- Email: [email protected]
A study of the relativistic electronic state of disordered systems
J. Phyr. Chem. So/t& Vol. 51. No. 2. pp. 147-155. N‘XI Printed in Great Britain.
0022~x93/90 s3.00 + 0.00 Pergamon Press plc
A STUDY OF THE RELATIVISTIC ELECTRONIC STATE OF DISORDERED SYSTEMS C. L. ROY and CIUNDAN BASU Department of Physics, Indian Institute of Technology, Kharagpur 721 30.2, India (Received 21 June 1989; accepted in revised form 15 September 1989)
Abstract-We have carried out a quantitative study of the relativistic density of states (DOS) of a one-dimensional system, using analytical results of a related treatment previously reported by Roy (J. Phys. Chem. So&h SO, 1II,-1989).-We have examined various features of the qua&tat&e resultsof thk relativistic DOS of our model of a ID disordered svstem. in liaht of(i) non-relativistic CNR) DOS of the same disordered system, (ii) relativistic and non-relkvistk DC% of frk electrons, and (iii) klativistic and non-relativistic band structure of 1D periodic systems relevant to our disordered systems. Among many other things, our investigation brings out clearly the circumstances under which the relativistic effects on the DOS of disordered systems can become substantially larger; in this regard, it is found that the relativistic effects on the DOS of disordered systems are very large in the simultaneous presence of low concentration of scatterers and low values of height (depth) of potential barriers (wells) which represent the potentials due to scatterers in our model of disordered systems. Keywords: Dirac equation, density of states, band structures, propagator, Simpson’s 1/3rd rule.
1. INTRODUCTION Theoretical treatments of the electronic properties of condensed matter consisting of heavy atoms need to be relativistic. During the last three decades or so, relativistic investigation of various electronic properties of condensed matter evoked continuous attention. So far, relativistic studies of condensed matter have been concerned with (i) bulk states of threedimensional (3D) crystalline systems [l-4], (ii) bulk states of one-dimensional (I D) crystalline systems [Ml, (iii) surface states [P-12] and interface states [13], and (iv) electronic states of disordered systems [ 14-181. Any rigorous relativistic study of electrons in condensed matter would require exact use of the (time-independent) Dirac equation with adequate account of the relevant potentials. Relativistic studies of electronic properties of 3D condensed matter, with exact use of the 3D (time-independent) Dirac equation, involve enormous complications. As a result, relativistic studies of electrons in 3D condensed matter are often required to take recourse to approximate methods [4]. On the other hand, for 1D condensed matter exact use of the 1D Dirac equation is possible for properties such as the bulk states of crystalline systems and surface states [5-131. However, for the relativistic treatment of the density of states (DOS) of disordered systems, the use of the exact Dirac equation creates serious difficulties even when the system is one-dimensional and it becomes necessary to make use of the approximate methods. Compared with the relativistic studies of items (i)-(iii) mentioned earlier, those [14-181 of (iv) are quite recent. These (relativistic) studies dealt with
some 1D models of disordered systems. Since about 1960, several workers have been using 1D models [ 191 very fruitfully for the purpose of non-relativistic investigations of the electronic properties of disordered systems. We hope that relativistic results about the electronic properties of ID disordered systems will form a basis for relativistic studies of more realistic disordered systems. Of the works in Refs 11118, the ones under Refs 17 and 18 are concerned with the relativistic DOS of a particular model of ID disordered systems. The treatments of [ 171 and [18] are based essentially on relativistic generalizations [21] of two non-relativistic approaches due to Klauder [20]. The purpose behind both [17] and [18] is to d&e formulae for the relativistic DOS by evaluating the relevant dressed electron propagator. In [17], the dressed electron propagator is evaluated without propagator modification, while in [ 181,propagator modification is taken into account. Thus, the treatment of [18] is more realistic than that in [17]. In order to see the impact of relativistic corrections, it is necessary to carry out quantitative investigations based on relevant analytical results. While quantitative investigations of relativistic effects have often been carried out on items (i)-(iii) mentioned above, no quantitative study has yet been made about the relativistic electronic states in disordered systems. The purpose of this paper is to report such a quantitative study based on the results of the analytical treatment in (181. The treatment in [18] is concerned with the derivation of the relativistic DOS of a 1D disordered system in a fairly realistic manner, and the quantitative results now being reported would, conse-
147
118
C. L. ROY and CHASDAN Btic
quently, throw much light on the nature of relativistic effects on the electronic states of disordered systems. The formulae and related issues concerned with our quantitative study of the relativistic DOS, are presented in Section 2. For a better comprehension of the relativistic effects, we compared our (quantitative) results with allied results of several other cases such as (I) the non-relativistic DOS of the same ID disordered systems as studied relativistically, (II) the non-relativistic and relativistic DOS of free electrons, and (III) the relativistic and non-relativistic band structures of periodic systems relevant to our model of disordered systems. The comparisons of (I), (II) and (III) are presented in Sections 3, 4, and 5, respectively. The various numerical analyses that we have carried out, are discussed in Section 6, and the results are reviewed critically in Section 7. Finally, some features of our numerical analyses are elucidated in the Appendix. 2. FORMULA
AND
FOR THE
RELATIVISTIC
The relativistic (energy) DOS, p,(a), reported in [IS] was obtained by summing up the dressed electron propagator relevant to (3), and potentials (1) and (2), taking propagator modification into account. Explicitly, p,,,(c) appears as:
Pm(~) = rt’P(Vh where
(5)
x[nF -
L(~~tl)l~&(~~rl)d~ [G&P, tt)l Z, (p, ~11'+
G,(P, tt) & (PYr1)1” (6)
The symbols in (6) have the meanings:
DOS
SOME RELATED ISSUES
The model of disordered systems for our present study is, as mentioned before, the same as that of Ref. 18. It consists of an electron in the presence of “N” fixed atoms with random locations X, along the x-axis in a chain of length “L”. The atomic potentials, V(x - x,), are assumed to be rectangular barriers or wells of width 2b. The total potential, V,(x), appears as: V,(x) = 1 V(x -x,x
F = V;A V;=2bV,
V,, for (xi-b)
(7)
n = concentration of scatters (atoms) along the chain .L
= limit E ‘V-X 0 L
-I 1
L-Z
(1)
q+ -nF-&
G,(p,v)=nF V(x -xi)=
(4)
(8)
<(x,+6) * (2)
q+=q+iS,
6+0+
= 0, for x > (xi + b) or x -=z(xi - b) I The treatment in [18] of the relativistic DOS of the disordered system corresponding to potentials (I) and (2), makes use of the following approximately relativistic equation (211for a constant potential “V”: d2tj 2m s++tl-
&(P*V)=
- @?7-(P. 4b* --
mnF2 b
(
62
c+2mc’
c=(ER-mc’),
s,l
-ry+D: 1 (9) p’-T
(P2
W)$ -0,
where tl=
rl) + V-P,
1
> W= VA
s=Zm(q-nF-D,)
1 ER = relativistic energy eigenvalue c = velocity of light.
Q(P,v)=~
(13)
149
Relativistic electronic state of disordered systems
where
T, = exp( - 2gb)[sin 2cf - p)b] x U-(U-P)*
p(c) = NR limit of p(q).
-g2)
Explicitly, p(c) is given by:
-~~f-~l~21+~~~/2+p2-~pf-~*~ x [i
Q,-[1
-exp(-2gb)cos2Cf-pjb]
(14)
-exp(-2gb)cos2Cf-p)b] x ~f~cf-P~z-~2~--2cf-Pl~21
x
[G&J, ~11~ x2 (P. 0 dp [~~v~-(G~(P,~))Z,(P,~)I*+[G~(P,~)~?(P.E)I~’
-t?(3f2+PZ-w--g2)
(24)
xexp(-2gb)*sinZ(f-p)b
(1%
D= U---P)~+~~V*+~~I
(16)
f = [q; + D~“‘cos(~/~)
(17)
The meanings of the various symbols in eqn (24) are given below. G,,(p, 6) = NR limit of G,(p, V)
1
c+ -nVi-P
=nV;
g = (4: + DfJ”2 sin(4/2)
(18)
tan 4 = VU&)
2
2m
1 -I
(25)
Z, (p, 6) = NR limit of X, (p, 7) 2
q=nF+$D,
(19) =
D, = -$lT(qo,
tt) + T(-qo,
rt)l
(20)
D 2=~[Q(4o’“)+Q(-“‘sll+~.
-~[S(p,r)+S(-p,c)] --
mn( Vi)2 b
p2-a (p2 - a)2 + y2
1
(26)
Z, (p, 6) = NR limit of Z, (p, q)
(21) p,,,(c) is the DOS with respect to energy “6”. The entity “6” is the relativistic analogue of NR energy of the electron. Thus p,(a) can be conveniently used for a comparative study of relativistic and non-relativistic DOS, with inclusion of ail salient features of the relativistic DOS.
+y[J(p,c)+J(-p,r)] +
mn(W2 b
Y (P2 -a)2+
Y2
1
(27)
a = NR limit of T 3. NON-RELATIVISTIC DOS OF OUR
=2m(6
MODEL AND RELATED ITEMS
The non-relativistic DOS of the present model of disordered systems, was reported previously by Bhattacharya and Roy [22]. The formula for the non-relativistic DOS follows as the limit of p,(e) when “c” is allowed to become infinitely large. We thus have: Mc)lNR = non-relativistic
DOS of our model
= NR limit of [~‘p(rl)].
(22)
Taking the NR limit on the RHS of eqn (22) in the way mentioned above, we have
bA~)INR = P(r). PCSSI,2--E
(23)
S-NR
-nVh-X)
(28)
limit of T(p,q)=Z
S,
(2%
J(p, c) = NR limit of Q(p, IJ) = z
(30)
S, = NR limit of T,(p, 9) =exp(-2&)[sin2(r
-p)b][r((r
-~)~-f.i~)
- 2(r - p)d7 + d(3r2 + p2 - 4pr - Cf2)
x [l -exp(-2rb)cos2(r
-p)b]
(31)
I50
C.
L. ROYand
of free electrons; its relation to momentum follows:
J, = NR limit of Q, =[I -exp(-2db)cosZ(r
- 2(r -PM21
-d(3rz+p2-4pr
-d*)
x exp( - 2db). sin 2(r - p)b
for (39) for (40),
5. EQUATIONS
= [(r - p)* + d212[r2+ d’] [q: + Y2]li4.cos8/2
(33) (34)
d = NR limit of g = [q: + Y2]“4. sin O/2 (35) tan 0 = Y 4; c=NRhmitofq=nV;+g+X
(36)
X= NR limit of D,
~(6 + 2mc*) =pV.
FOR BAND STRUCTURES
KP equation for
V, > 0 and
(37) cos ~(a + 26) = cosh(2k,b). k:--kf + [ 2k,k,
mn( V&)2
(38)
+by.
(42)
As mentioned in the Introduction, we require for our present study some aspects of relativistic as well as non-relativistic band structures in the crystalline systems related to our model of disordered systems. The aspects of band structures for our purpose would be obtained on the basis of relativistic and nonrelativistic Kronig-Penney equations related to crystalline systems of periodicity (a + 26) where 2b is the width of potential barriers (wells) and a is the zero potential region between potential barriers (wells). The KP equations to be used are as given below (in all cases, p is the wave vector).
Non-relativistic 6 < v,
-~[S(q,,f)+S(-qO.L)l
6 = -&
(32)
Z= NR limit of D
=
p is as
-p)b]
x ]r((r -PY -8)
r = NR limit off-
CHANDAN Btiu
cos(k, a)
1
. sinh&b)
. sin(k, a),
(43)
where
4. DOS OF RELATIVISTIC AND NON-RELATIVISTIC FREE ELECTRONS
The relativistic as well as non-relativistic DOS of 1D free electrons can be found by noting that, for both cases, (i) the elementary phase-cell is “II” and (ii) there is two-fold momentum degeneracy. With the use of these features and the dispersion relations relevant to non-relativistic and relativistic free electrons, we can derive the following results:
Non-relativistic 6 > v,
pR =
(k: + S:) sin(hb)
- -.
(44)
where
1 I’*
(3%
(a + mc’)
= hc * [L(L +
sin(k, a),
2k, 81
DOS per unit length for 1D relativistic free electrons 2
V, > 0 and
cos(p(n + 26)) = cos(2/?,b)cos(k,a)
pNR= DOS per unit length of ID NR free electrons 1 2 =5; [ z
KP equation for
2mcZ)]“* .
Relativistic KP equation for V, > 0 and 6 < V, (40)
In eqns (39) and (40), L stands for the kinetic energy
cos(p(u + 26)) = cos(k,a) cosh@k,,b) + P, sin(k,a) sinh(2k,,b),
(45)
151
Relativistic electronic state of disordered systems
where k;==
C(C-t 2mc2) h2c2
k:l=(v,-s)(~
P‘Relativistic
- v,+~c’) h2c2
k;,E2 - (V, - c)2k; 2a(V,,-a)k,,k,
*
KP equation for V,, B 0 and L > V,
cos(p(a + 26)) = cos(k,a) cos(tj?,,b) - P2 sin(k,a) sin(2&6),
(46)
where B2 _ (C - V&6 - v, + 2mc2) II h2c2
Equations (45) and (46) are taken from Ref. 5, while eqns (43) and (44) are available in standard books.
6. NUMERICAL
ANALYSES
Our numerical analyses are concerned with (i) evaluation of the relativistic DOS, p,(a), on the basis of eqn (4) and related formulae, (ii) evaluation of the non-relativistic DOS [p,(c)],,, on the basis of eqn (23) and other related entities, (iii) evaluation of the non-relativistic and relativistic DOS of free electrons on the basis of eqns (39) and (40) respectively, and (iv) widths of some allowed and forbidden regions of energies on the basis of the KP eqns (43x46). We discuss below the salient features concerned with these numerical analyses. Evaluation of p,,,(c) To start with evaluation of p,(c), we are required to find D, and D2 of eqns (20) and (21). These equations constitute a set of coupled equations in terms of D, and D2; it is not possible to solve them analytically, and we consequently have taken recourse to a (numerical) iteration method to solve them. For our purpose, we choose the values of n, Vi, b . . . ) etc. and vary the values of 6; the value of qO corresponding to a particular c is obtained from eqn (19). Inserting D, and D, in eqns (9) and (10) we next obtain Z, (p, q) and Z,(p, q). Substituting Z, (p, q) and Z2(p, q) in (6). we then obtain p(q) by carrying out a numerical integration. We have used Simpson’s 1/3rd rule in regard to numerical integration in eqn (6). Some particulars about numerical integration in
Fig. 1. Variations of p,(c), [~,&)]~a and pSR, with e:. Graphs IA and 2A correspond to [p,(r)],,, while graphs 19 and 29 correspond to p,(c). The values of the parameters are as follows. For IA and 19: V, = 0.001. b = 258.14, n = 9.68 x IO-‘; for 2A and 29: V, =O.OOl, b = 258.14, n = 9.68 x 1O-4.
(6) are discussed in the Appendix. Making use of p(q) and eqn (4), we finally obtain p,,,(c). The results for p,,,(e) are shown in Figs 1-5, and Tables 1 and 2.
r
o.ool
I a002
I
111
I
aoo4 4006’ 4008 sot
I o.ol2
1 0.014
t
Fig. 2. Variations of [~_(~)],,a and p,(e) parameters indicated klow. For 1A and b = 258.14, n = 9.68 x 10mJ; for 2A and b = 258.14, n -9.68 x IO-‘. As in Fig. while 19 and correspond to [~&&e to P,(f ).
with c, for the 19: V, = 0.0001, 29: V, = 0.0001, 1, IA and 2A 29 correspond
C. L. ROY and CHANDANBAW
0,ol -
soot
I
I
I
I
I
I
1
coo2 a004 0006 oooa a01 o.o2 a014 c Fig. 3. Same kind of graphs as in Figs 1 and 2. The parameters now are as follows. For IA and 1B: Y = 0.001, b = 25.8, n = 9.68 x IO-‘. IA corresponds to [p,(a)],a and IB to p,(c).
Evaluation of [pm (c)lNR The non-relativistic DOS [&,(~)]~a, is evaluated on the basis of eqn (23), by evaluating p(a). The evaluation of p(c) is dependent on the use of eqn (24). It can be seen that the integral in (24) is formally
Fig. 5. Same kind of graphs as in Figs l-4, for the parameters given below: For IA and 18: V,,= -0.0001, b = 258.14, n = 9.68 x 10-r; for 2A and 28: V, = -0.0001, b = 258.14, n = 9.68 x IO-‘. IA and 2A refer to [p,(c)],,, while 19 and 2B correspond to p&).
the same as that for p(q) in (6). Hence, the formal steps in the evaluation of p(e) are the same as those for P(rt)l To evaluate p(e), we have to proceed as follows. We obtain first X and Y using eqns (37) and (38), in the same way as we evaluated D, and D,. The values
Table 1. p,(c) and [p,(c)],, for some low values of 6, in the range of e greater than zero+ d
b&llNR
0.00025 O.COO5 0.0907s 0.001 0.00125 0.0015
4.71440 1.47250 0.7803 1 0.50167 0.35656 0.27028
P&l
4.76076 I.47663 0.78049 0.501 IS 0.35621 0.26987
7 For;11 re;;l;rqin this table, V, = 0.001, n = 9.68 x 10-f, table, results this For in and 0.0075 < (;)
-0.002
0
o.ooa o.oo4 oooa ooce c
aoi
aot2
Fig. 4. Variations of p,(s) and [~~(e)]na with c, for the parameters given below, For iA and IB: V, = -0.001. b = 258.14, R = 9.68 x 10-s; for 2A and 2B: V, = -0.001, b = 258.14, n = 9.68 x 10”. As in Figs 1-3, 1A and 2A correspond to [P&&+, while 1B and 29 correspond to p,(a).
O.oooO25 0.000675 0.0001 0.000125 0.00015 0.000175 0.0002 0.0003 0.0005 t For
this
IPA~LR 29.2842 24.6553 15.9553 11.3291 8.5663 6.7684 5.5201 2.4923 I.3871
Pm(~)
66.6377 37.4643 16.0932 11.3991 8.6056 6.7886 5.5347 2.9957 I .3586
table, V,,=O.OOOl, n = 9.68 x 10-s and Further, for results of this table, 0.0003 < (cc) < 0.0005. b P 258.14.
Relativistic electronic state of disordered systems of X and I’ are used to obtain Xi (p, q) and Z,(p, q) from (26) and (27). Z,(p, r)) and Z,(p. q) are then substituted in (24), and [~,(c&.,a is obtained by carrying out the integration in (24) using Simpson’s 1/3rd rule. The results for [p,(a)],, are shown in Figs l-5, and Tables 1 and 2. Evaluation of pNR and pR The non-relativistic and relativistic DOS of ID free electrons, given by pNR and pR, respectively, are evaluated by eqns (39) and (40). We find that p,, is always greater than pNR, but the difference between pR and pNR is not large enough to be displayed graphically. We have chosen to only show pNR graphically. For comparing conveniently the DOS of disordered systems with the DOS of free electrons, we have shown graphically the c-dependence of pNRin all the Figs l-5. Further, to show effectively the extent of the difference between the pR and pNa, we have computed the results shown in Table 3, besides presentmg graphs of pNRin Figs 1-S. Evaluation of band edges, band gaps and related features For the sake of examining how disorder disturbs band structures, we have computed the widths of the first few allowed and forbidden regions of the energies of a crystalline system typically related to our model. The equations used for this purpose are (43)-(46). The results are shown in Table 4. It is worth noting here that, in all our numerical analyses, we have taken 2m = 1, ti = 1, and c = 1. 7. REVIEW OF RESULTS, DISCUSSION AND CONCLUSIONS Our quantitative investigations of p,,,(c) and [p,(c)lNR cover quite a wide range of relevant parameters, namely n, Vi and b. Consequently, we are able to see pretty clearly, from graphs in Figs 1-5, how these parameters influence the difference be-
Table 3. The relativistic and non-relativistic DOS of free electrons (denoted by the pRand pNR,respectively) for some values of E c 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.15
PNII
5.03292 3.55881 2.90575 2.51646 2.25079 2.05468 1.90226 1.77940 1.67764 1.59155 1.51748 1.45288 1.39588 1.34510 1.29949
PR
5.04046 3.56948 2.91881 2.53153 2.61631 2.07312 1.92218 1.80069 1.70020 1.61532 1.54241 1.47890 1.42295 1.37319 1.32855
153
Table 4. The band widths and band gaps for a periodic system
corresponding to the following parameters: I’. = 0.001. b = 258.14. n = 9.68 x IO-?
Serial No. of band width or band gap 8 9 10 11 12 13 14
Bandgap 6.33 x 5.91 x 5.32 x 5.32 x 3.50 x 2.73 x 1.61 x
10-j 10-5 IO-5 IO-’ 10-J lO-5 lO-5
Band width 9.11 x 9.55 x 1.33 x 1.33 x 1.61 x 1.68 x 2.11 x
10-J 10-J lo-’ 10-4 lo-’ IO-’ 10-4
t’fhe band width and band gaps of this table span roughly the range 6 = 0 to 0.0035.
tween p,(c) and [pm(a)lNR.The principal results for p,(c) and [p,,,(c)]Na, and other related items (such as the DOS of free electrons, and band structures) are summarized below: (I) Looking at the graphs in Figs l-5, we find that both p,(c) and [p,,,(c)INRdecrease with the increase of 6. in almost the same qualitative way as the free electronic DOS (relativistic as well as non-relativistic) decreases with increasing 6. (II) For positive V, (barrier-type potential), c obviously takes only positive values while, for negative V, (well-type potential), c can take negative as well as positive values. It is also obvious that the free electronic DOS is concerned only with positive values of c. A comparison of the DOS of disordered systems with the DOS of free electrons is thus meaningful only for positive values of c. Looking at Figs 1-5, we see that, for very small positive values of t, both the relativistic and non-relativistic DOS of disordered systems tend to merge with the DOS of free electrons under all circumstances. On the other hand, for large positive values of c, both the relativistic and nonrelativistic DOS of disordered systems are smaller than the DOS of free electrons under all circumstances. These results seem to indicate that, for our model of 1D disordered systems, the states at large c are less degenerate than those of free electrons. (III) Looking at the graphs in Figs 4 and 5, we find that, for c 0, we are required to pay attention to the graphs in Figs l-5 for L > 0, and to the results in Tables 1 and 2. The graphs for p,(t) and [pm(c)lNR in Figs l-5 indicate that they start differing from each other for large positive values of c. There is also the same difference between p,,,(f) and [pn(c)lNRfor small positive values of a; this diflbrence is, however, small from a quantitative point of view and is elucidated in Tables 1 and 2. We indicate below some details about the difference between p,(a) and [pm(c)JNRfor
154
C. L. ROY and CHANDAN BAXJ
positive values of c, large as well as small; we Brst consider the case of large L. Looking at Figs l-5, we see that, for large 6, is greater than p,(c) under all circum[Pm(~h stances. This feature is the opposite of what is seen in the case of the relativistic and non-relativistic DOS of free electrons; the results of Table 3 indicate that, for 1D free electrons, the relativistic DOS is larger than the non-relativistic one. It can also be seen from Figs l-5 that the difference between p,(e) and [~,,,(a)]~~ increases with (i) decrease of 1V,,f, (ii) decrease of concentration (n), and (iii) decrease of b. We now discuss some special features regarding the difference between p,(e) and [pm(e)]uR for low values of 6, with particular attention to the values of L around V,. In Ref. 18, we discussed that, for a barriertype potential (positive VO), F becomes equal to its NR counterpart Vi for c = V,. As a result, the difference between p,,,(e) and [p,(e)]uu due to F alone, would vanish for a barrier-type potential for c = I’,,, and this difference would increase for L > I’, and c < V,,. Actually, the formula for p,(c) is involved with other c-dependent terms such as r!’ and r]. The disappearance of the difference between p,(e) and [p,(c)]uu may not consequently occur exactly at c = Va, but at some other values of es around I’,, as illustrated by Tables 1 and 2. For Table 1, Vs = 0.001 and 0.00075 c e,, < 0.001; for Table 2, V, = 0.0001, and 0.0003 < t0 c 0.005. (V) The question as to how far a band structure persists in a disordered system is a very pertinent one.
To examine the question for our present case, we have inserted in Table 4 the NR band widths and band gaps for a periodic system corresponding to the parameters given in the table; the equations used for this purpose are (43) and (44). We find that the relativistic band widths and band gaps, evaluated from (45) and (46), are nearly the same as those in Table 4, with the general features that the relativistic band widths are smaller than the NR band widths and the band gaps are the same in both cases. Table 4 shows that the band gaps are, as expected, larger for a smaller L. At the same time, both p,(e) and [p,(r)],, are very large for small L, and do not vanish for any interval of L. Thus we may say that the presence of the kind of disorder we have treated destroys the band structure altogether, and that the quantitative effect of this disappearance of the band structure is stronger for lower values of e. (VI) Finally, we would like to mention the following two points: (a) by looking at Figs 2 and 5, we can say that the difference between p,(r) and [p,(e)],, is very large when we have the simultaneous presence of low concentration and low 1V,l; (b) there are realistic circumstances under which the difference between [p,,,(e)]uu and p,(e) can be as large as 50%; the value of L = 0.002 for graphs 1A and 1B in Fig. 2, provides one such case. This order
of difference between (p,,,(e)]un and p,(e) is much larger than the difference between relativistic and non-relativistic results for other aspects, say, the relativistic shrinkage of allowed bands [5].
REFERENCES 1. Conklin J. 9.. Jr, Johnson L. E. and Pratt G. 9.. Jr, Phys. Rev. 137. Al292 (1965). 2. Seven P., Phys. Rev. 137, A!717 (1965). 3. Animalu A. 0. E., Phil. Mug. 13, 53 (1968). 4. Loucks T. L., Augmented Plane Wave Merhod, Chap. 4.
Benjamin, New York (1967). 5. Glasser M. L. and Davison S. G., Int. J. quantum Chem. IIIS, 867 (1970). 6. Sen Gupta N. D., Phys. Status Solidi (b) 65,351 (1974). 7. Mladenov I. M., Phys. Status Solidi (6) 145, Kll5 (1988). 8. Dominauez-Adame F.. J. Phvs.: Cond. Matter 1. 109 (!989j.9. Davison S. G. and Steslicka M., J. Phys. C: Solid St. Phys. 2, 1802 (1969). 10. Roy C. L. and Pandey J. S.. Physica 137A, 389 (1986). I!. Fairbaim W. M., Glasser M. L. and Steslicka M., Surf. I
Sci. 36, 462 (1973). 12. Srinivasan U. and Davison S. G., Physica 64, 114 (1973). 13. Roy C. L. and Roy G., Phys. Status Solidi (b) 121, 717 (1984). 14. Roy C. L., Physica 9103, 275 (1981). 15. Roy C. L., Physica 9113, 94 (1982). 16. Roy C. L., Indian J. Phys. 57, 375 (1983). 17. Roy C. L., Phys. Lett. A 118, 32 (1986). 18. Roy C. L., J. Phys. Chem. Solids SO, 1I! (1989). 19. Erdos P. and Hemdon R. C., Adv. Phys. 31,65 (1982). 20. Klauder J. R., Ann. Phys. 14, 43 (1961). 21. Steslicka M. and Davison S. G., Phys. Rev. 91, 1858
(1970). 22. Bhattacharya A. K. and Roy C. L., Indian J. pure appl. Phys. 10, 253 (1972).
APPENDIX As mentioned in Section 6, we used Simpson’s !/3rd rule to evaluate the integrals in eqns (6) and (24). The purpose of this Appendix is to discuss how we handled Simpson’s 1/3rd rule and some other related aspects, so as to achieve high accuracy with convenience. The integrals in (6) and (24) are of the genera! form, I, given by: I= I
_+;/(P)
(A!)
dp.
The basic definition of such an integral is: f(p)
If -co *. . -rr < -r, numbers, then:
(A2)
dp.
c 0 < r, < r2 . . . co, is a sequence of
+ j;,,/(p)
dp + l&)
dp.
(A3)
Each of the integrals on the RHS of (A3) is proper and can be evaluated without any difficulty. We terminate the
Relativistic electronic state of disordered systems
155
Table Al. Value of p,(c) for a few intervals of p and a number of subdivisions Lower limit of p
Upper limit of p
No. of divisions
P,(C )
-0.2 -0.2
+0.2 +0.2
3000 6000
1.33031 1.33030
+0.2
+0.4
3000
4.2 x lo-’
-0.4
+0.2
3000
4.31 x lo-’
evaluation when
)-;"fW+~ l~;;J(~)d+.
(A4)
(AS)
where, 6 is a preassigned small positive number. The termination conditions corresponding to eqns (A4) and (A5) are useful in evaluating an integral such as (A3). The integrals in eqns (6) and (24) decrease rapidly with p and we used the termination criteria (A4) and (AS), considering the interval ofp to be p = -0.2 top = +0.2 for
Remarks Contribution almost the same as for the preceding case Contribution negligible Contribution negligible
this purpose. This interval is’subdivided into 3000 divisions and the relevant integration is carried out with the application of Simpson’s 1/3rd rule. There is no significant contribution to integrals (6) and (24) if we increase the number of divisions beyond 3000, or if we extend the limit of p beyond p = 0.2 and p = + 0.2. To substantiate these points, we show in Table Al some results for p,(s) corresponding to a few intervals ofp and a number of divisions. Table Al shows clearly that, extending the interval of p beyond p = -0.2 to +0.2 and increasing the number of divisions beyond 3000, does not bring any significant change in the value of p,(c) obtained for the first case, namely the interval p = -0.2 to +0.2 and 3000 divisions.
0022~x93/90 s3.00 + 0.00 Pergamon Press plc
A STUDY OF THE RELATIVISTIC ELECTRONIC STATE OF DISORDERED SYSTEMS C. L. ROY and CIUNDAN BASU Department of Physics, Indian Institute of Technology, Kharagpur 721 30.2, India (Received 21 June 1989; accepted in revised form 15 September 1989)
Abstract-We have carried out a quantitative study of the relativistic density of states (DOS) of a one-dimensional system, using analytical results of a related treatment previously reported by Roy (J. Phys. Chem. So&h SO, 1II,-1989).-We have examined various features of the qua&tat&e resultsof thk relativistic DOS of our model of a ID disordered svstem. in liaht of(i) non-relativistic CNR) DOS of the same disordered system, (ii) relativistic and non-relkvistk DC% of frk electrons, and (iii) klativistic and non-relativistic band structure of 1D periodic systems relevant to our disordered systems. Among many other things, our investigation brings out clearly the circumstances under which the relativistic effects on the DOS of disordered systems can become substantially larger; in this regard, it is found that the relativistic effects on the DOS of disordered systems are very large in the simultaneous presence of low concentration of scatterers and low values of height (depth) of potential barriers (wells) which represent the potentials due to scatterers in our model of disordered systems. Keywords: Dirac equation, density of states, band structures, propagator, Simpson’s 1/3rd rule.
1. INTRODUCTION Theoretical treatments of the electronic properties of condensed matter consisting of heavy atoms need to be relativistic. During the last three decades or so, relativistic investigation of various electronic properties of condensed matter evoked continuous attention. So far, relativistic studies of condensed matter have been concerned with (i) bulk states of threedimensional (3D) crystalline systems [l-4], (ii) bulk states of one-dimensional (I D) crystalline systems [Ml, (iii) surface states [P-12] and interface states [13], and (iv) electronic states of disordered systems [ 14-181. Any rigorous relativistic study of electrons in condensed matter would require exact use of the (time-independent) Dirac equation with adequate account of the relevant potentials. Relativistic studies of electronic properties of 3D condensed matter, with exact use of the 3D (time-independent) Dirac equation, involve enormous complications. As a result, relativistic studies of electrons in 3D condensed matter are often required to take recourse to approximate methods [4]. On the other hand, for 1D condensed matter exact use of the 1D Dirac equation is possible for properties such as the bulk states of crystalline systems and surface states [5-131. However, for the relativistic treatment of the density of states (DOS) of disordered systems, the use of the exact Dirac equation creates serious difficulties even when the system is one-dimensional and it becomes necessary to make use of the approximate methods. Compared with the relativistic studies of items (i)-(iii) mentioned earlier, those [14-181 of (iv) are quite recent. These (relativistic) studies dealt with
some 1D models of disordered systems. Since about 1960, several workers have been using 1D models [ 191 very fruitfully for the purpose of non-relativistic investigations of the electronic properties of disordered systems. We hope that relativistic results about the electronic properties of ID disordered systems will form a basis for relativistic studies of more realistic disordered systems. Of the works in Refs 11118, the ones under Refs 17 and 18 are concerned with the relativistic DOS of a particular model of ID disordered systems. The treatments of [ 171 and [18] are based essentially on relativistic generalizations [21] of two non-relativistic approaches due to Klauder [20]. The purpose behind both [17] and [18] is to d&e formulae for the relativistic DOS by evaluating the relevant dressed electron propagator. In [17], the dressed electron propagator is evaluated without propagator modification, while in [ 181,propagator modification is taken into account. Thus, the treatment of [18] is more realistic than that in [17]. In order to see the impact of relativistic corrections, it is necessary to carry out quantitative investigations based on relevant analytical results. While quantitative investigations of relativistic effects have often been carried out on items (i)-(iii) mentioned above, no quantitative study has yet been made about the relativistic electronic states in disordered systems. The purpose of this paper is to report such a quantitative study based on the results of the analytical treatment in (181. The treatment in [18] is concerned with the derivation of the relativistic DOS of a 1D disordered system in a fairly realistic manner, and the quantitative results now being reported would, conse-
147
118
C. L. ROY and CHASDAN Btic
quently, throw much light on the nature of relativistic effects on the electronic states of disordered systems. The formulae and related issues concerned with our quantitative study of the relativistic DOS, are presented in Section 2. For a better comprehension of the relativistic effects, we compared our (quantitative) results with allied results of several other cases such as (I) the non-relativistic DOS of the same ID disordered systems as studied relativistically, (II) the non-relativistic and relativistic DOS of free electrons, and (III) the relativistic and non-relativistic band structures of periodic systems relevant to our model of disordered systems. The comparisons of (I), (II) and (III) are presented in Sections 3, 4, and 5, respectively. The various numerical analyses that we have carried out, are discussed in Section 6, and the results are reviewed critically in Section 7. Finally, some features of our numerical analyses are elucidated in the Appendix. 2. FORMULA
AND
FOR THE
RELATIVISTIC
The relativistic (energy) DOS, p,(a), reported in [IS] was obtained by summing up the dressed electron propagator relevant to (3), and potentials (1) and (2), taking propagator modification into account. Explicitly, p,,,(c) appears as:
Pm(~) = rt’P(Vh where
(5)
x[nF -
L(~~tl)l~&(~~rl)d~ [G&P, tt)l Z, (p, ~11'+
G,(P, tt) & (PYr1)1” (6)
The symbols in (6) have the meanings:
DOS
SOME RELATED ISSUES
The model of disordered systems for our present study is, as mentioned before, the same as that of Ref. 18. It consists of an electron in the presence of “N” fixed atoms with random locations X, along the x-axis in a chain of length “L”. The atomic potentials, V(x - x,), are assumed to be rectangular barriers or wells of width 2b. The total potential, V,(x), appears as: V,(x) = 1 V(x -x,x
F = V;A V;=2bV,
V,, for (xi-b)
(7)
n = concentration of scatters (atoms) along the chain .L
= limit E ‘V-X 0 L
-I 1
L-Z
(1)
q+ -nF-&
G,(p,v)=nF V(x -xi)=
(4)
(8)
<(x,+6) * (2)
q+=q+iS,
6+0+
= 0, for x > (xi + b) or x -=z(xi - b) I The treatment in [18] of the relativistic DOS of the disordered system corresponding to potentials (I) and (2), makes use of the following approximately relativistic equation (211for a constant potential “V”: d2tj 2m s++tl-
&(P*V)=
- @?7-(P. 4b* --
mnF2 b
(
62
c+2mc’
c=(ER-mc’),
s,l
-ry+D: 1 (9) p’-T
(P2
W)$ -0,
where tl=
rl) + V-P,
1
> W= VA
s=Zm(q-nF-D,)
1 ER = relativistic energy eigenvalue c = velocity of light.
Q(P,v)=~
(13)
149
Relativistic electronic state of disordered systems
where
T, = exp( - 2gb)[sin 2cf - p)b] x U-(U-P)*
p(c) = NR limit of p(q).
-g2)
Explicitly, p(c) is given by:
-~~f-~l~21+~~~/2+p2-~pf-~*~ x [i
Q,-[1
-exp(-2gb)cos2Cf-pjb]
(14)
-exp(-2gb)cos2Cf-p)b] x ~f~cf-P~z-~2~--2cf-Pl~21
x
[G&J, ~11~ x2 (P. 0 dp [~~v~-(G~(P,~))Z,(P,~)I*+[G~(P,~)~?(P.E)I~’
-t?(3f2+PZ-w--g2)
(24)
xexp(-2gb)*sinZ(f-p)b
(1%
D= U---P)~+~~V*+~~I
(16)
f = [q; + D~“‘cos(~/~)
(17)
The meanings of the various symbols in eqn (24) are given below. G,,(p, 6) = NR limit of G,(p, V)
1
c+ -nVi-P
=nV;
g = (4: + DfJ”2 sin(4/2)
(18)
tan 4 = VU&)
2
2m
1 -I
(25)
Z, (p, 6) = NR limit of X, (p, 7) 2
q=nF+$D,
(19) =
D, = -$lT(qo,
tt) + T(-qo,
rt)l
(20)
D 2=~[Q(4o’“)+Q(-“‘sll+~.
-~[S(p,r)+S(-p,c)] --
mn( Vi)2 b
p2-a (p2 - a)2 + y2
1
(26)
Z, (p, 6) = NR limit of Z, (p, q)
(21) p,,,(c) is the DOS with respect to energy “6”. The entity “6” is the relativistic analogue of NR energy of the electron. Thus p,(a) can be conveniently used for a comparative study of relativistic and non-relativistic DOS, with inclusion of ail salient features of the relativistic DOS.
+y[J(p,c)+J(-p,r)] +
mn(W2 b
Y (P2 -a)2+
Y2
1
(27)
a = NR limit of T 3. NON-RELATIVISTIC DOS OF OUR
=2m(6
MODEL AND RELATED ITEMS
The non-relativistic DOS of the present model of disordered systems, was reported previously by Bhattacharya and Roy [22]. The formula for the non-relativistic DOS follows as the limit of p,(e) when “c” is allowed to become infinitely large. We thus have: Mc)lNR = non-relativistic
DOS of our model
= NR limit of [~‘p(rl)].
(22)
Taking the NR limit on the RHS of eqn (22) in the way mentioned above, we have
bA~)INR = P(r). PCSSI,2--E
(23)
S-NR
-nVh-X)
(28)
limit of T(p,q)=Z
S,
(2%
J(p, c) = NR limit of Q(p, IJ) = z
(30)
S, = NR limit of T,(p, 9) =exp(-2&)[sin2(r
-p)b][r((r
-~)~-f.i~)
- 2(r - p)d7 + d(3r2 + p2 - 4pr - Cf2)
x [l -exp(-2rb)cos2(r
-p)b]
(31)
I50
C.
L. ROYand
of free electrons; its relation to momentum follows:
J, = NR limit of Q, =[I -exp(-2db)cosZ(r
- 2(r -PM21
-d(3rz+p2-4pr
-d*)
x exp( - 2db). sin 2(r - p)b
for (39) for (40),
5. EQUATIONS
= [(r - p)* + d212[r2+ d’] [q: + Y2]li4.cos8/2
(33) (34)
d = NR limit of g = [q: + Y2]“4. sin O/2 (35) tan 0 = Y 4; c=NRhmitofq=nV;+g+X
(36)
X= NR limit of D,
~(6 + 2mc*) =pV.
FOR BAND STRUCTURES
KP equation for
V, > 0 and
(37) cos ~(a + 26) = cosh(2k,b). k:--kf + [ 2k,k,
mn( V&)2
(38)
+by.
(42)
As mentioned in the Introduction, we require for our present study some aspects of relativistic as well as non-relativistic band structures in the crystalline systems related to our model of disordered systems. The aspects of band structures for our purpose would be obtained on the basis of relativistic and nonrelativistic Kronig-Penney equations related to crystalline systems of periodicity (a + 26) where 2b is the width of potential barriers (wells) and a is the zero potential region between potential barriers (wells). The KP equations to be used are as given below (in all cases, p is the wave vector).
Non-relativistic 6 < v,
-~[S(q,,f)+S(-qO.L)l
6 = -&
(32)
Z= NR limit of D
=
p is as
-p)b]
x ]r((r -PY -8)
r = NR limit off-
CHANDAN Btiu
cos(k, a)
1
. sinh&b)
. sin(k, a),
(43)
where
4. DOS OF RELATIVISTIC AND NON-RELATIVISTIC FREE ELECTRONS
The relativistic as well as non-relativistic DOS of 1D free electrons can be found by noting that, for both cases, (i) the elementary phase-cell is “II” and (ii) there is two-fold momentum degeneracy. With the use of these features and the dispersion relations relevant to non-relativistic and relativistic free electrons, we can derive the following results:
Non-relativistic 6 > v,
pR =
(k: + S:) sin(hb)
- -.
(44)
where
1 I’*
(3%
(a + mc’)
= hc * [L(L +
sin(k, a),
2k, 81
DOS per unit length for 1D relativistic free electrons 2
V, > 0 and
cos(p(n + 26)) = cos(2/?,b)cos(k,a)
pNR= DOS per unit length of ID NR free electrons 1 2 =5; [ z
KP equation for
2mcZ)]“* .
Relativistic KP equation for V, > 0 and 6 < V, (40)
In eqns (39) and (40), L stands for the kinetic energy
cos(p(u + 26)) = cos(k,a) cosh@k,,b) + P, sin(k,a) sinh(2k,,b),
(45)
151
Relativistic electronic state of disordered systems
where k;==
C(C-t 2mc2) h2c2
k:l=(v,-s)(~
P‘Relativistic
- v,+~c’) h2c2
k;,E2 - (V, - c)2k; 2a(V,,-a)k,,k,
*
KP equation for V,, B 0 and L > V,
cos(p(a + 26)) = cos(k,a) cos(tj?,,b) - P2 sin(k,a) sin(2&6),
(46)
where B2 _ (C - V&6 - v, + 2mc2) II h2c2
Equations (45) and (46) are taken from Ref. 5, while eqns (43) and (44) are available in standard books.
6. NUMERICAL
ANALYSES
Our numerical analyses are concerned with (i) evaluation of the relativistic DOS, p,(a), on the basis of eqn (4) and related formulae, (ii) evaluation of the non-relativistic DOS [p,(c)],,, on the basis of eqn (23) and other related entities, (iii) evaluation of the non-relativistic and relativistic DOS of free electrons on the basis of eqns (39) and (40) respectively, and (iv) widths of some allowed and forbidden regions of energies on the basis of the KP eqns (43x46). We discuss below the salient features concerned with these numerical analyses. Evaluation of p,,,(c) To start with evaluation of p,(c), we are required to find D, and D2 of eqns (20) and (21). These equations constitute a set of coupled equations in terms of D, and D2; it is not possible to solve them analytically, and we consequently have taken recourse to a (numerical) iteration method to solve them. For our purpose, we choose the values of n, Vi, b . . . ) etc. and vary the values of 6; the value of qO corresponding to a particular c is obtained from eqn (19). Inserting D, and D, in eqns (9) and (10) we next obtain Z, (p, q) and Z,(p, q). Substituting Z, (p, q) and Z2(p, q) in (6). we then obtain p(q) by carrying out a numerical integration. We have used Simpson’s 1/3rd rule in regard to numerical integration in eqn (6). Some particulars about numerical integration in
Fig. 1. Variations of p,(c), [~,&)]~a and pSR, with e:. Graphs IA and 2A correspond to [p,(r)],,, while graphs 19 and 29 correspond to p,(c). The values of the parameters are as follows. For IA and 19: V, = 0.001. b = 258.14, n = 9.68 x IO-‘; for 2A and 29: V, =O.OOl, b = 258.14, n = 9.68 x 1O-4.
(6) are discussed in the Appendix. Making use of p(q) and eqn (4), we finally obtain p,,,(c). The results for p,,,(e) are shown in Figs 1-5, and Tables 1 and 2.
r
o.ool
I a002
I
111
I
aoo4 4006’ 4008 sot
I o.ol2
1 0.014
t
Fig. 2. Variations of [~_(~)],,a and p,(e) parameters indicated klow. For 1A and b = 258.14, n = 9.68 x 10mJ; for 2A and b = 258.14, n -9.68 x IO-‘. As in Fig. while 19 and correspond to [~&&e to P,(f ).
with c, for the 19: V, = 0.0001, 29: V, = 0.0001, 1, IA and 2A 29 correspond
C. L. ROY and CHANDANBAW
0,ol -
soot
I
I
I
I
I
I
1
coo2 a004 0006 oooa a01 o.o2 a014 c Fig. 3. Same kind of graphs as in Figs 1 and 2. The parameters now are as follows. For IA and 1B: Y = 0.001, b = 25.8, n = 9.68 x IO-‘. IA corresponds to [p,(a)],a and IB to p,(c).
Evaluation of [pm (c)lNR The non-relativistic DOS [&,(~)]~a, is evaluated on the basis of eqn (23), by evaluating p(a). The evaluation of p(c) is dependent on the use of eqn (24). It can be seen that the integral in (24) is formally
Fig. 5. Same kind of graphs as in Figs l-4, for the parameters given below: For IA and 18: V,,= -0.0001, b = 258.14, n = 9.68 x 10-r; for 2A and 28: V, = -0.0001, b = 258.14, n = 9.68 x IO-‘. IA and 2A refer to [p,(c)],,, while 19 and 2B correspond to p&).
the same as that for p(q) in (6). Hence, the formal steps in the evaluation of p(e) are the same as those for P(rt)l To evaluate p(e), we have to proceed as follows. We obtain first X and Y using eqns (37) and (38), in the same way as we evaluated D, and D,. The values
Table 1. p,(c) and [p,(c)],, for some low values of 6, in the range of e greater than zero+ d
b&llNR
0.00025 O.COO5 0.0907s 0.001 0.00125 0.0015
4.71440 1.47250 0.7803 1 0.50167 0.35656 0.27028
P&l
4.76076 I.47663 0.78049 0.501 IS 0.35621 0.26987
7 For;11 re;;l;rqin this table, V, = 0.001, n = 9.68 x 10-f, table, results this For in and 0.0075 < (;)
-0.002
0
o.ooa o.oo4 oooa ooce c
aoi
aot2
Fig. 4. Variations of p,(s) and [~~(e)]na with c, for the parameters given below, For iA and IB: V, = -0.001. b = 258.14, R = 9.68 x 10-s; for 2A and 2B: V, = -0.001, b = 258.14, n = 9.68 x 10”. As in Figs 1-3, 1A and 2A correspond to [P&&+, while 1B and 29 correspond to p,(a).
O.oooO25 0.000675 0.0001 0.000125 0.00015 0.000175 0.0002 0.0003 0.0005 t For
this
IPA~LR 29.2842 24.6553 15.9553 11.3291 8.5663 6.7684 5.5201 2.4923 I.3871
Pm(~)
66.6377 37.4643 16.0932 11.3991 8.6056 6.7886 5.5347 2.9957 I .3586
table, V,,=O.OOOl, n = 9.68 x 10-s and Further, for results of this table, 0.0003 < (cc) < 0.0005. b P 258.14.
Relativistic electronic state of disordered systems of X and I’ are used to obtain Xi (p, q) and Z,(p, q) from (26) and (27). Z,(p, r)) and Z,(p. q) are then substituted in (24), and [~,(c&.,a is obtained by carrying out the integration in (24) using Simpson’s 1/3rd rule. The results for [p,(a)],, are shown in Figs l-5, and Tables 1 and 2. Evaluation of pNR and pR The non-relativistic and relativistic DOS of ID free electrons, given by pNR and pR, respectively, are evaluated by eqns (39) and (40). We find that p,, is always greater than pNR, but the difference between pR and pNR is not large enough to be displayed graphically. We have chosen to only show pNR graphically. For comparing conveniently the DOS of disordered systems with the DOS of free electrons, we have shown graphically the c-dependence of pNRin all the Figs l-5. Further, to show effectively the extent of the difference between the pR and pNa, we have computed the results shown in Table 3, besides presentmg graphs of pNRin Figs 1-S. Evaluation of band edges, band gaps and related features For the sake of examining how disorder disturbs band structures, we have computed the widths of the first few allowed and forbidden regions of the energies of a crystalline system typically related to our model. The equations used for this purpose are (43)-(46). The results are shown in Table 4. It is worth noting here that, in all our numerical analyses, we have taken 2m = 1, ti = 1, and c = 1. 7. REVIEW OF RESULTS, DISCUSSION AND CONCLUSIONS Our quantitative investigations of p,,,(c) and [p,(c)lNR cover quite a wide range of relevant parameters, namely n, Vi and b. Consequently, we are able to see pretty clearly, from graphs in Figs 1-5, how these parameters influence the difference be-
Table 3. The relativistic and non-relativistic DOS of free electrons (denoted by the pRand pNR,respectively) for some values of E c 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.15
PNII
5.03292 3.55881 2.90575 2.51646 2.25079 2.05468 1.90226 1.77940 1.67764 1.59155 1.51748 1.45288 1.39588 1.34510 1.29949
PR
5.04046 3.56948 2.91881 2.53153 2.61631 2.07312 1.92218 1.80069 1.70020 1.61532 1.54241 1.47890 1.42295 1.37319 1.32855
153
Table 4. The band widths and band gaps for a periodic system
corresponding to the following parameters: I’. = 0.001. b = 258.14. n = 9.68 x IO-?
Serial No. of band width or band gap 8 9 10 11 12 13 14
Bandgap 6.33 x 5.91 x 5.32 x 5.32 x 3.50 x 2.73 x 1.61 x
10-j 10-5 IO-5 IO-’ 10-J lO-5 lO-5
Band width 9.11 x 9.55 x 1.33 x 1.33 x 1.61 x 1.68 x 2.11 x
10-J 10-J lo-’ 10-4 lo-’ IO-’ 10-4
t’fhe band width and band gaps of this table span roughly the range 6 = 0 to 0.0035.
tween p,(c) and [pm(a)lNR.The principal results for p,(c) and [p,,,(c)]Na, and other related items (such as the DOS of free electrons, and band structures) are summarized below: (I) Looking at the graphs in Figs l-5, we find that both p,(c) and [p,,,(c)INRdecrease with the increase of 6. in almost the same qualitative way as the free electronic DOS (relativistic as well as non-relativistic) decreases with increasing 6. (II) For positive V, (barrier-type potential), c obviously takes only positive values while, for negative V, (well-type potential), c can take negative as well as positive values. It is also obvious that the free electronic DOS is concerned only with positive values of c. A comparison of the DOS of disordered systems with the DOS of free electrons is thus meaningful only for positive values of c. Looking at Figs 1-5, we see that, for very small positive values of t, both the relativistic and non-relativistic DOS of disordered systems tend to merge with the DOS of free electrons under all circumstances. On the other hand, for large positive values of c, both the relativistic and nonrelativistic DOS of disordered systems are smaller than the DOS of free electrons under all circumstances. These results seem to indicate that, for our model of 1D disordered systems, the states at large c are less degenerate than those of free electrons. (III) Looking at the graphs in Figs 4 and 5, we find that, for c
154
C. L. ROY and CHANDAN BAXJ
positive values of c, large as well as small; we Brst consider the case of large L. Looking at Figs l-5, we see that, for large 6, is greater than p,(c) under all circum[Pm(~h stances. This feature is the opposite of what is seen in the case of the relativistic and non-relativistic DOS of free electrons; the results of Table 3 indicate that, for 1D free electrons, the relativistic DOS is larger than the non-relativistic one. It can also be seen from Figs l-5 that the difference between p,(e) and [~,,,(a)]~~ increases with (i) decrease of 1V,,f, (ii) decrease of concentration (n), and (iii) decrease of b. We now discuss some special features regarding the difference between p,(e) and [pm(e)]uR for low values of 6, with particular attention to the values of L around V,. In Ref. 18, we discussed that, for a barriertype potential (positive VO), F becomes equal to its NR counterpart Vi for c = V,. As a result, the difference between p,,,(e) and [p,(e)]uu due to F alone, would vanish for a barrier-type potential for c = I’,,, and this difference would increase for L > I’, and c < V,,. Actually, the formula for p,(c) is involved with other c-dependent terms such as r!’ and r]. The disappearance of the difference between p,(e) and [p,(c)]uu may not consequently occur exactly at c = Va, but at some other values of es around I’,, as illustrated by Tables 1 and 2. For Table 1, Vs = 0.001 and 0.00075 c e,, < 0.001; for Table 2, V, = 0.0001, and 0.0003 < t0 c 0.005. (V) The question as to how far a band structure persists in a disordered system is a very pertinent one.
To examine the question for our present case, we have inserted in Table 4 the NR band widths and band gaps for a periodic system corresponding to the parameters given in the table; the equations used for this purpose are (43) and (44). We find that the relativistic band widths and band gaps, evaluated from (45) and (46), are nearly the same as those in Table 4, with the general features that the relativistic band widths are smaller than the NR band widths and the band gaps are the same in both cases. Table 4 shows that the band gaps are, as expected, larger for a smaller L. At the same time, both p,(e) and [p,(r)],, are very large for small L, and do not vanish for any interval of L. Thus we may say that the presence of the kind of disorder we have treated destroys the band structure altogether, and that the quantitative effect of this disappearance of the band structure is stronger for lower values of e. (VI) Finally, we would like to mention the following two points: (a) by looking at Figs 2 and 5, we can say that the difference between p,(r) and [p,(e)],, is very large when we have the simultaneous presence of low concentration and low 1V,l; (b) there are realistic circumstances under which the difference between [p,,,(e)]uu and p,(e) can be as large as 50%; the value of L = 0.002 for graphs 1A and 1B in Fig. 2, provides one such case. This order
of difference between (p,,,(e)]un and p,(e) is much larger than the difference between relativistic and non-relativistic results for other aspects, say, the relativistic shrinkage of allowed bands [5].
REFERENCES 1. Conklin J. 9.. Jr, Johnson L. E. and Pratt G. 9.. Jr, Phys. Rev. 137. Al292 (1965). 2. Seven P., Phys. Rev. 137, A!717 (1965). 3. Animalu A. 0. E., Phil. Mug. 13, 53 (1968). 4. Loucks T. L., Augmented Plane Wave Merhod, Chap. 4.
Benjamin, New York (1967). 5. Glasser M. L. and Davison S. G., Int. J. quantum Chem. IIIS, 867 (1970). 6. Sen Gupta N. D., Phys. Status Solidi (b) 65,351 (1974). 7. Mladenov I. M., Phys. Status Solidi (6) 145, Kll5 (1988). 8. Dominauez-Adame F.. J. Phvs.: Cond. Matter 1. 109 (!989j.9. Davison S. G. and Steslicka M., J. Phys. C: Solid St. Phys. 2, 1802 (1969). 10. Roy C. L. and Pandey J. S.. Physica 137A, 389 (1986). I!. Fairbaim W. M., Glasser M. L. and Steslicka M., Surf. I
Sci. 36, 462 (1973). 12. Srinivasan U. and Davison S. G., Physica 64, 114 (1973). 13. Roy C. L. and Roy G., Phys. Status Solidi (b) 121, 717 (1984). 14. Roy C. L., Physica 9103, 275 (1981). 15. Roy C. L., Physica 9113, 94 (1982). 16. Roy C. L., Indian J. Phys. 57, 375 (1983). 17. Roy C. L., Phys. Lett. A 118, 32 (1986). 18. Roy C. L., J. Phys. Chem. Solids SO, 1I! (1989). 19. Erdos P. and Hemdon R. C., Adv. Phys. 31,65 (1982). 20. Klauder J. R., Ann. Phys. 14, 43 (1961). 21. Steslicka M. and Davison S. G., Phys. Rev. 91, 1858
(1970). 22. Bhattacharya A. K. and Roy C. L., Indian J. pure appl. Phys. 10, 253 (1972).
APPENDIX As mentioned in Section 6, we used Simpson’s !/3rd rule to evaluate the integrals in eqns (6) and (24). The purpose of this Appendix is to discuss how we handled Simpson’s 1/3rd rule and some other related aspects, so as to achieve high accuracy with convenience. The integrals in (6) and (24) are of the genera! form, I, given by: I= I
_+;/(P)
(A!)
dp.
The basic definition of such an integral is: f(p)
If -co *. . -rr < -r, numbers, then:
(A2)
dp.
c 0 < r, < r2 . . . co, is a sequence of
+ j;,,/(p)
dp + l&)
dp.
(A3)
Each of the integrals on the RHS of (A3) is proper and can be evaluated without any difficulty. We terminate the
Relativistic electronic state of disordered systems
155
Table Al. Value of p,(c) for a few intervals of p and a number of subdivisions Lower limit of p
Upper limit of p
No. of divisions
P,(C )
-0.2 -0.2
+0.2 +0.2
3000 6000
1.33031 1.33030
+0.2
+0.4
3000
4.2 x lo-’
-0.4
+0.2
3000
4.31 x lo-’
evaluation when
)-;"fW+~ l~;;J(~)d+.
(A4)
(AS)
where, 6 is a preassigned small positive number. The termination conditions corresponding to eqns (A4) and (A5) are useful in evaluating an integral such as (A3). The integrals in eqns (6) and (24) decrease rapidly with p and we used the termination criteria (A4) and (AS), considering the interval ofp to be p = -0.2 top = +0.2 for
Remarks Contribution almost the same as for the preceding case Contribution negligible Contribution negligible
this purpose. This interval is’subdivided into 3000 divisions and the relevant integration is carried out with the application of Simpson’s 1/3rd rule. There is no significant contribution to integrals (6) and (24) if we increase the number of divisions beyond 3000, or if we extend the limit of p beyond p = 0.2 and p = + 0.2. To substantiate these points, we show in Table Al some results for p,(s) corresponding to a few intervals ofp and a number of divisions. Table Al shows clearly that, extending the interval of p beyond p = -0.2 to +0.2 and increasing the number of divisions beyond 3000, does not bring any significant change in the value of p,(c) obtained for the first case, namely the interval p = -0.2 to +0.2 and 3000 divisions.