Manifestation of Coulomb gap in luminescence spectra of amorphous chalcogenides

)OURNA

L OF

NON-CRYSTALLIN SO ELIDS ELSEVIER

Journal of Non-Crystalline Solids 171 (1994) 172-181

Manifestation of Coulomb gap in luminescence spectra of amorphous chalcogenides L.P. G i n z b u r g * Department of Physics, Technical University of Communications and Informatics, 105855 Moscow, Russian Federation Received 7 May 1993; revised manuscript received 21 February 1994

Abstract

By a modification of the Coulomb gap theory is is shown that this gap gives an appreciable contribution to the luminescence spectra of amorphous chalcogenides. This result is confirmed by most of the available experimental data.

1. Introduction

The concept of the Coulomb gap is applicable to any system that contains randomly distributed localized defects with filled and empty electron states. Pollak [1] first pointed out that, owing to the long-range Coulomb interactions, the density of states (DOS) of such systems must be reduced near the Fermi level. Since then this concept has been analyzed in many papers and it seems that now two trends are present in this field of interest. The first was proposed by Efros and Shklovskii [2] and received detailed conceptional confirmation in several works (see, for instance, Refs. [3-6]). The main idea of this trend comes from the principle of ground-state energy minimization with respect to any one-electron transitions from filled states to empty ones. This

* Corresponding author. Tel: + 7-095 274 0032.

+7-095 924 3474. Telefax:

proposition leads to the conclusion that near the Fermi level, /x, the DOS should follow the quadratic law

g(e) ct ( e - l.z)2.

The other trend (see, for instance, Refs. [7,8] and references therein), criticises the first trend. From its point of view, g(e) should depend strongly upon many-electron processes and thus become more sharp than in Eq. (1). It should be pointed out that, despite the large number of discussions which give credit to one or another trend, most of the arguments are based only on computer simulations. Verifications that are based on experiments concerning, say, tunneling spectroscopy or variable-range hopping conduction are rare and do not lead to precise conclusions [6,8]. This lack seems surprising because many materials are known to contain charged defects in appreciable concentrations. For example, it is known [9-12] that many features of amorphous chalcogenides are attributed to va-

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SSDI 0 0 2 2 - 3 0 9 3 ( 9 4 ) 0 0 1 8 2 - M

(1)

L.P. Ginzburg/Journal of Non-Crystalline Solids 171 (1994) 172-181

lence-alternation pairs (VAP). These are the positively charged threefold-coordinated C~- centers and negatively charged singly-coordinated C 1 centers. Also in materials containing As there may be positive pnictide P4+ centers [11]. The concentration of these defects is about 1018 cm -3. Thus, at first glance, this defect system can serve as an excellent object for testing the presence of the Coulomb gap and all its peculiarities. To the author's knowledge, no such investigations have yet been undertaken. The main purpose of this p a p e r is to discover the contribution of the Coulomb gap to the luminescence spectra of amorphous chalcogenides. As shown below, this contribution has been discovered in the energy spectrum of VAP defects. The gap is consistent with the quadratic law given by the first trend. (In the following we call this the 'quadratic' Coulomb gap (QCG).) However, this result required some modification of the original theory given in Refs. [2-6]. This modification was needed in order to make it applicable to defects in amorphous chalcogenides. As a result, the Q C G is displayed in two 'shifted' energy regions, above and below/x. This problem is discussed in the next section. In Section 3, the main formulas which govern the contribution of Q C G (if present) to the low-energy part of amorphous chalcogenide luminescence spectra are derived. In Section 4, a procedure is described which permits a comparison of the formulas with experimental data. It is shown that in most of the available cases concerning materials such as As2Se3, As2S 3, AszSel.5Tel.5, experiments demonstrate the presence of QCG. Some preliminary discussions of these results are given in Section 5.

173

empty. In particular the occupied centers can be negatively charged and the empty centers can be positively charged. In any case, all the centers interact by long-range Coulomb forces and the system as a whole is assumed to be neutral. Now it is assumed that, when the system is in its ground state, there can be introduced an energy level, /x (the Fermi level), such that all the occupied levels lie below /x and the empty levels lie above it. If an electron is transferred from site i with energy e i #, then the total energy change should follow the relation

ej-ei-(eZ/Krij)>O,

(2)

where K is the dielectric constant, rij is the distance between sites and e 0 is the electronic charge. The inequality (2) can be understood in the following way. First, the inequality sign comes from the fact that, prior to the electron transition, the system was assumed to be in its ground state with minimum energy. Next, e i is the ground state energy of an empty site when the center i is filled. Thus it incorporates the Coulomb repulsion from this center. Thus when an electron leaves site i this Coulomb interaction must be subtracted. Now, let us consider a small energy interval [/x - g,/x + g]. If the centers with energies inside this interval are distributed at random then, according to the Poisson rule, the typical distance between a filled site and the nearest empty one can be estimated as ri~ -~ 0 . 5 5 ( U ( g ) / 2 ) -1/3,

(3)

where N(g) is the total concentration of centers under consideration. According to Fq. (3), the relation (2) can be transformed to an inequality g3K3 (ey -- e i ) 3 K ~3 < 0.33 e6 ~< 2.64 e~-)

2. C o u l o m b gap in amorphous chalcogenides

N(g)

We begin with a brief review of the main statements concerning the QCG. Let us assume that we have a system of randomly distributed centers with localized electron states. The centers can be divided into two parts: centers which are occupied by an electron and centers which are

The relation (4) means that if g - , 0 then the DOS, g(g)= dN(g)/dg, must tend to zero and not slower than in accordance with Eq. (1). A faster tendency of g(g) to zero (when g ~ 0) can be forced only by some special source which makes the centers n e a r / x more distant than due

(4)

L.P. Ginzburg /Journal of Non-Crystalline Solids 171 (1994) 172-181

174

to simple randomization conditions. If we ignore secondary electron-electron effects which are shown to affect only an extremely small energy interval n e a r / z [5,6], this assumption seems unrealistic. Thus we come to the conclusion that, in the vicinity of /x, there should be an interval A (gap) such that if I Ei. j -- ~Z I ~ A / 2 then Eq. (1) is satisfied. The above considerations are the basic propositions of the Q C G concept as they are described in literature (see, for instance, Ref. [6]). Application of these principals ~o concrete materials such as amorphous chalcogenides demands some corrections. First, these corrections concern the energies ci and Ej. It is known [9-12] that owing to strong e l e c t r o n - p h o n o n interaction the release or capture of an electron by some point defect is accompanied by lattice distortions. In Refs. [3] and [6] a similar (in some sense) situation was treated: the electron transition was accompanied by hops of other nearby electrons. By analogy, instead of Ei and Ej the polaron energies E i and E i are introduced which have the following meaning: -E~ is equal to the total work which must be done in order to remove an electron from site i to infinity and to change the atomic structure around the remaining hole. Correspondingly, E i is the work which must be done in order to bring an electron from infinity to site j and also to change the atomic structure surrounding it. Thus the left-hand side of Eq. (2) should be replaced by

ej

-E,-

e2 --

K rij

with E / < / ~ and Ej >/x. However this correction is not the only one that is needed. We must also take into account the effect of local forces which play an essential role in determining the ground state of the glass. It is known [9-12] that in amorphous chalcogenides these forces place the positive C~-(P4+) levels near the bottom of the conduction band and the negative C 1 levels near the top of the valence band. Thus the defect subsystem already contains a host energy gap which has nothing to do with the Coulomb gap. It

C.B. "f--T- c3 W #+Era

--

26

d

-~

___--~

#-Era

6E

V.B.

t

Fig. 1. Energy level scheme for defect subsystem in amorphous chalcogenides. The thin curves represent the DOS incorporating the polaronic shifts, W. The thick lines reproduce the QCG part of the DOS.

seems likely that, even after the energy shifts due to lattice distortions or energy broadening produced by randomness, there still remains a minimum energy interval, say, 28 (Fig. 1) that must be overcome in order to replace an electron from a Ci- center and bring it to a C~-(P4+) center. In this case, the Fermi level, /x, must be placed in the middle of this interval. Hence we can say that prior to electron transitions all the filled polaron states lie below/z - 8 and all the empty polaron states lie above /x + 8. From these conditions, it follows that instead of Eq. (2) we should write Ej-E

i - (e2/Krij)

> 28.

(5)

Introducing new energies A E i = E i - 8,

AE i =

E i + 6

(6)

for empty and filled states respectively (Fig. 1) we shall have A E j - A E i - ( e ~ / K r i j ) > 0.

(7)

Eq. (7) can be treated in the same manner as in original papers concerning the Q C G (see Ref. [6] for instance). The only difference is that instead

L.P. Ginzburg/ Journal of Non-Crystalline Solids 171 (1994) 172-181 of [/z - g, # + g] we shall consider states within two intervals: [/z + 6,/z + E], where E > 3, and [/.t - 6,/.t - E], where E < 8 (see Fig. 1). Utilizing Eqs. (3) and (7) we have N ( E ) < 0.33

e6

~<2.64 ( E

-

8)3K 3

e6

(8) and thus directly come to a conclusion that there must be a value E m such that if

]Ei. j - al
sideration can radiatively recombine with another state, q~i, within the energy interval, [E i, E i + dEi], such that E~ - E / = hoJ can be estimated as

Pij( Ei, Ej) d E i dE~ = Q(hw0, Ej)Wijg(Ei) dEi dEj,

(12)

where Wq is the probability of a q~j ~ q~i transition (per unit volume and unit time)• According to the Fermi 'golden rule',

(9)

h,.), (13)

(10)

where 2 ' is the electron-photon interaction Hamiltonian. The common quantum expression for 2;%' has the form (see for instance Ref. [13])

then

g(Eid) at ( Ei,j- a) 2.

175

Wij = (2rr/h)I(,pe

2' -

12 ' I,pj> 12 a(Ej

ie°hm ,[ 2"rrh

[--0--~w( U + 1)

) 1/2

exp(iq-r).

3. Contribution of the QCG to luminescence spectra

First, the defect subsystem which is supposed to be connected with the low-energy part of the luminescence spectra is specified. This subsystem is a system of C{ and C~-(P4+) centers which are distributed in space almost at random. According to Refs. [11,12] this means that intimate VAP (IVAP) modifications, where the mentioned centers form associated pairs, are not taken into account. Thus in our model there cannot be any space correlations between the densities of occupied and empty states as well as any excitonic-like effects. (The posible behaviour of IVAPs is discussed in Section 5.) Now let us introduce a quantity

Q(hw o, Ej) d E i = F ( h w o, Ej)g(Ej) dEj,

(11)

which is the probability that an electron after absorbing a quantum ho) 0, and after possible accompanying processes (tunneling, hopping, lattice distortions and so on) finds itself in some (polaronic) state, %, within an energy interval [Ej, Ej + dE~]. In the right-hand side of Eq. (11), F(hcoo, Ej) is the number of Cj states available for the electron and g(Ej) is their density. Now, the joint probability that the electron under con-

(14) Here 12 is the volume of the system, m is the effective mass (which incorporates the polaronic effect), q is the photon wave vector and N is the number of photons which are initially present in the system. Now, in the conditions under consideration, ho2 < 1 eV. If q~i, Cj represent localized states with dimensions about several Angstroms, then in this case the dipole approximation becomes justified. Hence, putting N = 0, we shall have o

ieoh(2~hl'/2 m k ]

( il2tl J)--

(15) Utilizing the relation i

(q)i I V I

=

h(q)i I P I,pj> moJ

h (v~ ~1 vj),

(16)

we can write • [ 2"rrhw)X/2

(qgil~'~'l~j)=ICo[~

(rij),

(17)

L.P. Ginzburg/ Journal of Non-Crystalline Solids 171 (1994) 172-181

176

where (rij) ~ (qgi I r I @j). Thus, introducing Eq. (17) into Eq. (13) and taking into account Eqs. (12) and (11), we come to the equality

It is easy to verify that in this case we shall come to an equation I( h(.o ~ h(.om) = ( h o ) ) 2 ( A - Bhw + C(ho.))2), %

gij( Ei, Ej) d E i d g

(21)

4 ,rre ~o) - I(Pij) 12F(hwo, Ej)g( Ej)g( Ei) I2 X(5( E j - E i - h a ) )

dEi d E j.

where Em

Aoc fa dEj F(hwo, Ej)(Ej-6)4; (18)

We want to emphasize that, in the case of localized q?i and q~j states, the quantity, (t:ij), cannot essentially depend upon hw. If we integrate Eq. (18) by dEj from (5 to and by dEi from - ~ to -(5, we obtain the quantity which represents the number of hoJ photons emitted per unit time (and unit volume). In order to calculate the emitted energy (intensity), this quantity must be multiplied by ho~. Thus, according to Eq. (18), the luminescence intensity spectra can be represented by the relation (/x = 0)

I(hw) cx (hoJ) 2

dEj

f-jdEi

F(hwo, Ej)

g oc 2 ~

Em

dEj F(hwo, Ej)(Ej - 6)3;

Em

C ~ fa dEj F(hwo, E j ) ( E j - a ) 2.

4. Comparison with experiment Eq. (21) enables the next procedure for verification of the QCG contribution to the luminescence spectra. First, having an experimental I(hw) curve, we can construct another curve

~( hw) = I(hoo)/(hw) 2. x g ( Ej)g( Ei)(5( g -- E i - ho))

(19)

Let us now assume that we are interested only in some low-energy part of I(hto), say, 0 ~ hw m. This means that, if the QCG in the sense represented by Eqs. (9) and (10) is present, then we may choose such a value, htom, that the intervals +[(5, E m] will coincide with the gap. In this case, restricting the integration in the right-hand side of Eq. (19) by (5 and Em, we, according to Eq. (10), can utilize the relations

g(Ej) ct (Ej - (5)2, g ( E j - hoo) ~ ( E j - ( 5 -

h6o) 2.

(23)

According to Eq. (21), if the QCG is present then the low-energy part of this curve should satisfy the relation

= (hw)Zfa dEj F(hw o, Ej)g(Ei)

x g( Ey - hoo).

(22)

(20)

~ ( h ~ ~ hoJm) = A - B h o o + C ( h o J ) 2.

(24)

Now, having drawn the ~(hoJ) curve, with the help of tangent lines we can draw a curve for the function

rT( hw ) = d~( h~o) / d( hw ) .

(25)

According to Eq. (24), in the presence of QCG the low-energy part of this curve should satisfy the equality "r](ho) ~ hOJm) = 2Chw - B .

(26)

Thus, if drawing the rl(hoJ) curve we observe that its low-energy part becomes linear, this can serve as an indication of the QCG contribution. The described procedure has been applied to most of the available experimental data concerning luminescence in amorphous chalcogenides [14-19]. In the overwhelming number of cases,

L.P. Ginzburg / Journal of Non-Crystalline Solids 171 (1994) 172-181

177

5,0

3,5

i

3.0

4.5

/

./

4.0

2.5

3.5

2.0

l

I

I

I

I

0.75

0.80

0.85

0.90

0.95

hw,(ev)

---

Fig. 2. An example of the ~(hoJ)= l(hw)/(hto) 2 dependence with a typical low-energy part. Here l(h~o) is the luminescence intensity. Points are deduced from the data of Bishop and Mitchell for a-As2S 3 [16].

the results were beyond doubt: the low-frequency parts of the r/(hw) curves converted into well-defined straight lines. Some typical examples of ~:(hto) and r/(hoJ) dependences are shown in Figs. 2-5. All the curves for materials listed in Table 1 had the same character. Only in three cases was it impossible to observe the mentioned results. These are the cases of a-As2Se 3 and c-As2S 3 in

16

3.0 i

i

0,725

i

hW, ( e V )

1

0,750

Fig. 4. The function r/(h~o) for a-As2Se 3. The straight line results from the Q C G contribution. Points are deduced from the appropriate ~C(hto) curve with the help of the tangent lines. L u m i n e s c e n c e data are from Street et al. [15].

Ref. [14] and a-Se in Ref. [19]. The only reason for this, in the author's opinion, is the small range of the low energy parts in the I(hto) curves. It should also be emphasized that, as seen from Table 1, the QCG was discovered also in some crystalline materials. This observation is in accordance with the suppositions [9,10] that the lumi-

/

/

//

/ / /

14

2.5C

/ /

? oJ

12

? >

2.25

IO 2.00

1.75

I

0.725

I

I

0.750

0,775

hw,

0.800

(eV)

Fig. 3. The function "q(hto) for a-As2S 3. The straight line results from the QCG contribution. Points are deduced from the appropriate ~:(hto) curve with the help of tangent lines. Luminescence data are from Bishop and Mitchell [16].

I

I 0.575

I

hw,(ev)

I

I

0.600

Fig. 5. The function -q(hoJ) for a-As2S 3. The straight line results from the QCG contribution. Points are deduced from the appropriate ~:(hto) curve with the help of the tangent lines. Luminescence data are from Mollot et al. [18].

178

L.P. Ginzburg /Journal o f Non-Crystalline Solids 171 (1994) 172-181

Table 1 Parameters defining the QCG contribution to the luminescence spectra in amorphous chalcogenides Material

a-As2Se 3 a-AszSe 3 a-AszSe 3 c-AszSe 3 c-As 2Se 3 a-AszS 3 a-AszS 3 a-AszS 3 c-As2S 3 a-AszSel.sTel. 5

Ref. for luminescence data

hto m

A

B

C

(eV)

(eV 2)

(eV-3)

(eV-4)

[16] [17] [15] [15] [ 14] [16] [18] [14] [18] [16]

0.702 0.560 0.730 0.840 0.746 0.765 0.600 0.760 0.820 0.465

27.51 166.96 3.67 3.58 1174.9 46.93 2.01 783.6 3.96 197.1

97.36 855.36 11.79 10.82 3495.6 128.91 8.62 2302.0 11.01 903.7

92.71 1063.51 10.53 8.33 2606.9 91.72 9.29 1704.0 7.78 1052.8

nescence properties in amorphous and crystalline chalcogenides are governed by the same defect

tErn-t3

A O~Jo

}2

an(hwo,

6 ) ( E m - ~) n

(31) n+p+l

n=l

From Eq. (31) it follows that 7(P)Y(P - 2 ) ( y ( p - 1)) 2

dUF(htoo, U + 6 ) U 4,

E r

B (x 2foem-~dUF(htoo, U+ 6)U 3,

0.27 0.24 0.28 0.25 0.25 0.26 0.25 0.25 0.25 0.25

where

centers.

However the discovery of a straight line in the r/(hto) curve is not the only indication of the QCG contribution. Let us rewrite Eq. (22) in the form

AC/B 2

aZ(Em - 6) 2r

(r+p)2-1

V,V~

aras(Em - 6) r+s

+ rZ"~@ ( r + p + l ) ( s + p - 1 )

a2(Em_6)2r

(27) Er

(r+P) 2

aras(Em_6) ....

+ Y_.,F., r#~s (r + p)(s + p) (32)

C ~ foEm-~dUF(hwo, U+ 6)U e and expand the function F(hwo, U + 8) into a series

where a r - ar(htoo, 3) and, according to Eq. (31), r, s >/1. In the numerator of the the right-hand side of Eq. (32), the coefficient at ( E m - 6) 2r is

oo

F(hto o, U + 6 ) = E an(ht°o, 6) Un.

(28)

n-1

(29)

Introducing Eq. (28) into Eq. (27), we have

C~'y(2)(em-~)

3,

2 (1 2

)-'

(r+p)

2

If p >i 4, the deviation from 1 in the right-hand side bracket is ~ 0.04. The coefficient at ( E m t~)r+s in the numerator of the right-hand side of Eq. (32) is 1

a~a~ ( r + p + l ) ( s + p - 1 )

(30)

,

a~r

(33)

(

A ~ y ( n ) ( E m - 3 ) 5,

B a 2~,(3) (Era - 6 ) 4

-1 -

(r+p)

Note that the summation in Eq. (28) begins from n = 1. This is because in accordance with the sense of F(hto o, Ej) we can choose 6 to satisfy the condition

V(hwo, Ej ~< ~) = 0.

a 2 ( ( r + p ) 2 _ 1)

=2

ara ,

1

)

+ (s+p+l)(r+p-1)

1 - [ 1 / ( r + p ) ( s + p)]

(r + p)(~ + p) (1- [1/(r + p)~])(1- [1/(s + p)q)"

(34)

L.P. Ginzburg/Journal of Non-Crystalline Solids 171 (1994) 172-181

Again if p >t 4 the deviations from 1 that are seen in the right-hand side of Eq. (34) are ~< 0.04. Thus, comparing the right-hand sides of Eqs. (33) and (34) with the coefficients in the denominator of Eq. (32), we can concludes that, if p >~ 4, then Y ( P ) 3'(P - 2 ) / ( y ( P - 1))2 _ 1.

(35)

From Eqs. (30), (31) and (35) it follows that the coefficients in Eq. (24) must satisfy the relation A C / B 2 = ¼ [ 7 ( 4 ) 7 ( 2 ) / ( 7 ( 3 ) ) 2] - 0.25.

179

3.5

3.0

2.5

2.0

(36) 1.5

This relation means that, if we discover a linear dependence in the low-energy side of the +/(hw) curve, then Eq. (36) should serve as another and, in our opinion, essential indication of the QCG contribution to the luminescence spectra. This is because at first glance one might have an apprehension that the mentioned linear dependence is accidental, or one may view Eq. (24) as an ordinary general small-hw expansion having no relation to the QCG. However, representing a consequence of Eqs. (19)-(22), Eq. (36) leaves no doubts that its origin is the QCG. The verification of Eq. (36) can be carried out as follows. Having drawn the linear function, defined by the low-energy part of the ~/(ho~) curve, and utilizing Eq. (26), we can deduce the experimental values of B and C. These values can be introduced into Eq. (24). Using the data of the ~(hoJ) curve, we can thus define the value of A. Having determined the three values A, B and C, we can verify Eq. (36). As seen from Table 1, in all investigated cases this relation is confirmed within a range < ]0.03]. This is despite the large variation of A, B, C and the arbitrary units in which the intensity was measured (in the case of Ref. [15], where these units are not explicitly shown, we normalized to unity the entire range of I ( h w ) variance).

5. Some preliminary discussions

1.0

I 0.80

I 0.85

i 0.90

~ 0.95 ~LO, ( e V )

Fig. 6. The ~:(ho~)= I(ho~)/(hw) 2 dependence for a-Si. Points are deduced from Engemann and Fisher [20].

(the data are taken from Ref. [20]). As is known [9] a-Si does not contain charged defects and it is seen that the shape of the curve in Fig. 6 differs from that in Fig. 2. The former curve could never satisfy a relation like Eq. (26). At the same time we see that the mentioned results are observed by considering only the VAP sub-system without taking into account the IVAP modification. It is not yet clear what modification really prevails in the number density of charged defects in amorphous chalcogenides (see Refs. [9-11]). Thus the further considerations must be treated as only preliminary. In our model, the Q C G is displayed in two separate energy intervals: above ~ + 3 and below / z - & Thus it becomes interrupted by an additional gap equal to 23. This circumstance seems important for the following reasons. First it enables a reasonable estimate of the number of defects which participate in Q C G formation. According to Fig. 8 of Ref. [5], computer simulations show that there should be a relation like E m - 6 ~ 0 . 5 ( e Z / K ) ( 2 U v ) ~/3,

From the results described above, it seems obvious that the Q C G is present in the energy spectra of amorphous chalcogenides. In Fig. 6 we show the £ ( h w ) curve for luminescence in a-Si

I 1.00

(37)

where 2 N v is the total concentration of charged defects. In our case from Eq. (37) it follows that 0 . 6 3 ( e Z / K ) N v 1/3 > (ho0m/2) - 3,

(38)

180

L.P. Ginzburg/Journal of Non-CrystaUineSolids 171 (1994) 172-181

where hw m is the maximum energy for which the linear dependence of ~7(ho~) holds. The values of hto m can be deduced from corresponding curves similar to those in Figs. 3-5 and are listed in Table 1. Utilizing these data, it is not difficult to verify that if we put 6 = 0 then the value of N v, deduced from Eqs. (37) and (38), will be much larger than the experimentally established density of charges in amorphous chalcogenides (which is about 1017-1018 cm -3 [10]). At the same time, according to Fig. 1, we can write 6 ~ ( E g / 2 ) - ~E - W - 5e.

(39)

Here Eg is the band gap, ~E is the departure of defect levels from corresponding band edges, W the polaronic shift and ~e is the energy broadening. If we put for AszSe 3 Eg = 1.8 eV [21], ~E + W = 0.55 eV [22], ~E = 0.1 eV [23] and K = 10 [12], then we shall have 6 ~ 0.35 eV. Thus, adopting the value hw m = 0.716 eV, which is the mean for nearby spectras shown in Refs. [15] and [16], we shall have N v ~ 6.9 × 1017 cm -3.

(40)

Although the above number is only an estimate, it shows that the number of VAPs must be at least comparable with the number of IVAPs. The concept of a 26 gap also helps to understand why the IVAPs (if present) do not affect the low-energy part of the luminescence spectra. This is important because in principle the IVAPs can be affected by the field produced by long hops between VAP centers. Owing to energy relaxation related to intra-IVAP electron transitions, the Coulomb gap can become sharper than predicted by Eqs. (1) or (10) (see the Introduction). However, as seen from above, the quadratic law is in our case retained. The situation can probably be explained in terms of a model discussed by Chicon et al. [24]. This model takes into account the presence of a large number of dipole-like pair (analogous to IVAPs). The probability of an energy relaxation less than, say, x 0 is put equal to zero. Then, as a consequence, this assumption results in a displaced DOS [24]

g( E) o~ (E - ( X o / 2 ) ) z,

(41)

which, if we put x 0 = 26, does not differ from Eq. (10). (In this sense our Q C G can be treated as 'completely hard' [24].) Thus we can say that the local forces which determine the ground state of the glass not only interrupt the ordinary QCG by introducing an additional 26 gap, but at the same time make the IVAPs incapable of influencing the low-energy luminescence. The latter is mainly due to long-range transitions between VAP defects. Of course such transitions are much better coupled to phonons. Strictly speaking, according to Street [25], the right-hand side of Eq. (19) should be multiplied by a factor: e x p ( - / 3 T ) , where/3 = constant. However it is not difficult to verify that this will not affect any of the subsequent considerations. Concluding this section, we shall draw attention to another problem that shows the importance of the 26 gap. This concerns the variablerange hopping conduction. It is supposed [3,6] that the main indication of the presence of QCG comes from the low-temperature conduction law ~r = ~r0 exp( - To~T) 1/2,

(42)

which appears instead of the common Mott law o- = ~r0 exp( - T6/T) 1/4.

(43)

As far as the author knows, neither Eq. (42) nor Eq. (43) have been observed in materials such as As2Se3, As2S 3 and As2Sel.sTel.5. Concerning this contradiction, it is pointed out that in a model incorporating the 26 gap the conduction due to defect centers can never lead to a dependence such as Eq. (42). This is because such a model will always retain an activation term having the form e x p ( - 26/kBT).

6. Conclusions

It has been shown that a Q C G in the energy spectra of amorphous chalcogenides contributes to the low-energy part of the luminescence intensity. The origin of this gap is a system of VAP defects and is displayed in two separate energy

L.P. Ginzburg / Journal of Non-Crystalline Solids 171 (19941 172-181

intervals being interrupted by an additional gap. The latter is created by the action of local forces. The results of the paper are confirmed by most of the available experimental data concerning luminescence in amorphous chalcogenides. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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181

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