonic and then to strongly anharmonic, including double-well, soft con®gurations (k k0 ), and ®nally to con®gurations with
double-well potentials and rigid potential wells, as g decreases from g P 1 to g < 0 at |g| 1 and then to |g| P 1. Thus, the `tunneling states' in double-well potentials are not likely to occur at 0.1 6 E 6 1 K, if the other types of soft con®gurations (soft-mode excitations) do not [4]. The SCM predicts that, for any type of glass, the generalized susceptibility, v, of the soft con®gurations to distortions is anomalously higher v k ÿ1 v0 k0ÿ1 ;
2
the general scale, v0 , characterizing the vast majority of atoms in a glass. Thus, the soft con®gurations are more aected by forces such as pressure giving rise to distortions, than the standard con®gurations at k k0 . Therefore, the soft con®gurations of the glass structure can be essential for properties related to structure distortions, even though the (relative) atomic concentration of these con®gurations, ca º Na a3 , is rather low (ca (0) 2 ´ 10ÿ2 ) at ambient pressure (p º 0). The concentration is estimated from some general considerations or from comparing the SCM properties depending on ca , to the appropriate data [4,7,8]. Since v is 100 P v0 P 30, the role of the soft con®gurations for the mentioned properties is comparable at ambient pressure to that of the vast majority of atoms with v v0 , at ca (0)v ctot v0 v0 . It is worth adding that ca (0) 10ÿ2 for `non-glassy' materials (e.g., a-Si), not exhibiting glassy low (<50 K) temperature properties. A principal consequence of the occurrence of the soft con®gurations in glasses at ambient pressure was found to be the existence of localized `negative-U centers' as basic charge carriers in semiconducting and related glasses (SGs) with moderately wide mobility gaps, of width, Eg (0), 1±4 eV [4,5]. The negative-U centers actually determine the main electronic properties of the glasses at ambient pressure [2±4], as well as many of those at high pressure (Section 3). In the SCM, the charge carriers in question are identi®ed as singlet electron (hole) pairs, occupying the negative-U states in a rather wide energy band around the Fermi level (per particle) near the middle of the mobility gap. The basic gap states, exhibiting a negative pair correlation energy (inter-electron attraction), U(g), are formed due to a new type of
M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
self-trapping in the soft con®gurations at ambient pressure (but not at high pressure, see Section 3) [4,5]. Hybridization of states in the mobility gap is the decisive feature of this type of self-trapping, providing a negative pair-correlation energy which, in general, is |U(g)| |Ueff | º |U(geff )| @ Eg /2 and, in particular, is large in magnitude at ambient pressure, |Ueff (0)| @ Eg (0)/2 P 1 eV. Here U(g) W2 (g)/2 < 0 with W2 (g), the pair self-trapping energy (energy gain), increasing in magnitude with decreasing g P 0 for the most numerous quasiharmonic soft con®gurations. The extremely large magnitude of W2 (g) for the soft con®gurations is the main cause of the decisive role of the hybridization of states in the self-trapping process. The parameter, geff , introduced above is the eective value of the softness (g) characteristic of most of the negative-U centers at the Fermi level, f(0), near the gap middle, Eg (0)/2. The related thermal-equilibrium concentration of the negative-U centers is also considerable, 10ÿ3 P ca (0) P 10ÿ4 , in accordance with experimental data. It is worthy of note that the mobility gap width, Eg Eg (p) of SGs, is observed to decrease with growing pressure [12], Eg 0 P Eg p P Eg pg 0
3
at Eg (p) @ 2f(p). Here pg is the critical pressure for the semiconductor (p < pg ) ± metal (p > pg ) transition, 105 bar 6 pg < 106 bar, and f is the Fermi level referred to the valence-band mobility edge Ev . High-pressure induced macroscopic changes in the dynamical (and elastic) properties of glasses and in the electron dynamics of semiconducting and related glasses are discussed, respectively, in Sections 2 and 3. The conclusions are presented in Section 4.
259
P 107 bar. Here L is the typical length of the quasimolecular structure unit, L 1.5 nm, and EB is the typical covalent or ionic bonding energy, EB P 1 eV, while z is the average coordination number. At accessible lower p p0 , macroscopic changes in properties associated with phase transitions may occur only in crystals in which extended soft-phonon modes are available. One may suggest that, in general, some softness is needed for macroscopic changes in properties of solids at p p0 . In glasses, at ambient pressure, softness is generally distributed in the random soft con®gurations for jgj 1, so that pressure-induced macroscopic changes in properties may occur even at p p0 . The glassy features discussed below are related at ambient pressure to the soft con®gurations (Eq. (1)) and the probability distribution density, H(g), of the random `softness' (g) at 0 < g 6 g* 0.1, for most of the con®gurations with quasiharmonic soft modes. The main problem is the affects of pressure on H(g) and the atomic concentration, ca º ca (p). A plausible model identi®es H(g) with the appropriate Gaussian - like function Z1 2 H g dn U g; n H0 g expÿgÿ1 c 1 ÿ g=g0 ÿ1
4 at H0 (g) @ H0 const. Here g0 const @ 1 characterizes the standard rigid harmonic con®gurations at k @ k0 for the vast majority of atoms of the glass; generally speaking, H0 ¹ H0 (g > g*). Then, at least at not too high T < Tg , Zg ca
dg H g:
5
ÿ1
2. Pressure-induced macroscopic changes in dynamical properties 2.1. Problem and results The eect of (hydrostatic) pressure on properties is dierent for glasses and crystals. For the latter, generally speaking, macroscopic changes are expected to occur at very high p P p0 EB /zL3
The basic parameter, which may depend on pressure, is gc gc (p), whereas both g and g0 , by de®nition, are actually constants. The approximate expression obtained, which describes the features and scale of gc (p), is as follows [13]: 0 ÿ1 gc gc p g1 c gc pcr ÿ ppg H pcr ÿ p
6
with h(x) the step function (see Eq. (10) and (11) 0 for parameters g1 c , gc and pcr , pg ).
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M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
2.2. Discussion Two arguments are useful for deriving Eq. (6). One is that glassy low-temperature properties at ambient pressure are observed, in some types of glasses at least, to be suppressed with increasing average coordination number, z. This suppression is certainly the case for amorphous semiconductors, as a representative example, that are covalent glasses for zmin 6 z < zcr (e.g., zmin z(g-Se) 2 6 z 6 z(g-GeSe2 ) 2.67 < zcr ), and become nonglassy covalent materials (without glassy low-temperature properties) for zcr < z 6 zmax 4(a-Si), with 2.67 < zcr < 3(a-As) for the `critical' value zcr [13,4]. Furthermore, a change in the elastic (bulk) modulus, B[z], was observed from B[z 6 zcr ] 105 bar for SGs to B[zcr < z] 6 B[zmax ] 106 bar for a-Si. As noted in [14,13], these features are related to the remarkable dichotomy of bonding forces, strong (covalent) and `weak' (van der Waals) at z < zcr while actually strong at zcr < z 6 zmax . I assume that z z0 + Dz, with z0 and Dz the contributions of the strong bonding in each quasimolecular unit related to the MRO and of the weak bonding between the random units, respectively, at 0 < Dz z0 and z < zcr . The eective dimensionality, d0eff , of the `decoupled' units is expected to be low (e.g., d0eff < 1) compared to that, deff , of the extended random structures consisting of cross-linked units, d0eff < deff 6 3. The theory indicates that, on average, the weaker bonds increase in strength while the other bonds remain unchanged (z0 const) with growing coordination (Dz), the eective dimensionality deff increasing self-consistently. Actually Dz and deff might be found as self-consistent solutions of certain equations (to be investigated elsewhere). Within the framework of the soft-con®guration model of glasses [4], the suppression of the glassy low-temperature properties with increasing z at ambient pressure means that for the amorphous semiconductors mentioned ca [z < zcr ] ca [z 3 > zcr ] ca [zmax 4], at zmin 6 z 6 zmax . Here ca [z < zcr ] is considerable, around 10ÿ2 in scale, for the SGs, whereas generally speaking ca [z] is negligible at zcr < z 6 zmax for the non-glassy materials mentioned. In this sense, the dierence between ca [3] and ca [4] may be neglected. Actually in the
whole range of z in question, dca /dz < 0, i.e., also dgc /dz < 0, but rather {jdca /dzj at z zcr } < {jdca /dzj at z P zcr }, in accord with the expected sharp, though continuous, transformation of the glasses into non-glassy materials at z zcr . Then, the linear approximation 0 gc z g1 c gc zcr ÿ zH zcr ÿ z;
7
extrapolating the expansion around zcr to both zmin and zmax , may be applied, where 0 g1 c gc zmax gc fjdgc =dzj at z zcr g
g1 c
0.1g0c ,
8
g0c
with 0.1 for typical e.g., ca (0) 10ÿ2 at z < zcr . Another argument for estimating gc (p) is that some empirical data, e.g., concerning pressure effects on the optical-absorption edge, for the glasses in question, may give rise to the following estimate [14]: 0 < dz=dp 6 10ÿ5 =bar;
9
so that (0 <) Dz(p) º z(p) ) z(0) < 1 at the applied pressure, 0 < p 6 105 bar, in question. This relation may be interpreted, within the framework of the previously mentioned model, as follows. The strong (e.g., covalent) bonds actually remain intact while the weak bonding forces may increase in strength, continuously transforming into a kind of strong interactions, with increasing pressure up to p P pcr at zcr z(pcr ). This transformation, as well as the related transformation of the glass into a dierent kind of amorphous material at z z(p) > zcr , is due to the pressure-induced decrease of the average distance between the quasimolecular units (densi®cation) until the dichotomy of forces disappears at a pressure, pcr , and only strong bonding forces exist in the resulting `continuous random network' structure of higher eective dimensionality deff (e.g., 2 6 deff < 3), at p P pcr and Dz P 1. In other words, one may assume that dz/dp @ dDz/dp jdz0 /dpj and 105 bar 6 pcr p0 and, from continuity reasons, that z z(p) is a simple function. The related continuous transformation of the random structure may be characterized in a more accurate way by Dz(p) z(p) ) z(0) and deff (p) as self-consistent solutions of the appropriate equations. Then, in a linear approximation, at p 6 pmax and z(pmax ) zmax ,
M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
zcr ÿ z p z pcr ÿ z p pcr ÿ ppgÿ1
10
with gc (p) and ca (p) monotonically decreasing with p, at dca (p)/dp < 0 and {jdca /dpj at p < pcr } {jdca /dpj at p P pcr }. In Eq. (10): ÿ1
106 bar P pmax > pg f dz=dp at p pcr g ÿ1
P pcr zcr ÿ z 0 dz=dp0 P 105 bar
11
with 0 < zcr ) z(0) < 1 for the glasses in question. The very existence and scale of the characteristic pressure pcr and pg are essential, rather than their precise value. Taking into account Eqs. (7)±(10), I establish Eq. (6). The latter means that the distribution density, H(g), is decreased at gc , decreasing from g0c to gc , so the soft con®gurations are suppressed, and the related glassy features are eventually lost, as the atomic concentration, ca (p), is diminished by orders of magnitude when p > pcr . The analytical expression, specifying Eq. (6), may be found by taking into account the self-consistent solutions of the noted equations for Dz(p) and deff (p) as functions of p. Actually the macroscopic changes in the dynamic properties of glasses at high p ( 6 106 bar) may be rather general, as generally glasses have competing stronger and weaker bondings. Macroscopic changes in the elastic properties of non-metallic glasses also may be predicted from Eqs. (6)± (10) at high p P pcr : the elastic modulus increases in scale, e.g., from B 105 bar at p < pcr to B 106 bar at p > pcr . The resulting amorphous material at high p ( P pcr ) is expected to be either a higher-density non-glassy material, which does not exhibit glassy low-temperature properties, or at least at not too high p (possibly, at pcr 6 p < pg ), still a glassy substance though more dense and fragile than the initial glass. Actually the glassy low-temperature properties appear to be less evident in fragile glasses than in strong ones (see Section 1 and, e.g., Ref. [10]). This macroscopic change in structure and properties of glasses, due to the predicted pressure-induced phenomenon of suppression of soft atomic con®gurations, is similar to the remarkable `vitreous polymorphism', or `polyamorphism', recently revealed and studied in real compression-in-
261
duced vitri®cation experiments and computer simulations (see Ref. [15] and references therein). The question is whether or not such a similarity is indeed due to essential correlations between the polyamorphism, implying that the glass may exist in more than one distinct metastable state separated by barriers in the con®guration space, and the pressure-induced suppression of soft con®gurations of a glass. At present the question cannot be answered comprehensively. 3. Pressure-induced transformations in electron dynamics and transport 3.1. Problem and results High pressures are predicted, in the theory under discussion, to give rise to transformations in the electronic properties of glasses, as recently shown by Klinger and Taraskin [5] for the electron dynamics and transport processes of semiconducting and related glasses (SGs), actually threedimensional systems, as representative examples. In this connection, two recently obtained principal results are presented and discussed in this section. One result is that the negative-U centers as basic Fermi-degenerate charge carriers, localized at ambient pressure in SGs (Section 1), become non-localized for high enough p < pg (Eq. (3)). Another result is that, due to the delocalization of the negative-U centers, the electron transport at high enough p < pg and not too high T < Tg is of a metallic type, unlike the thermally activated transport which is not related to negative-U centers at ambient pressure. In accordance with the well-known criterion of the Anderson±Mott localization±delocalization transition, a quantum particle (quasiparticle) or its state in a three-dimensional system is localized or non-localized, respectively, at [3,16] c q 2zJav q=Dav q < ccr const 1 or c > ccr :
12
Then, c(qc ) ccr corresponds to such a transition, at a critical value, qc , of the `ruling' parameter, q, which commonly is considered as a continuous
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M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
one. Variations of the latter may give rise to changes in both the eective (average) amplitude, Jav (q), of the particle tunneling between the relevant localized states, or `sites', and the mean ¯uctuation, Dav (q), of the respective energy levels, in the disordered system in question. As shown in what follows, pressure, p, takes the part of the parameter q for the negative-U centers, so that the Anderson±Mott transition occurs at p pc2 , from localized (p < pc2 ) to non-localized (p > pc2 ) states, in the `random lattice' of relevant sites. The sites are suggested to be the negative-U states formed due to the revealed `non-polaronic' selftrapping in appropriate MRO atomic con®gurations (Section 1). The latter are rather rigid ones (g g0 @ 1), typical of the vast majority of atoms, at high pressures p P pcn under discussion (or soft con®gurations (Eq. (1)), at ambient and low pressures). A similar transition may be predicted for single-particle excitations created by dissociation or ionization of the negative-U centers, at p pc1 . Both transitions are of the same type as the wellknown Anderson±Mott transition at ci ccr in an `impurity band' in the inter-band (mobility) gap of a doped semiconductor, for which the impurity concentration, ci , is the `ruling' parameter q. Then, the Anderson±Mott transition has to occur at the critical pressure, pcn (n 2 or n 1), as the relevant solution of equation c 2zav Jn pcn =Dav pcn ccr 1
13
in the semiconducting phase of the glass for pcn < pg , with J n av (p) Jn (p) and z 6 zav . Numerical calculations of pcn from Eq. (13) have been performed, by applying the analytical expressions obtained for Jn (p) and Dav (p) (Section 3.2) at n 2 and n 1, for a basic set of typical values of the parameters used [5,17]: Eg 0 2 eV 10Ud ;
q=a 3;
hX0 0 200K hxD =2
and
c2 0 10ÿ3 14
with the valence band width, D0 , equal to 2.5Eg (0) and ccr 1. Here q stands for the localized state size, X0 (p) is the characteristic frequency of the most important atomic motion mode for the formation of the negative-U centers, and c2 (p) is the thermal-equilibrium concentration of the nega-
tive-U centers, c2 (p) c1 (p, T), the concentration of the single-particle excitations (see Section 3.2). The critical pressures pc1 and pc2 are indeed found to be less than pg : 0:75 e1 6 pc1 =pg 6 0:85 e2
and
0 6 pc2 ÿ pc1 =pg 6 0:02
15
with pc2 /pg < 1 at 0 6 e2 6 e1 6 0.1, and the dependence of pcn on the parameters (Eq. (14)) is weak. Thus, the pressure-induced phenomenon of delocalization of the negative-U centers, at pc2 6 p < pg , and of their single-particle excitations, at pc1 6 p < pg , indeed, appears to be predicted theoretically. Generally speaking, the delocalization phenomenon gives rise to essential transformations in nonequilibrium properties, particularly in transport processes and related phenomena. The representative example is dc conductivity, rdc (p, T), as an essential transport coecient. The resulting expressions for rdc (p,T) are as follows. At the highest pressures, pc2 6 p < pg , and not too high T < {Tg ; Tk (p)}, rdc (p, T) is basically related to the non-localized negative-U centers as main current carriers for which c2 (p) is rather independent of T (see [17]). Then, rdc (p, T) is of a metallic type 2
rdc p; T rdc p; T 2
4jejN0 c1=3 2 plIR p; T 24r0 c2 pzJ2 p=jh
16 2 lIR
the related with N0 º aÿ3 , j 1 ) 3, and (drift) mobility. Here the parameter of the generalized Einstein formula, h n(dn/df)ÿ1 , is h @ f(p) @ Eg (p)/2 > T for the main charge carriers, negativeU centers around the Fermi level f (Eq. (3)), of concentration n c2 (p), whereas it is h @ T for other, non-degenerate, charge carriers such as standard electron±hole pairs and single-particle excitations of the negative-U centers. Generally speaking, rdc (p, T) increases here with growing p, whereas it does not noticeably change with T at T < Tk (p), exhibits a pronounced kink at T Tk (p), and increases by the Arrhenius law (18), with an activation energy, Wr (p) @ Eg (p)/2, which decreases with pressure at T > Tk (p) > Tg . If, however, Tg < Tk (p) at pc2 6 p < pg , rdc (p,T) is of a metallic type, basically independent of T,
M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
over the whole range of the glass for T < Tg . This case is realized (in numerical calculations) for SGs with typical Tg < 103 K. For intermediate pressure, pc1 6 p < pc2 , rdc (p, T) is related mainly to the non-localized single-particle excitations of the negative-U centers, and is thermally activated. Then, for not too high 2=3 T < Td (p) Eg (p)/ln [c2 (p)D0 /2zJ1 (p)], 1
rdc p; T rdc p; T 1
jejN0 c1 p; T lIR p; T r0 T1 p=T expÿWr p=T ;
17 1=3
where Wr (p) |Ueff (p)| @ Eg (p)/2 and T1 (p) 2c2 (p)zJ1 (p). Typically Td (p) is rather high, Td (pc1 ) P 103 K > Tg (SGs) at c2 (pc1 ) 10ÿ2 P 10c2 (0), so Eq. (17) holds for all T < Tg at pc1 6 p < pc2 . Both Td (p) and T1 (p) increase with p. In the alternative case for Td (pc1 ) < Tg (if realizable in SGs), rdc (p, T) at Td (pc1 ) < T < Tg , as well as rdc (0, T) at 0 < T < Tg and ambient pressure [3], would be related to the contributions of standard electron±hole pairs, and described by the Arrhenius law rdc p; T r0 expÿWr p=T :
18
Here r0 const 103 ) 104 /X cm at T < Tg and the activation energy Wr (p) @ Eg (p)/2. Note that Eqs. (16) and (17) hold for p not too close to the semiconductor±metal transition at pg (Eq. (3)), with |Ueff (p)| @ Eg (p)/2 T, for which the thermal-equilibrium concentration of the single-particle charge carriers is low, c1 (p, T) c2 (p). 3.2. Discussion The main results, Eqs. (13)±(17) may be interpreted as follows. As recently shown (see Ref. [17]), the concentration, c2 (p), of the basic charge carriers, the negative-U centers (Section 1), in SGs depends on pressure in a non-monotonic manner. It has a minimum for p º pmin , 0.35 6 pmin /pg 6 0.45, and increases with pressure to values, c2 (p) 10ÿ1 c2 (0) 10ÿ4 , at 0 < pg ) p pg . For the latter c2 (p) actually characterizes weak negative-U centers, of which the eective negative pair-correlation energy, Ueff (p), typical
263
of most of the centers is small in magnitude compared to that at ambient pressure, |Ueff (p)| |Ueff (0)| (Eq. (19)). Two basic eects and their competition determine the non-monotonic dependence of c2 (p) on increasing pressure [17]: (1) The probability distribution density, H(g), for the random con®guration softness becomes more narrow, with a decrease in its width, Dg(p) g1=2 c (p), at growing densi®cation of the glass at T const (Eqs. (4) and (6)). The related pressure induced suppression of the soft con®gurations manifests, in particular, as a decrease in the atomic concentration of the localized soft modes that are associated with the electron pair self-trapping and to the resulting formation of the negative-U centers (Section 1). (2) The eective softness, geff (p), of the local atomic con®gurations of nanometric size, in which most of the negative-U centers are formed with energies (per particle) around the Fermi level near the mobility gap middle, and the related eective negative pair-correlation energy, Ueff (p), are characterized as follows [5]: geff p c =Eg p Ud and Ueff p U geff Eg p=2
19
with Eg (p) @ Eg (0)(1 ) p/pg ) (Eq. (3)) and the practically pressure independent eective electronatomic-motion mode coupling energy c 0.1Eg (0) and Hubbard pair-repulsion energy Ud 0.1Eg (0). Thus, geff (p) increases while |Ueff (p)| decreases with growing pressure, so that geff (p) 0.1 and Ueff (p) Eg (0)/2 P 1 eV at p 6 0.1pg (strong negative-U centers), whereas geff (p) 1 and Ueff (p) Eg (p)/ 2 6 0.1Eg (0)/2 at high p, 0 < pg ) p 6 0.1pg (weak negative-U centers). Since the distribution density, H(g), monotonically increases with g at least for the most numerous quasiharmonic soft con®gurations (Eq. (1)) at 0 < g < g0 @ 1, the concentration, c2 (p), of the weak negative-U centers also rapidly increases. In fact, these negative-U centers can be created because of self-trapping even in common `rigid' atomic con®gurations when geff (p) @ g0 º geff (p0 ) @ 1 (and p0 6 p < pg ), as Eg (p) is small enough to be close to the pair selftrapping energy, W2 (g) @ c /g [5], at g @ g0 .
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M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
Let us note that the localized states of the negative-U centers and their single-particle excitations are expected to be of the standard type (v(r;xn ) v0 (xn ) exp()r/q), with xn the equilibrium displacement in the atomic-motion mode most essential for the self-trapping, and q const (2 ) 3)a at p < pg ) and that their mean separation, ÿ1=3 R2 (p) ac2 (p), decreases from R2 (0) P 10a to R2 (pg ) a. Then, the average overlap, Iav , of the states and the related tunneling amplitude, Jn , strongly increase with growing p. Thereby, the Anderson±Mott delocalization of most of the negative-U centers, around the Fermi level, could indeed occur in the mobility gap of SGs at high enough p pcn , if the critical pressure pcn < pg . The average energy ¯uctuation in Eq. (13) is actually determined by random interactions of the electron and hole negative-U centers, 2
Dav p 4e =jeff R2 p 4Ud q=R2 p;
20
where jeff is the eective dielectric susceptibility and the above mentioned eective Hubbard repulsion energy, Ud e2 /jeff q. Actually the eective Hubbard repulsion energy for two ground-state singlet negative-U electron pairs on the same `site' 2 2 is UH 6 4Ud and so UH 6 Dav < 2zav J2 (p) at pc2 6 p < pg . Similar relationships also hold for two single-particle excitations or for one excitation and one ground-state pair, on the same `site', with 1 UH < Dav < 2zav J1 (p), and c1 (p,T) c2 (p), at pc1 6 p < pg . In this connection, in the present calculations dynamic correlations of ground-state singlet negative-U pairs and their excitations (`many-body' eects) are not explicitly taken into account in the approximation used, the correlations being assumed less important than the basic Anderson±Mott transitions under discussion. In order to estimate the critical pressure, pcn , from Eq. (13), the expressions describing the average tunneling amplitudes have to be considered both for the negative-U centers (n 2) and for their single-particle excitations (n 1). Note that the eective size of both kinds of states is actually the same, q1 @ q2 º q, as their structure is similar. Two kinds of the ground-singlet-pair tunneling are available (see, e.g., [3,4]): `direct' tunneling of the pair as a whole (J2 ) and `indirect' tunneling
due to two virtual single-particle transitions (J2 ). In general, Jn p J1 pfn J1 p=DE p;
21
0
0
where: J1 (p) J1 exp[)W1 (p)] and J2 (p) J2 exp[)W2 (p)]. Moreover, f1 J1 =DE 1 while f2 J1 =DE J12 p=J1 p a DE p at a const 1
22
0
0
with dierent J1 const and J2 const, so that J2 J1 at J1 DE but J2 J1 at J1 P DE. The contribution of the self-trapping to the tunneling amplitudes is described in Wn (p) nR2 (p)/ q + wn (p) by the following expressions: wn p n2 w1 p
and
w1 p Eg p=4hX0 p w2 p=4:
23
Taking into account the above considerations 1=2 (Eq. (19)), one can estimate that X0 (p) geff (p)xD xD for the weak negative-U centers in rather typical rigid con®gurations most important for ®nding the high pcn from Eq. (13), at 0 < pg ) pcn pg . The eective negative correlation energy, by its de®nition, determines the eective gap-width DE(p) in the single-particle excitation energy spectrum, DE(p) @ |Ueff (p)| @ Eg (p)/2. One may then conclude from Eqs. (21) and (23) that J2 (p) J2 (p), and that J2 (p) J1 (p) at low p < 0.1pg , whereas J2 (p) J1 (p) at high p, 0 < pg ) p pg . It follows from Eqs. (19) and (20) that both D(p) and Jn (p) increase with growing pressure, but (at not too low p at least) Jn (p) increases more rapidly so that c(p) in Eq. (13) also increases and the critical pressure, pcn , exists, being, in general, dierent for n 2 and n 1, as obtained in Eq. (15) from the numerical calculations of pcn . Both the negative-U centers and their single-particle excitations in the gap are localized at p < pc1 , but non-localized for p P pc2 as actually J1 (p) P J2 (p) J2 (p) and so pc1 6 pc2 . Only the single-particle excitations are non-localized at intermediate pressures, pc1 6 p < pc2 . By this means, the behavior of rdc (p, T) dramatically changes at the critical pressures, pc1 , and particularly, pc2 , compared to rdc (0, T).
M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
Actually, at the highest critical pressure pc2 (
and ÿ2=3
lIR jejDIR =h l0 c2
pXIR =h:
24
Here, the eective frequency of the tunneling transitions in this motion is XIR zJ2 (p)/jh zJ1 (p)/ jh at j 1 ) 3, and l0 |e|a2 /h 1 cm2 /V s. Actually, Eq. (24) may be obtained from the general Kubo±Greenwood formula for rdc , e.g., in the `random-phase' approximation (RPA) [20], with h zJn (p)/j h, j 3, and gn (f) XIR 2z2 Jn 2 (p)gn (f)/ the respective DOS at the Fermi level f, gn (f) 1/ 6zJn . In fact, some phase coherence in the carrier
265
wave function on the nearest neighbours, neglected in the RPA, may increase lIR (T, p) at 1 6 j < 3 but hardly changes its behavior. We conclude from Eq. (24) that a simple relation ex IM ists, rdc (ce ccr ; d) e2 Ncr DIR /h r M (d), at low T between a characteristic conductivity dÿ2 [3] and the conductivity r M (d) 2pe2 /hacr IM rdc (ce ; d) of Fermi degenerate charge carriers of concentration ce ci in the vicinity of an Anderson±Mott type (continuous) metal±insulator transition, for a d-dimensional (2 < d 6 3) disordered system of `impurity centers' at their critical concentration, ci ccr . Here: h f ) constant zJ/2 and aÿd cr N0 ccr . We may then expect that AM rdc (ce ccr ;d 2) 2pe2 /h, a well-known universal constant for such a (quasi) two-dimensional system, if it exhibits the noted transition. 4. Conclusions Recently predicted high-pressure phenomenon, suppression of soft atomic con®gurations for which the sensitivity to distortions and so to pressure is anomalously high, is expected to give rise to macroscopic changes in low-temperature and lowfrequency (as well as in elastic) properties of glasses at high pressure. Delocalization of the negative-U centers, and associated changes in the electronic (e.g., transport) properties of semiconducting glasses, at high pressure have been predicted and discussed. The basic parameters of the theory, such as J2 (p) for pc2 6 p < pg , J1 (p) for pc1 6 p < pg , and c2 (p), as well as 2|Ueff (p)|/Eg (p), may be estimated by comparing Eqs. (16)±(19) with the relevant data which is not readily available. Some problems also may be mentioned in conclusion, which are under investigation and concern the extent to what the soft con®gurations and their pressure-induced suppression may contribute to the following remarkable properties of glasses (i) `polyamorphism', or `vitreous polymorphism' [15]; (ii) a low-frequency `boson peak' in the picosecond range of the Raman and neutron scattering spectra, exhibiting anharmonic features (see, e.g. [21]), and, for still lower frequencies, a probably related quasielastic line; (iii) fragility of
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M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
glass-formers, and related aspects of the glass transition. If the boson peak at ambient pressure is partially associated with the low-energy soft-mode vibrational excitations of soft con®gurations (see, e.g. [4]), that part of the peak has to be suppressed at high p P pcr . This prediction might be tested at p P 105 bar.
References [1] S.R. Elliott, Physics of Amorphous Materials, Longman, Harlow, 1990. [2] P.W. Anderson, in: R. Balian (Ed.), Ill-Condensed Matter, North-Holland, Amsterdam, 1979, p. 161. [3] N.F. Mott, E.A. Davis, Electron Processes in Non-Crystalline Materials, Clarendon, Oxford, 1979. [4] M.I. Klinger, Phys. Rep. 94 (1983) 183; 165 (1988) 275; Commun. Condens. Mater. Phys. 37 (1992) 116. [5] M.I. Klinger, S.N. Taraskin, Phys. Rev. B 47 (1993) 10235; 52 (1995) 2557. [6] R. Gilmore, Catastrophe Theory, Wiley, New York, 1981. [7] U. Buchenau, Yu.M. Galperin, V.L. Gurevich, H.R. Schober, Phys. Rev. B 43 (1991) 5039.
[8] H.R. Schober, C. Oligschleger, Phys. Rev. B 53 (1996) 11469. [9] P.W. Anderson, B.I. Halperin, C.M. Varma, Philos. Mag. 25 (1972) 1; W.A. Phillips, Low Temper. Phys. 7 (1972) 351. [10] A.P. Sokolov, R. Calemczuk, B. Salce, A. Kisliuk, D. Quitmann, E. Duval, Phys. Rev. Lett. 78 (1997) 2405. [11] C.A. Angell, J. Non-Cryst. Solids 131±133 (1991) 13. [12] K. Tanaka, Phys. Rev. B 30 (1984) 4549; I. Berman, N. Brandt, Sov. Phys. Low Temper. (in Russian) 16 (1990) 1227. [13] M.I. Klinger, S.N. Taraskin, in: A.C. Wright, S.A. Feller, A.C. Hannon (Eds.), Borate Glasses, Crystals and Melts, Soc. Glass Technol., Sheeld, 1997, p. 357. [14] K. Tanaka, Sol. State Commun. 60 (1986) 295; 54 (1985) 867. [15] C.A. Angell, P.H. Poole, J. Shao, Nuovo Cim. 16D (1994) 993. [16] P.W. Anderson, Phys. Rev. 109 (1958) 1492. [17] M.I. Klinger, S.N. Taraskin, J. Phys.: Condens. Mater. 9 (1997) 4425. [18] A.F. Ioe, A.R. Regel, Progr. Semicond. 4 (1960) 237. [19] M.I. Klinger, Problems of Electron Transport Theory in Semiconductors, Pergamon, Oxford, 1979. [20] M.H. Cohen, J. Non-Cryst. Solids 4 (1970) 391; N.K. Hindley, J. Non-Cryst. Solids 5 (1970) 17; L. Friedman, J. Non-Cryst. Solids 6 (1971) 329. [21] U. Buchenau, Philos. Mag. B 71 (1995) 793.
Section 7. Interrelationship between structure and properties
High-pressure phenomena in glasses: The role of soft atomic con®gurations Michael Klinger * Department of Physics, and Jack and Pearl Resnick Institute of Advanced Technology, Bar Ilan University, 52900 Ramat Gan, Tel Aviv, Israel
Abstract The revealed suppression of the anomalous low-energy dynamical, and the related low-temperature, properties of glasses at high pressures is described. Some similarity of this phenomenon to recently observed pressure-induced transitions between `strong' and `fragile' glass-formers (`polyamorphism') is noted. Changes in the electron properties of semiconducting glasses, which are due to delocalization of negative-U centers as basic charge carriers, at high pressures are predicted and analyzed. The phenomena are related to soft atomic con®gurations available in glasses. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Two dierent theoretical approaches are mainly applied for describing properties of amorphous solids. One is based on detailed structural information (see, e.g., [1]). The other proceeds from general concepts that are rather independent of the structure details [2±4]. The purpose of the present paper is to discuss correlations between the variation of some structural features and macroscopic changes in dynamical properties and electronic processes in glasses, actually being 3-dimensional systems. The correlations result from the general concepts of `soft atomic con®gurations' [4] and `negative-U centers' [2±5] for pressure-induced phenomena in glasses as the representative example.
The uni®ed theoretical approach, the `soft con®guration model' (SCM) of glasses, has been developed for the quantitative description of both the low-energy excitations in atomic dynamics of glasses and the electron properties that are related to negative-U centers of non-metallic glasses [4]. The SCM has been derived from the observed similarity between the elastic properties of glasses and their respective crystals, from the macroscopic isotropy of glasses, and from mathematical `catastrophe theory' [6] applied to analyze features of the potential energy of spatially ¯uctuating mediumrange-order (MRO) structures. The basic result was that each soft atomic con®guration, of the MRO length scale LMRO between 1 and 2 nm, actually contains a single soft atomic-motion-mode (x) in which, generally speaking, the potential energy is anharmonic [4] V x V x; g; n A gx2 nx3 x4 :
* Tel.: +972-3 531 8424; fax: +972-3 535 3298; e-mail: [email protected]. 0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 4 0 0 - 1
1
The soft mode as basic random parameters {g > 0 or g < 0} and {n > 0 or n < 0}, with {|g|, n2 } 6
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M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
g* const 1, e.g., g* 0.1, and the resulting spring constant, k k(g,n;A) k0 2A 30±60 eV, the general range for (non-molecular) solids. The modes are localized states of cooperative atomic motion on the length scale LMRO , involving NS 30 (10 6 NS 6 102 ) atoms, at 1 NS 6 NMRO (LMRO /a)3 < 103 with mean atomic spacing a 0.3 nm [4,7,8]. Actually the SCM is the theory of low-energy (E) excitations (in addition to sound waves) in atomic dynamics, their spectra, E(g, n), in the Schroedinger equation with V(x) of Eq. (1), and of the related low-temperature (T) and lowfrequency (x) properties of glasses, at {E;T;hx} hxD , the Debye energy [4]. The new energy scale w 1±2 meV ( 6 0.1 hxD ) obtained in the SCM, is the lowest vibrational energy characteristic of strongly anharmonic soft modes (Eq. (1)), which arises in the quartic anharmonic potential (Ax4 ). Excitations of substantially lower energy (E w) are the well-known `tunneling states' (cf. [9]) while those of higher E, w E hx D , are quasiharmonic soft vibrations. The basic spectral properties, the density-ofstates (DOS), n(E), of the soft-mode excitations with energy spectrum, E(g, n), is related to the probability density distribution, U(g, n). The DOS appears to exhibit similar features, including the main maximum at E EM , for any type of glass. In general, the maximum of n(E) is located either at EM w, the quasiharmonic vibrational DOS nqh (E) growing with E, for w Eb < E hxD at Eb 3w, in a Debye-like manner (Ea , at a 2), or at higher EM , Eb < EM hxD , with nqh (E) increasing, at w 6 E < EM , more rapidly (Ea , at a 3±4) [4,7,8]. Correspondingly, the glassy low-temperature properties are qualitatively similar while quantitatively dierent for different types of glass. Some recent data (see, e.g., [10]) appear to indicate a quantitative dierence in the low-temperature properties of `strong' (e.g., g-SiO2 ) and `fragile' (e.g., Ca0:4 K0:6 (NO3 )1:4 ) glasses, in Angell's classi®cation [11]. Actually, the SCM presents a continuous transition from standard rigid harmonic single-well con®gurations (k k0 ) to quasiharmonic and then to strongly anharmonic, including double-well, soft con®gurations (k k0 ), and ®nally to con®gurations with
double-well potentials and rigid potential wells, as g decreases from g P 1 to g < 0 at |g| 1 and then to |g| P 1. Thus, the `tunneling states' in double-well potentials are not likely to occur at 0.1 6 E 6 1 K, if the other types of soft con®gurations (soft-mode excitations) do not [4]. The SCM predicts that, for any type of glass, the generalized susceptibility, v, of the soft con®gurations to distortions is anomalously higher v k ÿ1 v0 k0ÿ1 ;
2
the general scale, v0 , characterizing the vast majority of atoms in a glass. Thus, the soft con®gurations are more aected by forces such as pressure giving rise to distortions, than the standard con®gurations at k k0 . Therefore, the soft con®gurations of the glass structure can be essential for properties related to structure distortions, even though the (relative) atomic concentration of these con®gurations, ca º Na a3 , is rather low (ca (0) 2 ´ 10ÿ2 ) at ambient pressure (p º 0). The concentration is estimated from some general considerations or from comparing the SCM properties depending on ca , to the appropriate data [4,7,8]. Since v is 100 P v0 P 30, the role of the soft con®gurations for the mentioned properties is comparable at ambient pressure to that of the vast majority of atoms with v v0 , at ca (0)v ctot v0 v0 . It is worth adding that ca (0) 10ÿ2 for `non-glassy' materials (e.g., a-Si), not exhibiting glassy low (<50 K) temperature properties. A principal consequence of the occurrence of the soft con®gurations in glasses at ambient pressure was found to be the existence of localized `negative-U centers' as basic charge carriers in semiconducting and related glasses (SGs) with moderately wide mobility gaps, of width, Eg (0), 1±4 eV [4,5]. The negative-U centers actually determine the main electronic properties of the glasses at ambient pressure [2±4], as well as many of those at high pressure (Section 3). In the SCM, the charge carriers in question are identi®ed as singlet electron (hole) pairs, occupying the negative-U states in a rather wide energy band around the Fermi level (per particle) near the middle of the mobility gap. The basic gap states, exhibiting a negative pair correlation energy (inter-electron attraction), U(g), are formed due to a new type of
M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
self-trapping in the soft con®gurations at ambient pressure (but not at high pressure, see Section 3) [4,5]. Hybridization of states in the mobility gap is the decisive feature of this type of self-trapping, providing a negative pair-correlation energy which, in general, is |U(g)| |Ueff | º |U(geff )| @ Eg /2 and, in particular, is large in magnitude at ambient pressure, |Ueff (0)| @ Eg (0)/2 P 1 eV. Here U(g) W2 (g)/2 < 0 with W2 (g), the pair self-trapping energy (energy gain), increasing in magnitude with decreasing g P 0 for the most numerous quasiharmonic soft con®gurations. The extremely large magnitude of W2 (g) for the soft con®gurations is the main cause of the decisive role of the hybridization of states in the self-trapping process. The parameter, geff , introduced above is the eective value of the softness (g) characteristic of most of the negative-U centers at the Fermi level, f(0), near the gap middle, Eg (0)/2. The related thermal-equilibrium concentration of the negative-U centers is also considerable, 10ÿ3 P ca (0) P 10ÿ4 , in accordance with experimental data. It is worthy of note that the mobility gap width, Eg Eg (p) of SGs, is observed to decrease with growing pressure [12], Eg 0 P Eg p P Eg pg 0
3
at Eg (p) @ 2f(p). Here pg is the critical pressure for the semiconductor (p < pg ) ± metal (p > pg ) transition, 105 bar 6 pg < 106 bar, and f is the Fermi level referred to the valence-band mobility edge Ev . High-pressure induced macroscopic changes in the dynamical (and elastic) properties of glasses and in the electron dynamics of semiconducting and related glasses are discussed, respectively, in Sections 2 and 3. The conclusions are presented in Section 4.
259
P 107 bar. Here L is the typical length of the quasimolecular structure unit, L 1.5 nm, and EB is the typical covalent or ionic bonding energy, EB P 1 eV, while z is the average coordination number. At accessible lower p p0 , macroscopic changes in properties associated with phase transitions may occur only in crystals in which extended soft-phonon modes are available. One may suggest that, in general, some softness is needed for macroscopic changes in properties of solids at p p0 . In glasses, at ambient pressure, softness is generally distributed in the random soft con®gurations for jgj 1, so that pressure-induced macroscopic changes in properties may occur even at p p0 . The glassy features discussed below are related at ambient pressure to the soft con®gurations (Eq. (1)) and the probability distribution density, H(g), of the random `softness' (g) at 0 < g 6 g* 0.1, for most of the con®gurations with quasiharmonic soft modes. The main problem is the affects of pressure on H(g) and the atomic concentration, ca º ca (p). A plausible model identi®es H(g) with the appropriate Gaussian - like function Z1 2 H g dn U g; n H0 g expÿgÿ1 c 1 ÿ g=g0 ÿ1
4 at H0 (g) @ H0 const. Here g0 const @ 1 characterizes the standard rigid harmonic con®gurations at k @ k0 for the vast majority of atoms of the glass; generally speaking, H0 ¹ H0 (g > g*). Then, at least at not too high T < Tg , Zg ca
dg H g:
5
ÿ1
2. Pressure-induced macroscopic changes in dynamical properties 2.1. Problem and results The eect of (hydrostatic) pressure on properties is dierent for glasses and crystals. For the latter, generally speaking, macroscopic changes are expected to occur at very high p P p0 EB /zL3
The basic parameter, which may depend on pressure, is gc gc (p), whereas both g and g0 , by de®nition, are actually constants. The approximate expression obtained, which describes the features and scale of gc (p), is as follows [13]: 0 ÿ1 gc gc p g1 c gc pcr ÿ ppg H pcr ÿ p
6
with h(x) the step function (see Eq. (10) and (11) 0 for parameters g1 c , gc and pcr , pg ).
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M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
2.2. Discussion Two arguments are useful for deriving Eq. (6). One is that glassy low-temperature properties at ambient pressure are observed, in some types of glasses at least, to be suppressed with increasing average coordination number, z. This suppression is certainly the case for amorphous semiconductors, as a representative example, that are covalent glasses for zmin 6 z < zcr (e.g., zmin z(g-Se) 2 6 z 6 z(g-GeSe2 ) 2.67 < zcr ), and become nonglassy covalent materials (without glassy low-temperature properties) for zcr < z 6 zmax 4(a-Si), with 2.67 < zcr < 3(a-As) for the `critical' value zcr [13,4]. Furthermore, a change in the elastic (bulk) modulus, B[z], was observed from B[z 6 zcr ] 105 bar for SGs to B[zcr < z] 6 B[zmax ] 106 bar for a-Si. As noted in [14,13], these features are related to the remarkable dichotomy of bonding forces, strong (covalent) and `weak' (van der Waals) at z < zcr while actually strong at zcr < z 6 zmax . I assume that z z0 + Dz, with z0 and Dz the contributions of the strong bonding in each quasimolecular unit related to the MRO and of the weak bonding between the random units, respectively, at 0 < Dz z0 and z < zcr . The eective dimensionality, d0eff , of the `decoupled' units is expected to be low (e.g., d0eff < 1) compared to that, deff , of the extended random structures consisting of cross-linked units, d0eff < deff 6 3. The theory indicates that, on average, the weaker bonds increase in strength while the other bonds remain unchanged (z0 const) with growing coordination (Dz), the eective dimensionality deff increasing self-consistently. Actually Dz and deff might be found as self-consistent solutions of certain equations (to be investigated elsewhere). Within the framework of the soft-con®guration model of glasses [4], the suppression of the glassy low-temperature properties with increasing z at ambient pressure means that for the amorphous semiconductors mentioned ca [z < zcr ] ca [z 3 > zcr ] ca [zmax 4], at zmin 6 z 6 zmax . Here ca [z < zcr ] is considerable, around 10ÿ2 in scale, for the SGs, whereas generally speaking ca [z] is negligible at zcr < z 6 zmax for the non-glassy materials mentioned. In this sense, the dierence between ca [3] and ca [4] may be neglected. Actually in the
whole range of z in question, dca /dz < 0, i.e., also dgc /dz < 0, but rather {jdca /dzj at z zcr } < {jdca /dzj at z P zcr }, in accord with the expected sharp, though continuous, transformation of the glasses into non-glassy materials at z zcr . Then, the linear approximation 0 gc z g1 c gc zcr ÿ zH zcr ÿ z;
7
extrapolating the expansion around zcr to both zmin and zmax , may be applied, where 0 g1 c gc zmax gc fjdgc =dzj at z zcr g
g1 c
0.1g0c ,
8
g0c
with 0.1 for typical e.g., ca (0) 10ÿ2 at z < zcr . Another argument for estimating gc (p) is that some empirical data, e.g., concerning pressure effects on the optical-absorption edge, for the glasses in question, may give rise to the following estimate [14]: 0 < dz=dp 6 10ÿ5 =bar;
9
so that (0 <) Dz(p) º z(p) ) z(0) < 1 at the applied pressure, 0 < p 6 105 bar, in question. This relation may be interpreted, within the framework of the previously mentioned model, as follows. The strong (e.g., covalent) bonds actually remain intact while the weak bonding forces may increase in strength, continuously transforming into a kind of strong interactions, with increasing pressure up to p P pcr at zcr z(pcr ). This transformation, as well as the related transformation of the glass into a dierent kind of amorphous material at z z(p) > zcr , is due to the pressure-induced decrease of the average distance between the quasimolecular units (densi®cation) until the dichotomy of forces disappears at a pressure, pcr , and only strong bonding forces exist in the resulting `continuous random network' structure of higher eective dimensionality deff (e.g., 2 6 deff < 3), at p P pcr and Dz P 1. In other words, one may assume that dz/dp @ dDz/dp jdz0 /dpj and 105 bar 6 pcr p0 and, from continuity reasons, that z z(p) is a simple function. The related continuous transformation of the random structure may be characterized in a more accurate way by Dz(p) z(p) ) z(0) and deff (p) as self-consistent solutions of the appropriate equations. Then, in a linear approximation, at p 6 pmax and z(pmax ) zmax ,
M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
zcr ÿ z p z pcr ÿ z p pcr ÿ ppgÿ1
10
with gc (p) and ca (p) monotonically decreasing with p, at dca (p)/dp < 0 and {jdca /dpj at p < pcr } {jdca /dpj at p P pcr }. In Eq. (10): ÿ1
106 bar P pmax > pg f dz=dp at p pcr g ÿ1
P pcr zcr ÿ z 0 dz=dp0 P 105 bar
11
with 0 < zcr ) z(0) < 1 for the glasses in question. The very existence and scale of the characteristic pressure pcr and pg are essential, rather than their precise value. Taking into account Eqs. (7)±(10), I establish Eq. (6). The latter means that the distribution density, H(g), is decreased at gc , decreasing from g0c to gc , so the soft con®gurations are suppressed, and the related glassy features are eventually lost, as the atomic concentration, ca (p), is diminished by orders of magnitude when p > pcr . The analytical expression, specifying Eq. (6), may be found by taking into account the self-consistent solutions of the noted equations for Dz(p) and deff (p) as functions of p. Actually the macroscopic changes in the dynamic properties of glasses at high p ( 6 106 bar) may be rather general, as generally glasses have competing stronger and weaker bondings. Macroscopic changes in the elastic properties of non-metallic glasses also may be predicted from Eqs. (6)± (10) at high p P pcr : the elastic modulus increases in scale, e.g., from B 105 bar at p < pcr to B 106 bar at p > pcr . The resulting amorphous material at high p ( P pcr ) is expected to be either a higher-density non-glassy material, which does not exhibit glassy low-temperature properties, or at least at not too high p (possibly, at pcr 6 p < pg ), still a glassy substance though more dense and fragile than the initial glass. Actually the glassy low-temperature properties appear to be less evident in fragile glasses than in strong ones (see Section 1 and, e.g., Ref. [10]). This macroscopic change in structure and properties of glasses, due to the predicted pressure-induced phenomenon of suppression of soft atomic con®gurations, is similar to the remarkable `vitreous polymorphism', or `polyamorphism', recently revealed and studied in real compression-in-
261
duced vitri®cation experiments and computer simulations (see Ref. [15] and references therein). The question is whether or not such a similarity is indeed due to essential correlations between the polyamorphism, implying that the glass may exist in more than one distinct metastable state separated by barriers in the con®guration space, and the pressure-induced suppression of soft con®gurations of a glass. At present the question cannot be answered comprehensively. 3. Pressure-induced transformations in electron dynamics and transport 3.1. Problem and results High pressures are predicted, in the theory under discussion, to give rise to transformations in the electronic properties of glasses, as recently shown by Klinger and Taraskin [5] for the electron dynamics and transport processes of semiconducting and related glasses (SGs), actually threedimensional systems, as representative examples. In this connection, two recently obtained principal results are presented and discussed in this section. One result is that the negative-U centers as basic Fermi-degenerate charge carriers, localized at ambient pressure in SGs (Section 1), become non-localized for high enough p < pg (Eq. (3)). Another result is that, due to the delocalization of the negative-U centers, the electron transport at high enough p < pg and not too high T < Tg is of a metallic type, unlike the thermally activated transport which is not related to negative-U centers at ambient pressure. In accordance with the well-known criterion of the Anderson±Mott localization±delocalization transition, a quantum particle (quasiparticle) or its state in a three-dimensional system is localized or non-localized, respectively, at [3,16] c q 2zJav q=Dav q < ccr const 1 or c > ccr :
12
Then, c(qc ) ccr corresponds to such a transition, at a critical value, qc , of the `ruling' parameter, q, which commonly is considered as a continuous
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one. Variations of the latter may give rise to changes in both the eective (average) amplitude, Jav (q), of the particle tunneling between the relevant localized states, or `sites', and the mean ¯uctuation, Dav (q), of the respective energy levels, in the disordered system in question. As shown in what follows, pressure, p, takes the part of the parameter q for the negative-U centers, so that the Anderson±Mott transition occurs at p pc2 , from localized (p < pc2 ) to non-localized (p > pc2 ) states, in the `random lattice' of relevant sites. The sites are suggested to be the negative-U states formed due to the revealed `non-polaronic' selftrapping in appropriate MRO atomic con®gurations (Section 1). The latter are rather rigid ones (g g0 @ 1), typical of the vast majority of atoms, at high pressures p P pcn under discussion (or soft con®gurations (Eq. (1)), at ambient and low pressures). A similar transition may be predicted for single-particle excitations created by dissociation or ionization of the negative-U centers, at p pc1 . Both transitions are of the same type as the wellknown Anderson±Mott transition at ci ccr in an `impurity band' in the inter-band (mobility) gap of a doped semiconductor, for which the impurity concentration, ci , is the `ruling' parameter q. Then, the Anderson±Mott transition has to occur at the critical pressure, pcn (n 2 or n 1), as the relevant solution of equation c 2zav Jn pcn =Dav pcn ccr 1
13
in the semiconducting phase of the glass for pcn < pg , with J n av (p) Jn (p) and z 6 zav . Numerical calculations of pcn from Eq. (13) have been performed, by applying the analytical expressions obtained for Jn (p) and Dav (p) (Section 3.2) at n 2 and n 1, for a basic set of typical values of the parameters used [5,17]: Eg 0 2 eV 10Ud ;
q=a 3;
hX0 0 200K hxD =2
and
c2 0 10ÿ3 14
with the valence band width, D0 , equal to 2.5Eg (0) and ccr 1. Here q stands for the localized state size, X0 (p) is the characteristic frequency of the most important atomic motion mode for the formation of the negative-U centers, and c2 (p) is the thermal-equilibrium concentration of the nega-
tive-U centers, c2 (p) c1 (p, T), the concentration of the single-particle excitations (see Section 3.2). The critical pressures pc1 and pc2 are indeed found to be less than pg : 0:75 e1 6 pc1 =pg 6 0:85 e2
and
0 6 pc2 ÿ pc1 =pg 6 0:02
15
with pc2 /pg < 1 at 0 6 e2 6 e1 6 0.1, and the dependence of pcn on the parameters (Eq. (14)) is weak. Thus, the pressure-induced phenomenon of delocalization of the negative-U centers, at pc2 6 p < pg , and of their single-particle excitations, at pc1 6 p < pg , indeed, appears to be predicted theoretically. Generally speaking, the delocalization phenomenon gives rise to essential transformations in nonequilibrium properties, particularly in transport processes and related phenomena. The representative example is dc conductivity, rdc (p, T), as an essential transport coecient. The resulting expressions for rdc (p,T) are as follows. At the highest pressures, pc2 6 p < pg , and not too high T < {Tg ; Tk (p)}, rdc (p, T) is basically related to the non-localized negative-U centers as main current carriers for which c2 (p) is rather independent of T (see [17]). Then, rdc (p, T) is of a metallic type 2
rdc p; T rdc p; T 2
4jejN0 c1=3 2 plIR p; T 24r0 c2 pzJ2 p=jh
16 2 lIR
the related with N0 º aÿ3 , j 1 ) 3, and (drift) mobility. Here the parameter of the generalized Einstein formula, h n(dn/df)ÿ1 , is h @ f(p) @ Eg (p)/2 > T for the main charge carriers, negativeU centers around the Fermi level f (Eq. (3)), of concentration n c2 (p), whereas it is h @ T for other, non-degenerate, charge carriers such as standard electron±hole pairs and single-particle excitations of the negative-U centers. Generally speaking, rdc (p, T) increases here with growing p, whereas it does not noticeably change with T at T < Tk (p), exhibits a pronounced kink at T Tk (p), and increases by the Arrhenius law (18), with an activation energy, Wr (p) @ Eg (p)/2, which decreases with pressure at T > Tk (p) > Tg . If, however, Tg < Tk (p) at pc2 6 p < pg , rdc (p,T) is of a metallic type, basically independent of T,
M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
over the whole range of the glass for T < Tg . This case is realized (in numerical calculations) for SGs with typical Tg < 103 K. For intermediate pressure, pc1 6 p < pc2 , rdc (p, T) is related mainly to the non-localized single-particle excitations of the negative-U centers, and is thermally activated. Then, for not too high 2=3 T < Td (p) Eg (p)/ln [c2 (p)D0 /2zJ1 (p)], 1
rdc p; T rdc p; T 1
jejN0 c1 p; T lIR p; T r0 T1 p=T expÿWr p=T ;
17 1=3
where Wr (p) |Ueff (p)| @ Eg (p)/2 and T1 (p) 2c2 (p)zJ1 (p). Typically Td (p) is rather high, Td (pc1 ) P 103 K > Tg (SGs) at c2 (pc1 ) 10ÿ2 P 10c2 (0), so Eq. (17) holds for all T < Tg at pc1 6 p < pc2 . Both Td (p) and T1 (p) increase with p. In the alternative case for Td (pc1 ) < Tg (if realizable in SGs), rdc (p, T) at Td (pc1 ) < T < Tg , as well as rdc (0, T) at 0 < T < Tg and ambient pressure [3], would be related to the contributions of standard electron±hole pairs, and described by the Arrhenius law rdc p; T r0 expÿWr p=T :
18
Here r0 const 103 ) 104 /X cm at T < Tg and the activation energy Wr (p) @ Eg (p)/2. Note that Eqs. (16) and (17) hold for p not too close to the semiconductor±metal transition at pg (Eq. (3)), with |Ueff (p)| @ Eg (p)/2 T, for which the thermal-equilibrium concentration of the single-particle charge carriers is low, c1 (p, T) c2 (p). 3.2. Discussion The main results, Eqs. (13)±(17) may be interpreted as follows. As recently shown (see Ref. [17]), the concentration, c2 (p), of the basic charge carriers, the negative-U centers (Section 1), in SGs depends on pressure in a non-monotonic manner. It has a minimum for p º pmin , 0.35 6 pmin /pg 6 0.45, and increases with pressure to values, c2 (p) 10ÿ1 c2 (0) 10ÿ4 , at 0 < pg ) p pg . For the latter c2 (p) actually characterizes weak negative-U centers, of which the eective negative pair-correlation energy, Ueff (p), typical
263
of most of the centers is small in magnitude compared to that at ambient pressure, |Ueff (p)| |Ueff (0)| (Eq. (19)). Two basic eects and their competition determine the non-monotonic dependence of c2 (p) on increasing pressure [17]: (1) The probability distribution density, H(g), for the random con®guration softness becomes more narrow, with a decrease in its width, Dg(p) g1=2 c (p), at growing densi®cation of the glass at T const (Eqs. (4) and (6)). The related pressure induced suppression of the soft con®gurations manifests, in particular, as a decrease in the atomic concentration of the localized soft modes that are associated with the electron pair self-trapping and to the resulting formation of the negative-U centers (Section 1). (2) The eective softness, geff (p), of the local atomic con®gurations of nanometric size, in which most of the negative-U centers are formed with energies (per particle) around the Fermi level near the mobility gap middle, and the related eective negative pair-correlation energy, Ueff (p), are characterized as follows [5]: geff p c =Eg p Ud and Ueff p U geff Eg p=2
19
with Eg (p) @ Eg (0)(1 ) p/pg ) (Eq. (3)) and the practically pressure independent eective electronatomic-motion mode coupling energy c 0.1Eg (0) and Hubbard pair-repulsion energy Ud 0.1Eg (0). Thus, geff (p) increases while |Ueff (p)| decreases with growing pressure, so that geff (p) 0.1 and Ueff (p) Eg (0)/2 P 1 eV at p 6 0.1pg (strong negative-U centers), whereas geff (p) 1 and Ueff (p) Eg (p)/ 2 6 0.1Eg (0)/2 at high p, 0 < pg ) p 6 0.1pg (weak negative-U centers). Since the distribution density, H(g), monotonically increases with g at least for the most numerous quasiharmonic soft con®gurations (Eq. (1)) at 0 < g < g0 @ 1, the concentration, c2 (p), of the weak negative-U centers also rapidly increases. In fact, these negative-U centers can be created because of self-trapping even in common `rigid' atomic con®gurations when geff (p) @ g0 º geff (p0 ) @ 1 (and p0 6 p < pg ), as Eg (p) is small enough to be close to the pair selftrapping energy, W2 (g) @ c /g [5], at g @ g0 .
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Let us note that the localized states of the negative-U centers and their single-particle excitations are expected to be of the standard type (v(r;xn ) v0 (xn ) exp()r/q), with xn the equilibrium displacement in the atomic-motion mode most essential for the self-trapping, and q const (2 ) 3)a at p < pg ) and that their mean separation, ÿ1=3 R2 (p) ac2 (p), decreases from R2 (0) P 10a to R2 (pg ) a. Then, the average overlap, Iav , of the states and the related tunneling amplitude, Jn , strongly increase with growing p. Thereby, the Anderson±Mott delocalization of most of the negative-U centers, around the Fermi level, could indeed occur in the mobility gap of SGs at high enough p pcn , if the critical pressure pcn < pg . The average energy ¯uctuation in Eq. (13) is actually determined by random interactions of the electron and hole negative-U centers, 2
Dav p 4e =jeff R2 p 4Ud q=R2 p;
20
where jeff is the eective dielectric susceptibility and the above mentioned eective Hubbard repulsion energy, Ud e2 /jeff q. Actually the eective Hubbard repulsion energy for two ground-state singlet negative-U electron pairs on the same `site' 2 2 is UH 6 4Ud and so UH 6 Dav < 2zav J2 (p) at pc2 6 p < pg . Similar relationships also hold for two single-particle excitations or for one excitation and one ground-state pair, on the same `site', with 1 UH < Dav < 2zav J1 (p), and c1 (p,T) c2 (p), at pc1 6 p < pg . In this connection, in the present calculations dynamic correlations of ground-state singlet negative-U pairs and their excitations (`many-body' eects) are not explicitly taken into account in the approximation used, the correlations being assumed less important than the basic Anderson±Mott transitions under discussion. In order to estimate the critical pressure, pcn , from Eq. (13), the expressions describing the average tunneling amplitudes have to be considered both for the negative-U centers (n 2) and for their single-particle excitations (n 1). Note that the eective size of both kinds of states is actually the same, q1 @ q2 º q, as their structure is similar. Two kinds of the ground-singlet-pair tunneling are available (see, e.g., [3,4]): `direct' tunneling of the pair as a whole (J2 ) and `indirect' tunneling
due to two virtual single-particle transitions (J2 ). In general, Jn p J1 pfn J1 p=DE p;
21
0
0
where: J1 (p) J1 exp[)W1 (p)] and J2 (p) J2 exp[)W2 (p)]. Moreover, f1 J1 =DE 1 while f2 J1 =DE J12 p=J1 p a DE p at a const 1
22
0
0
with dierent J1 const and J2 const, so that J2 J1 at J1 DE but J2 J1 at J1 P DE. The contribution of the self-trapping to the tunneling amplitudes is described in Wn (p) nR2 (p)/ q + wn (p) by the following expressions: wn p n2 w1 p
and
w1 p Eg p=4hX0 p w2 p=4:
23
Taking into account the above considerations 1=2 (Eq. (19)), one can estimate that X0 (p) geff (p)xD xD for the weak negative-U centers in rather typical rigid con®gurations most important for ®nding the high pcn from Eq. (13), at 0 < pg ) pcn pg . The eective negative correlation energy, by its de®nition, determines the eective gap-width DE(p) in the single-particle excitation energy spectrum, DE(p) @ |Ueff (p)| @ Eg (p)/2. One may then conclude from Eqs. (21) and (23) that J2 (p) J2 (p), and that J2 (p) J1 (p) at low p < 0.1pg , whereas J2 (p) J1 (p) at high p, 0 < pg ) p pg . It follows from Eqs. (19) and (20) that both D(p) and Jn (p) increase with growing pressure, but (at not too low p at least) Jn (p) increases more rapidly so that c(p) in Eq. (13) also increases and the critical pressure, pcn , exists, being, in general, dierent for n 2 and n 1, as obtained in Eq. (15) from the numerical calculations of pcn . Both the negative-U centers and their single-particle excitations in the gap are localized at p < pc1 , but non-localized for p P pc2 as actually J1 (p) P J2 (p) J2 (p) and so pc1 6 pc2 . Only the single-particle excitations are non-localized at intermediate pressures, pc1 6 p < pc2 . By this means, the behavior of rdc (p, T) dramatically changes at the critical pressures, pc1 , and particularly, pc2 , compared to rdc (0, T).
M. Klinger / Journal of Non-Crystalline Solids 232±234 (1998) 257±266
Actually, at the highest critical pressure pc2 (
and ÿ2=3
lIR jejDIR =h l0 c2
pXIR =h:
24
Here, the eective frequency of the tunneling transitions in this motion is XIR zJ2 (p)/jh zJ1 (p)/ jh at j 1 ) 3, and l0 |e|a2 /h 1 cm2 /V s. Actually, Eq. (24) may be obtained from the general Kubo±Greenwood formula for rdc , e.g., in the `random-phase' approximation (RPA) [20], with h zJn (p)/j h, j 3, and gn (f) XIR 2z2 Jn 2 (p)gn (f)/ the respective DOS at the Fermi level f, gn (f) 1/ 6zJn . In fact, some phase coherence in the carrier
265
wave function on the nearest neighbours, neglected in the RPA, may increase lIR (T, p) at 1 6 j < 3 but hardly changes its behavior. We conclude from Eq. (24) that a simple relation ex IM ists, rdc (ce ccr ; d) e2 Ncr DIR /h r M (d), at low T between a characteristic conductivity dÿ2 [3] and the conductivity r M (d) 2pe2 /hacr IM rdc (ce ; d) of Fermi degenerate charge carriers of concentration ce ci in the vicinity of an Anderson±Mott type (continuous) metal±insulator transition, for a d-dimensional (2 < d 6 3) disordered system of `impurity centers' at their critical concentration, ci ccr . Here: h f ) constant zJ/2 and aÿd cr N0 ccr . We may then expect that AM rdc (ce ccr ;d 2) 2pe2 /h, a well-known universal constant for such a (quasi) two-dimensional system, if it exhibits the noted transition. 4. Conclusions Recently predicted high-pressure phenomenon, suppression of soft atomic con®gurations for which the sensitivity to distortions and so to pressure is anomalously high, is expected to give rise to macroscopic changes in low-temperature and lowfrequency (as well as in elastic) properties of glasses at high pressure. Delocalization of the negative-U centers, and associated changes in the electronic (e.g., transport) properties of semiconducting glasses, at high pressure have been predicted and discussed. The basic parameters of the theory, such as J2 (p) for pc2 6 p < pg , J1 (p) for pc1 6 p < pg , and c2 (p), as well as 2|Ueff (p)|/Eg (p), may be estimated by comparing Eqs. (16)±(19) with the relevant data which is not readily available. Some problems also may be mentioned in conclusion, which are under investigation and concern the extent to what the soft con®gurations and their pressure-induced suppression may contribute to the following remarkable properties of glasses (i) `polyamorphism', or `vitreous polymorphism' [15]; (ii) a low-frequency `boson peak' in the picosecond range of the Raman and neutron scattering spectra, exhibiting anharmonic features (see, e.g. [21]), and, for still lower frequencies, a probably related quasielastic line; (iii) fragility of
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glass-formers, and related aspects of the glass transition. If the boson peak at ambient pressure is partially associated with the low-energy soft-mode vibrational excitations of soft con®gurations (see, e.g. [4]), that part of the peak has to be suppressed at high p P pcr . This prediction might be tested at p P 105 bar.
References [1] S.R. Elliott, Physics of Amorphous Materials, Longman, Harlow, 1990. [2] P.W. Anderson, in: R. Balian (Ed.), Ill-Condensed Matter, North-Holland, Amsterdam, 1979, p. 161. [3] N.F. Mott, E.A. Davis, Electron Processes in Non-Crystalline Materials, Clarendon, Oxford, 1979. [4] M.I. Klinger, Phys. Rep. 94 (1983) 183; 165 (1988) 275; Commun. Condens. Mater. Phys. 37 (1992) 116. [5] M.I. Klinger, S.N. Taraskin, Phys. Rev. B 47 (1993) 10235; 52 (1995) 2557. [6] R. Gilmore, Catastrophe Theory, Wiley, New York, 1981. [7] U. Buchenau, Yu.M. Galperin, V.L. Gurevich, H.R. Schober, Phys. Rev. B 43 (1991) 5039.
[8] H.R. Schober, C. Oligschleger, Phys. Rev. B 53 (1996) 11469. [9] P.W. Anderson, B.I. Halperin, C.M. Varma, Philos. Mag. 25 (1972) 1; W.A. Phillips, Low Temper. Phys. 7 (1972) 351. [10] A.P. Sokolov, R. Calemczuk, B. Salce, A. Kisliuk, D. Quitmann, E. Duval, Phys. Rev. Lett. 78 (1997) 2405. [11] C.A. Angell, J. Non-Cryst. Solids 131±133 (1991) 13. [12] K. Tanaka, Phys. Rev. B 30 (1984) 4549; I. Berman, N. Brandt, Sov. Phys. Low Temper. (in Russian) 16 (1990) 1227. [13] M.I. Klinger, S.N. Taraskin, in: A.C. Wright, S.A. Feller, A.C. Hannon (Eds.), Borate Glasses, Crystals and Melts, Soc. Glass Technol., Sheeld, 1997, p. 357. [14] K. Tanaka, Sol. State Commun. 60 (1986) 295; 54 (1985) 867. [15] C.A. Angell, P.H. Poole, J. Shao, Nuovo Cim. 16D (1994) 993. [16] P.W. Anderson, Phys. Rev. 109 (1958) 1492. [17] M.I. Klinger, S.N. Taraskin, J. Phys.: Condens. Mater. 9 (1997) 4425. [18] A.F. Ioe, A.R. Regel, Progr. Semicond. 4 (1960) 237. [19] M.I. Klinger, Problems of Electron Transport Theory in Semiconductors, Pergamon, Oxford, 1979. [20] M.H. Cohen, J. Non-Cryst. Solids 4 (1970) 391; N.K. Hindley, J. Non-Cryst. Solids 5 (1970) 17; L. Friedman, J. Non-Cryst. Solids 6 (1971) 329. [21] U. Buchenau, Philos. Mag. B 71 (1995) 793.