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COMPUTATION OF ϵ2-SPECTRA OF AMORPHOUS SEMICONDUCTORS
B. Kramer, K. Maschke, P. Thomas
To cite this version:
B. Kramer, K. Maschke, P. Thomas. COMPUTATION OF ϵ2-SPECTRA OF AMOR- PHOUS SEMICONDUCTORS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-157-C3-165.
�10.1051/jphyscol:1972323�. �jpa-00215057�
JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-157
COMPUTATION OF &,-SPECTRA OF AMORPHOUS SEMICONDUCTORS
B. KRAMER, K. MASCHKE and P. THOMAS Institut ftir ~heoretische Physik (11) der Universitiit Marburg
R6sum6. - Les spectres € 2 des semiconducteurs amorphes ne peuvent pas &re simplement expliquks par le lissage des structures aigues des spectres cristallins. On observe, en plus du lissage, des changements selectifs lorsque l'on passe du cas cristallin au cas amorphe. Pour comprendre ces effets, on a trouve deux mkthodes pour le calcul des spectres E Z des semiconducteurs amorphes.
Elles se basent sur le calcul des structures de bandes, de la densite des Btats et des spectres ~2 du cristal parfait correspondant. La premikre methode de la (( structure de bandes complexe >) comprend une fonction approximative de la distribution des atomes, qui tient compte de I'ordre a courte port&. Les situations des pBles de la fonction de Green moyenne sur les differentes configurations sont dkterminees par une equation skculaire generalide a pseudo-potentiel. La deuxikme methode est le modkle de la cr transition non directe >>, dans laquelle on tient kgalement compte d'elCments de matrice dkpendant de l'knergie. Les resultats pour Se, Ge, Si et les composb 111-V sont presentes.
On montre le rapport entre les deux mkthodes.
Abstract. - The 82-spectra of amorphous semiconductors cannot be explained by only smearing out the sharp structures of the crystalline spectra. In addition to the smearing out selective changes are observed when going from the crystalline to the amorphous case. To understand these effects two methods have been developed to compute €2-spectra of amorphous semiconductors. They are based on the calculation of the band-structure, density of states, and €2-spectrum of the cor- responding perfect crystal. The first method of (r Complex-Band-Structure >> involves an approxi- mative atomic correlation function which contains the short range order. The locations of poles of the configurational averaged Green's function are determined by a generalized pseudopotential secular-equation. The second method is the <( Non-Direct-Transition D model extended to include energy dependent matrix elements. Results for amorphous Se, Ge, Si and 111-V-compounds are presented. The connection of both methods is shown.
1. Introduction. - 1 . l . EXPERIMENTAL OBSERVA-
TIONS. - A number of &,-spectra of amorphous semi- conductors have been determined experimentally in the last few years. In contrast t o the spectra of the crystalline modifications they are characterized by a lack of sharp structure due t o the scattering induced by disorder, which smoothes out the Van Hove singularities. But besides this there are selective changes in E, when going from the crystalline t o the amor- phous modification. Figure 1 shows the experimental spectrum of Se and a typical IV-IV- or 111-V-spectrum.
(References for the experimental &,-spectra reported in this work are : Se, cryst. : Stuke [I], [2] ; Se, am. : Leiga [3] ; Ge, cryst. : tau^ [4] ; Ge, am. : Donovan et al. [5] ; Si, cryst. : Phillips [6] ; Si, am. : Beaglehole et al. [7] ; Gap, GaAs, GaSb, InP, InAs, InSb, am.
and cryst. : Zimmerer [8]). I n amorphous Se the first large structure A is more reduced than the second structure B with respect t o the isotropic averaged crystalline spectrum. For amorphous IV-IV- and 111-V-semiconductors one realizes the vanishing of the E,- and the E;-peaks of the crystalline spectra, the El-peak, however, seems to be enlarged and gives the maximum of the amorphous spectra. Since the short range order of amorphous semiconductors is very similar t o the crystalline short range order we hoped to explain these features by a theory which explicitely
FIG. 1. -Selective change of 82. Solid line : crystalline 82, dashed line : amorphous € 2 . A, B, El, E2, Ei see text.
takes into account the short range order. Two methods have been developed which start from calculations of crystalline band structures, densities of states, and
&,-spectra.
1.2. BASIC FORMULAS. - An observable Q of a disordered system of atoms which is sufficient homo- genious macroscopically, can be calculated as the configurational average
< Q > = Q ( R l . . . R N ) P ( R 1 . . . R N ) n d 3 ~ i (1)
i
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972323
C3-158 B. KRAMER, K. MASCHKE A N D P. THOMAS
where P(R, ... RN) is the probability density of finding atoms at R, ... RN.
The density of one-electron states n(E) and the imaginary part c2 of the dielectric function c (in one- electron approximation) can be written in terms of Green's functions :
The Green's functions are matrix elements of the operators G(E) = (E + ic - H, - V ) - I . P is the momentum operator. The Green's function depends on the configuration because of the potential
V = C vi(r - Ri) .
i
The energy integration in eq. (3) may be easily carried out if the Green's functions are written in spectral representation. One arrives at the well known Golden Rule :
const
E ~ ( O ) = -- Z I P i f I26(Ei + co - E f ) .
0' if
The name << non-direct-transition >> method is somewhat misleading. Assuming that the amorphous densities of states do not differ from the crystalline ones which is reasonable if the short range order is preserved, no,, may approximately be written as the convolution of the individual crystalline densities of states. Then the transitions described in this way might be indicated as arrows in the crystalline band structure connecting valence- and conduction-band energies with different k-vectors. Only in this sense the name NDT is justified. In eq. (6) where no quantum number k is involved there is no sense in speaking of direct or non-direct transitions any longer.
How have we to handle the matrix elements ? For Ge Herman et al. 1121 and Donovan et al. [13]
assumed them to be energy independent, which leads to c2 f f noPt(w). But even using amorphous densities of states taken from photoemission measure- ments there is no good agreement with the experi- mental spectra. Especially in the high energy region they decrease too slowly [13]. Therefore we tried to get some insight into the energy dependence of the amorphous average transition probabilities I M(w) (A,
Maschke et al. 1141. The crystalline I M(w) 1' were
taken from Sandrock's [15] pseudopotential calcula- tions for crystalline Se and from our pseudopotential calculations on crystalline Ge.
If we write I M(o) 1' for the arithmetic mean of all w2 cz(@),,yst
1 M(o) (&st =
transition probabilities belonging to a given photon n j(m> (7)
energy w, eq. (4) becomes
They are shown in figure 2 (solid line).
const
&2(0) = -- I M(m) l2 no,,(o>
w2
where no,,(o) is the optical density of states which in the crystalline case is equal to the joint density of states nj(o).
We start from eq. (5) and review in section 2 the non- direct-transition method (NDT) which first was used by Tauq [9] to interprete the &,-spectra of amorphous semiconductors. In section 3 we present the complex- band-structure method (CBS) due to Kramer [lo], [l 1) which uses eq. (3) as a starting point and approxi- mately takes into account the short range order.
In section 4 we see how the simpler NDT-method may be justified by the CBS-method.
2. The NDT-method. - 2.1. THE MATRIX ELE-
MENTS. - The simplest way to consider the lack of long range order is to assume that there is no more k-selection rule involved in optical dipole transitions.
It is always possible to reach from a given initial state a final state if there is one. Then the optical density of states may be written as the convolution of the amorphous valence- and conduction-band density of states.
FIG. 2. - Left : Average crystalline (solid line) and amorphous (dashed line) transition probabilities of Se. Right : Average crystalline (solid line) and amorphous transition probabilities of Ge. The amorphous curves are obtained by means of convo- luted crystalline (dotted line) and amorphous (dashed line)
densities of states.
The dashed line gives the energy dependence of the amorphous transition probabilities, which are given by
COMPUTATION OF e2-SPECTRA OF AMORPHOUS SEMICONDUCTORS They were obtained from experimental amorphous
&,-curves and convoluted crystalline densities of states.
(At this time the density of states of amorphous Se was still unknown). Except for the large peak at about 4 eV the energy dependence of both curves is similar.
(Note that the zero is suppressed.) For Se this peak may be explained by umklapp enhanced transitions which are localized at top and bottom of the hexagonal Brillouin zone. There are only little contributions to
E, at this energy from other parts of the Brillouin zone.
The umklapp peak obviously vanishes when going from the crystalline to the amorphous phase.
For Ge things are a little more complicated (Fig. 2, right). The I M(o) obtained from the experimental
E, and calculated crystalline densities of states show no good agreement with the 1 M(co) I:,,, in the
overall energy dependence. The umklapp peak at 4 eV has vanished as in Se (dotted line). Since Dono- van et al. [13] improved their NDT-curves by using an experimentally determined valence band density of states of the amorphous modification, we convo- luted our calculated amorphous densities of states (see section 111) and obtained the dashed line for the
I M(o) . 1; Now the agreement, except for the umklapp peak, is much better. So we have learned that we may use the crystalline I M(o) 12,,, if we smear out the umklapp peak.
Why does the umklapp peak vanish ? This may be understood at least qualitatively if one expands the initial and final amorphous states in eq. (4) in terms of Bloch functions of the corresponding crystal.
FIG. 3. - Irreducible part of the Brillouin zone of Se (left) and the IV-IV and III-V semiconductors (right). The shaded area is
the A-region.
for 12 bands. These values have been interpolated for 50 000 random points of the irreducible wedge. With this n(E) and E,(CO) were calculated.
For IV-IV and III-V semiconductors we used the formfactors of Cohen and Bergstresser [16] and calculated eigenvalues and matrix elements in about 1 000 points of the Brillouin zone for 8 bands. With this the values for 20 000 points of the irreducible part of the Brillouin zone (Fig. 3, right) were interpolated.
Similar to the Se-case n(E) and E ~ ( W ) were found easily by summation.
2.3. RESULTS OF THE NDT-METHOD. - Figure 4 shows the crystalline / M(m) ,, I:, of Ge, Si, and III-V compounds ( ~ r a m e r et al. [17]). By qualitatively smearing out we got the amorphous 1 M(o) . 1;
Their product with 0-2 n,,,(o), properly normalized,
const
E,(CO) = ---- gi f(n, ..., n"', k, k') x
C02 if nn'k
nr'n"'k'
x I P n n * k I I Pn,,n,,,k, 1 6(Ei + co - Ef) . (9) The real function g is the product of the four expansion coefficients and the phase factors of the I Pnn' k 1.
g should have a maximum if wnnrk = En, - E n , is near coif = Ef - E, and should otherwise be indepen- dent of k, k' consistent with our assumption of a relaxed k-selection rule. The calculation of g is equivalent to calculating the Green's functions. This is done in section 3. If umklapp enhanced contributions to
E, in the crystalline case at an energy CO, are localized somewhere in the Brillouin zone, then in the amorphous case also smaller pairs of crystalline I P n n , I contri- bute, stemming from the volume of the Brillouin zone, if on,, w w,. Contributions to E, at other energies, especially at higher energies, are less influenc- ed by this averaging procedure. Therefore the gross energy dependence of the I M(o) 1' is preserved in the amorphous phase.
-
2.2. TECHNIQUE OF COMPUTATION. - Using form- factors of Sandrock [15], we have calculated eigen-
values and matrix elements of crystalline Se in 48 points 0 1 2 3 4 5 6 7 8 9 1 0 of the irreducible part of the Brillouin zone (Fig. 3) FIG. 4. - Average crystalline transition probabilities.
C3-160 B. KRAMER, K. MASCHKE AND P. THOMAS is shown in figure 5 for Ge and Si and in figure 6 for
GaAs, which is representative for the other 111-V compounds [17]. a) is the experimental spectrum ; b) our result for g2, and c) M,,,(w) alone. In contrast to b), which is in good agreement with the experimental curves, the assumption of constant matrix elements leads to rather unsatisfactory results.
The slight shift of our curves to higher energies is partly due to the fact that the crystalline densities of states were convoluted.
FIG. 5. - EZ of amorphous Ge (left) and Si (right). a) experi- mental, b) calculated by means of the NDT-method using energy dependent, c) without using energy dependent matrix elements.
GaAs
FIG. 6. - €2 of amorphous GaAs. a, b, c as in figure 5.
3. The CBS-method. - 3 . 1 . THE CONFIGURATIO- NAL AVERAGING. - TO calculate the configurational average of eq. (2) and (3), it is necessary to use an appro- ximative correlation function which enables one to resum the Born series for the Green's function and which takes into account the short range order. In the crystalline case the correlation function is exactly
given by a sum of products of 6-functions centered at the periodically distributed lattice points. The resum- mation is easily done and the zeros of the denominator of the Green's function for a given k are situated on the real axis of the energy plane. The band structure allows to calculate the density of states and gives insight into the general behaviour of 8,. If one knows the residues, in the crystalline case given by the Bloch functions, one is able to calculate E ~ .
In the amorphous case it is possible under some approximations to establish a scheme similar to the crystalline band structure. For each k-vector of the Brillouin zone there is a <( Quasi-crystal-electron >> with a certain energy and lifetime. The lifetime is inversely proportional to the imaginary part of the position of the pole in the energy plane. In analogy to the crys- talline case this complex band structure (CBS) gives information about n(E) and g2. An additional assump- tion concerning the residues enables one to calculate E,.
We approximate P(Rl ... R,) by a sum of products of Gaussian-like functions centered at the lattice points of the corresponding periodic crystal. The single functions are two-body correlation functions. Their half widths are proportional to an order parameter a and increase proportional to the distance from a given lattice point. This guarantees the vanishing of the long range order.
The integral equation for the Green's function G(r, r') = Go(r - r') +
+ Go(r - r") V(r") G(r", r') d3rr' (10) is iterated, multiplied by P(R, ... RN), and Fourier transformed. Here V is a sum of atomic pseudopo- tentials which are used in crystalline calculations.
The terms of the Born's series have the form
Here h:'(k) are Gaussian functions centered at reciprocal lattice points. Their half-widths increase with increasing distance in k-space and are propor- tional to cc. For small cc the integration over k t + , and
ki+, is restricted to small regions around Km, + k and K,,,, + k, respectively. Therefore we may alter the pseudo-potentials so that
Now it is possible to resum the series for G(k, kt), because each term factorizes and the series becomes geometric. The denominator of G(k, k') gives the secular determinant of the amorphous case.
COMPUTATION OF EZ-SPECTRA OF AMORPHOUS SEMICONDUCTORS C3-161
with The CBS's for amorphous Ge and Si are given in
figure 8 together with the crystalline band structure (k - q - K,),'
@F'(k, E) = ) x (Kramer 1191).
n3I2 a3 Kn3
The zeros of the determinant (12) can be calculated numerically. This costs by far the most computer time of all the calculations. For a = 0 we arrive back at the crystalline case.
3.2. THE APPROXIMATE REPRESENTATION OF THE
GREEN'S FUNCTION. - TO calculate 8, it is necessary to know also the residues. It should be noticed that we have to average the product of the two Green's func- tions of eq. (3). Then for a single function kt = k + K, as in the crystalline case. The product of the Green's functions is diagonal in k due to the homogeneity.
This feature suggests to use the crystalline residues in the amorphous case, too. Now we have a closed form for the Green's functions in terms of the poles which are given by the zeros of the determinant (12) and the crystalline residues belonging to the k-vector and band number of the pole.
3 . 3 THE COMPLEX BANDSTRUCTURES. - Figure 7 shows the complex bandstructures (CBS) of amor- phous Se (Kramer [ll]). Solid lines are the real parts E,("'@), the imaginary parts ~ ? ) ( k ) are given by the shaded area. a = 0.075 is in good agreement with structural data by Kaplov et al. [18].
r A H K . r
FIG. 7. - Complex band structure of amorphous Se.
The selective change in the &,-spectrum of Se can be seen already in the CBS. The structure A comes from transitions from the uppermost valence band triplett into the lowest conduction band triplett TI ... T,, both initial and final states are broadened. The struc- ture B due to transitions from the lower valence band triplett into the first conduction band triplett, e. g. T,, is less affected by disorder since the initial states are nearly unbroadened.
o( - 0 e.0.07
GERMANIUM
FIG. 8. - Crystalline and amorphous (CBS-) band structure of Ge and Si.
For Ge a = 0.07 and for Si a = 0.05. These values are determined as in Se to adjust the maximum of the
8, curves to the experimental height. Also for Si and Ge a is in good agreement with structural measurements (Richter et al. [20], Moss et al. [21]). For amorphous Ge the real parts E:'@) are nearly unchanged with respect to the crystalline band structure whereas in Si there is a considerable shift A?'@) = E,'"'(k) - E?'(k).
The CBS's of the 111-V compounds behave more like Ge. Figure 9 shows GaAs, results for the other 111-V
FIG. 9. - Crystalline and amorphous (CBS-) band structure of GaAs.
C3-162 B. KRAMER, K. MASCHKE AND P. THOMAS semiconductors are given in Kramer e t al. [17]. The
small imaginary parts of the valence bands and of the first conduction band along the A-axis are common to all the IV-IV and 111-V CBS's. This leads to the selec- tive changes in the &,-spectra of these materials.
3.4. TECHNIQUE OF COMPUTATION. - From eq. (2) and (3) one obtains the expressions for n(E) and E~
suitable for computation :
comparison of the fine structure of these curves with results of photoemission measurements is rather satisfactory (Laude et al. [23]). The structure of the uppermost valence band and the conduction bands is smeared out, the lower valence band, however, remains nearly unchanged. In figure 11 the densities of states of Ge and Si are given, in figure 12 those of GaAs.
The histogram is the crystalline, the solid line the amorphous curve. Here the valence bands are only little smeared out because of their small imaginary parts. The conduction band shows only the rest of the strong first crystalline peak which comes from states and
C;~(W) c C,
1
d ~ ! E~~ &gYst(~', kL, m, mt) xm,m
x dErt LmjkL(Et - E") Lm,,k(Eu - W) . (15) 2
Here ncmt(E, k,, m) and E$"(E, k,, m, m') are contributions to the crystalline n(E) and E,, respecti- vely, from bands m, m' out of regions around ki of the Brillouin zone where the imaginary parts and
energy shifts are taken to be constant. These regions are Energy (evJ the A-and X-regions for the cubic semiconductors,
a sphere around H and a region around the A-axis (r-A) for Se (Fig. 3). For Se the crystalline spectrum has to be averaged isotropically. The contributions are broadened by means of the Lorentzian curves
1 r:'(ki)
L,,kz(E - E') = -
n (E - E' + ~:)(ki))~ + ( ~ $ ) ( k
and summed up. Similar equations were used by Brust [22]. He calculated 8, of amorphous Ge using
the imaginary parts proportional to a fitting parameter 4 - 6 - 4 - 2 o 2 4 6
and the density of states. Energy /rVI ---r
FIG. 11. - n(E) of amorphous and crystalline (histogram) .4. THE OF STATES' - Ge and Si. The lower histogram gives the contribution of the Figure 10 shows the density of states of crystalline A-region. -
(dashed line) and amorphous (solid line) Se. A detailed
I I GaAs d-0.07 I
FIG. 10. - n ( E ) of amorphous and crystalline Se.
Energy bV1
-
FIG. 12. - n(E) of amorphous and crystalline (histogram) GaAs.
The lower histogram gives the contribution of the A-region.
COMPUTATION O F E ~ S P E C T R A O F AMORPHOUS SEMICONDUCTORS C3-163
L . . I
FIG. 14. - 82 of crystalline and amorphous Ge and Si. Left :
1 2 3 4 5 6 7 8 9
.nu (eV) experimental, right : calculated. The histograms are the results of the pseudopotential calculation for the crystalline case, the solid lines are the results of the CBS-calculation for the amor- FIG. 13. - €2 of amorphous Se. Dashed line : experimental, solid phous case.
line : calculated by means of the CBS-method.
near the A-axis, where in the vicinity of T the imaginary
parts are small. Ge exp.
40 - 3.3. RESULTS FOR e2. - The &,-spectrum of amor- phous Se is given in figure 13 (solid line) and compared with the experimental curve of Leiga [3] (dashed line). 30 ' The overall agreement is rather good. The selective
change is correctly reproduced. The weak kink near 20-
3 eV is a remainder of the crystalline peak stemming from transitions T I near A. Inspection of figure 7 shows that near A the imaginary parts are smaller than those of other regions of the Brillouin zone.
const
In figure 14 and 15 the experimental crystalline and &2NDT(o) = o ( M(w) x amorphous &,-spectra (left) and the calculated (right)
spectra are given for Ge, Si, and the 111-V semicon-
ductors. The selective changes are well reproduced due x dE dE' a(E - En,,k, + En$)
n,k n',k'
to the small imaginary parts for transitions near the
A-axis. In Si there is the additional large shift of the x Ln,ki(E - E') Ln,,,;(Ef - w) (17)
real parts which has the same effect as the non-direct
&CBS const
transitions across the indirect gap in the framework 2 (o) = -
of the NDT-method. w2 n,n',k n,n',kt
15- 40 -
€1 30 .
x - En,,, + En,,) 4. Comparison of the two methods. - The energy
shift of the NDT-curves with respect to the experimen- x L,,(E - E') Ln,,ki(E1 - 4 . (18)
tal has been assumed to-be partly due to the We first discuss the optical density of states alone.
fact that we convoluted the crystalline instead of the
The difference is that in the NDT-method there is an amorphous densities of states. Since we now have the
additional summation over k l , i. e. in the NDT- CBS-densities of states we can them and
method non-direct transitions are lifetime broadened, multiply with the amorphous transition probabilities
in the CBS-method only direct transitions. But the of Section 2. The result for &YDT now looks quite results may be seen to be not very different if one similar to the e:BS curves with an energy shift in the realizes that the imaginary parts of at least one band Correct Sense. Writing &FDT in terms of amorphous are nearly as large as the band widths. Then the densities of states and &EBS with the contributions to lifetime broadening of the joint density of states the crystalline E";" in the form of eq. (5) we can easily (CBS) and of the convoluted density of states P D T ) compare both methods and see why their results are (both eventually changed by the shift of the real so similar. parts) should yield similar results. In this case there
Si exp. Si colc.
I
cry. t 10 -
20 -
10 -
5-
J ,
o z i k 0 2 4 6
P H O T O N ENERGY (eVJ
C3-164 B. KRAMER, K. MASCHKE AND P. THOMAS
is obviously no sense in speaking of a totally relaxed or partly preserved k-selection rule.
The treatment of the matrix elements in the NDT- method may also be understood in terms of the CBS- method. If one regards the Lorentzian broadening as an averaging procedure and splits the averaging of the
I M(w) I&,, and optical density of states in the CBS formula into a product of separately averaged transi- tion probability and optical density of states, one arrives at the NDT-method.
Starting with an approximative correlation function
which allows for the short range order, we derived a complex band structure which is of similar importance as the crystalline bandstructure. Then the selective changes in the amorphous 8, with respect to the crys- talline curves may be interpreted as coming from a k- and band-dependent finite lifetime of quasi-crystal- electrons.
Acknowledgment. - One of us, P. T., is highly indebted to Prof. Dr. 0. Madelung for giving him the opportunity to give this talk at the conference.
References
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COMPUTATION OF ~2-SPECTRA OF AMORPHOUS SEMICOND UCTORS C3-165
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