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Energy-band tails and the optical absorption edge; the case of a-Si:H
~
Solid State Cormnunications, Printed in Great Britain.
ENERGY-BAND
Vol.44,No.9,
pp.1347-)349,
1982.
0038-IO98/82/451347-O3503.O0/0 Pergamon Press Ltd.
TAILS AND THE OPTICAL ABSORPTION EDGE; THE CASE OF a-Si:H* David Redfield RCA Laboratories, (Received
Princeton NJ
16 August
08540
1982 by J. Tauc)
For a disordered semiconductor having exponential energy-band tails and energy-independent matrix elements the optical absorption edge is calculated, including transitions to both bands. It is shown why the width of the absorption edge is controlled by only the broader of the two tails, whereas the strength is affected by both tails. For a-Si:H, comparison of the predicted width of the absorption edge to recently measured values results in a large discrepancy, which cannot be explained by the matrix element approximation.
It has proved difficult to quantify the properties of electronic states of energy-band tails in disordered semiconductors even though such states are known to affect a number of the material properties. Recently, however, intensive study of amorphous Si-H alloys has provided several results that can be used for quantitative evaluation of some features of the energy-band tails. The present work has the twofold purpose of (i) formulating a procedure for comparison of optical and electrical results of this type, and (ii) applying this procedure to some very recent data on a-Si:H. It is important to note that the models and assumptions used here are invoked only for consistency with those having widespread acceptance; they might well require modifications as the field advances. By an interpretation that invokes a model of multiple trapping of carriers whose transport occurs only when they are thermally excited past their mobility edge, recent measurements of drift mobility of both electrons and holes have led to quantitative descriptions of the tw~ band tails in the pseudo gap of undoped a-Si:H.The trapping states were assumed to be those of the band tail, and the distribution in energy of those states g(E) could be inferred from the observed characteristics of the mobility. It was concluded in Ref. ] that the energy dependence of both band tails is exponential; the characteristic widths of these exponential band tails was found to be W = 0.027 eV for the tail c below the electron mobility edge and W v = 0.043 eV for the tail above the hole mobillty edge. No temperature dependence of these values was discussed in Ref. l, and its analysis implies that any temperature dependence must be weak. Detailed measurements of the optical absorption spectrum of a-Si:H have been reported even more recently. 2 Because this work was done at the same laboratory as that of Ref. I, the same materials were used, and comparison of the two types of results is made later in this paper. The portion of such optical data of relevance for this kind of comparison is that having an exponential energy dependence of the
absorption coefficient ~ which is often observed over a range of photon energy ~w near, but below, that corresponding to the pseudo gap G. In Ref. 2 this spectral region was studied in detail, with particular emphasis on the characteristic width E of the exponential part of ~(~w). The object of the present work is to assess how such observed values of E might o
relate to the individual widths of the energy band tails W and W whose values are inferred v e from other measurements. This type of quantitative comparison is possible for the first time now because of the availability of sufficient experimental data. One of the widespread beliefs about exponential optical absorption edges is that they reflect the joint density of states of the two energy band tails. 3'4 In fact, the two kinds of tails--one in ~(hw) and the other in g(E)-are sometimes referred to interehangeably. 5 The implication in such usage, of course, is that any energy dependence of the matrix element of the optical transitions is negligible. That approximation can be justified for transitions involving extended states (i.e., normal bandband transitions), but it has been shown that for hw
1347
1348
A v
-
-
~c
e,0[
the conduction band. The solutions of the first and third integrals simplify further by the use of the very., good approximations [Iw>>W and c . flw>>W . When the various parts are comblned and V expressed in terms of the proper variable (G-~w), we find the general result
v
~ ~ k
I
Ev
I.
ENERGY (orb. units)
Av
Av exp[-(E-Ev)/W v]
I gf(E) =
v expI-(C-fiw)/Wv]
- W2 c exp [- (G-~w)/Wc]
~
I
Ac Ac exp[_(Ec_E)/Wc ]
,
EJE
,
Ev
,
E>_Ec
,
Ev
A V
~m
C
(2W+G-fiw) exp [- (G-15m)/W],
(5)
whose width is just W, not 2W as might be thought. If one of the tail widths is zero, only one of the three integrals survives. For example, if W = 0 C
~
A A v c ~ W v expI-(G-flw)/Wvl.
(6)
(1)
(2)
a ~
gi (E)gf(E+~w) dE
(4) When
One other result that will be of use here is the absorption spectrum that would result if the only transitions to be included had both initial and final states in a band tail. This is equivalent to setting g(E)=0 in both bands, thus leaving only transitions of type (2) in the discussion preceeding Eq. (4). In this case
where A and A are constants. Because con. c v , slderatlon is restrxcted here to photon energies less than the gap energy, all optical transitions to be included in this analysis require either the initial or final state, or both, to be tail states. Ignoring some constants of no interest here, the absorption coefficient is given by
~w
1"
Ec
Model densities of states for the initial (left) and final (right) states of optical transitions. For photon energy ~w less than (E -E ), • C V one state or both must be in a band tail.
I
gi(E) =
W2
Some special cases are of interest. W =W =W c v A
Fig.
c
5w~w~w c)
A ex [(Ec-E}]~ [
t 0,
~
Vol. 44, No. 9
THE CASE OF a-Si:H*
(3)
where conservation of energy is already incorporated, and the crystalline requirement of momentum conservation has been removed because 7 of the amorphous nature of these materials. Note that the variable of integration is the energy of the initial state of the optical transition. The implication that initial states are occupied and final states are empty is appropriate for intrinsic materials. The simple forms used here for all the portions of g(E) make the evaluation of the integral in Eq.(3) a simple matter. It is essential, however, to count correctly all possible transitions consistent with 5w
Av A c W r ~w
lexp [- (G-nm)/W v ]
- exp[-(G-Nm)/Wc] }
(7)
where W -I=w -1-W -1. This becomes z e r o a t ~ = G r c v o n l y b e c a u s e t h e r e a r e no s t a t e s t h a t a r e permitted to contribute to ft. Some features of these results to be noted are: (I) Although Eq. (4) has two exponential terms, the one with the larger width always dominates. This conclusion is not the same as that for the general sum of two exponential terms which can exhibit 2-segment behavior; the nature of the pre-exponential factors in Eq. (4) preclude such behavior. (These factors can only be obtained by correct inclusion of transitions involving the bands.) Therefore the width of the absorption edge reflects that of the wider tail only, not both of them. The strength of this absorption, however, is proportional to both A and A . (2) Provided that the spectral V. range is restrlcted to energles well below that of the pseudo gap, the same conclusions are reached from Eq. (7). Therefore, to this extent, the presence or absence in the calculation of extended states in the "bands" does not matter, and the crude approximation used for g(E) in the bands also does not matter. (3) The similarity of the extreme cases of Eqs. (5,6) indicates the validity of these conclusions for all cases. For quantitative purposes, of course, the general solution, Eq. (4), must be used. Further confirmation of these conclusions has been found in numerical evaluations with squareroot g(E) in the bands. The conclusion concerning the edge width is not universal; it is due to the exponential shapes of the tails. For
THE CASE OF a-Si:H*
Vol. 44, No. 9
linear tails -- which are more complicated to treat -- a different conclusion is reached. These ideas have been applied to the data of Refs. (1,2) with the results shown in Fig. 2.
!
I l"- Ec- Ev = 1 7 e V
/
Wc= 0 O 2 7 e V Wv = 0 . 0 4 3 eV
x
data of Ref. 2, we note that the values of W c and W found in Ref. I had no temperature d e p e n d e n c e . In fact,the values obtained there came from measurements between 125-350 K. The values of E reported in Ref. 2, however, are o temperature dependent, wlth E ~0.065eV at 250 K oand E ~0.070eV at room temperature. Since these otemperatures are well within the range used in Ref. 1, it is appropriate to compare our Wef f
-10in
o IX LIJ
,v Io-; LL
/
LU O LO Z O QnO O9
/
Wef f = 0 . 0 4 5 eV
,/ ,0-3
10- 4
1.4
1.5
1.6
1.7
1.8
ENERGY ( E v )
Fig. 2.
Values of Eq. (4), (points), and the fit to them by a single exponential (solid line). The width parameter of the single exponential is W ~ = 0.045 eV. The dashed line ~ the width parameter of 0.070eV reported as the slope of the optical absorption edge at 300 K in Ref. 2.
The points are computed by Eq. A
c
W =0.027eV, 1 and W =0.043eV. I It is obvious c v that in the spectral range covering the exponential absorption edge (l.3-1.7eV) the computed values of ~ can be fitted very well by a single exponential whose width is Weff=0.045eV , a value very close to that of the valence band tail. A similar conclusion was reached in Ref. 4 by means of a less general analysis and an incomplete representation. Before comparing this Wef f with the optical
/
/
1349
(4) (omitting A v 2
and o t h e r c o n s t a n t s ) w i t h G=I.7eV,
with a value of E in this range. In Fig. 2, the dashed line s~ows the room-temperature edge for comparison with the edge calculated by Eq.(4). Regardless of which temperature in this range is used, however, the optical value of E o is substantially larger than the calculated W ~ , a discrepancy that is too large to be ell consistent with the uncertainties in measurement. The neglect of a possible energy dependence of the optical matrix element appears have the wrong sense to be responsible for this discrepancy. That is, optical absorption edges include electron-hole localization and correlation effects which have been shown to produce steeper edges (i.e., to have smaller widths) than the 8 density of states. A different possible source of the discrepancy may be found in recent Monte Carlo simulations of dispersive transport by hopping among states of an exponential tail^ (instead of activation to a mobility edge). 9 If, as Ref. 9 suggests, the correct conduction mechanism is not yet known, the values obtained in Ref. 1 for W and W may not be correct. • v This dlscrepancy ~etween E and W :~ has a • .0 . . elI practical impact as well as sclentzflc xnterest because of the recent linkage between band tails and the maximum possible voltage that can be expected in an a-Si:H solar cell. ]0 If the width of the valence-band tail were equal to the value of E =0.07eV, then the formulas of Ref. I0 would lead°to a maximum voltage (0.90V) that is less than some observed values (0.93V).
References Research reported herein was supported by SERI under contract No. XG-0-9372 and by RCA Laboratories, Princeton, NJ 08540.
5.
6. I.
2. 3. 4.
T. Tiedje, J. Cebulka, D. Morel, and B. Abeles, Phys, Rev. Lett., 46, 1425
(1981). G.D. Cody, T. T i e d j e , B. A b e l e s , B. Brooks, and Y. G o l d s t e i n , Phys. Rev. L e t t . 47, 1480 (1981). G. Moddel, D. Anderson, and W. P a u l , Phys, Rev. B 22, 1918 (1980). D. J . D u n s t a n , S o l i d S t a t e Commun. 43, 341 (1982). R. S. C r a n d a l l , S o l a r C e l l s 2, 319 (1980).
7.
8. 9. I0.
W. Jackson and N. Amer, J. de Physique, Colloque C4, Supplement I0, Vol. 42, P. C4-293 (Oct. 1981). D. Redfield, Phys, Rev. 130, 916 (1963). J. Dow and D. Redfield, Phys. Rev. B 5, 594 (1972). N.F. Mort and E.A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd ed., (Clarendon, Oxford, 1979) p. 288. J.D. Dow and J.J. Hopfield, J. Non-Cryst. Solids 8-10, 664 (1972). M. Silver, G. Schoenherr, and Baessler, Phys. Rev. Lett. 48, 352 (1982). T. Tiedje, Appl. Phys. Lett. 40, 627(1982).
Solid State Cormnunications, Printed in Great Britain.
ENERGY-BAND
Vol.44,No.9,
pp.1347-)349,
1982.
0038-IO98/82/451347-O3503.O0/0 Pergamon Press Ltd.
TAILS AND THE OPTICAL ABSORPTION EDGE; THE CASE OF a-Si:H* David Redfield RCA Laboratories, (Received
Princeton NJ
16 August
08540
1982 by J. Tauc)
For a disordered semiconductor having exponential energy-band tails and energy-independent matrix elements the optical absorption edge is calculated, including transitions to both bands. It is shown why the width of the absorption edge is controlled by only the broader of the two tails, whereas the strength is affected by both tails. For a-Si:H, comparison of the predicted width of the absorption edge to recently measured values results in a large discrepancy, which cannot be explained by the matrix element approximation.
It has proved difficult to quantify the properties of electronic states of energy-band tails in disordered semiconductors even though such states are known to affect a number of the material properties. Recently, however, intensive study of amorphous Si-H alloys has provided several results that can be used for quantitative evaluation of some features of the energy-band tails. The present work has the twofold purpose of (i) formulating a procedure for comparison of optical and electrical results of this type, and (ii) applying this procedure to some very recent data on a-Si:H. It is important to note that the models and assumptions used here are invoked only for consistency with those having widespread acceptance; they might well require modifications as the field advances. By an interpretation that invokes a model of multiple trapping of carriers whose transport occurs only when they are thermally excited past their mobility edge, recent measurements of drift mobility of both electrons and holes have led to quantitative descriptions of the tw~ band tails in the pseudo gap of undoped a-Si:H.The trapping states were assumed to be those of the band tail, and the distribution in energy of those states g(E) could be inferred from the observed characteristics of the mobility. It was concluded in Ref. ] that the energy dependence of both band tails is exponential; the characteristic widths of these exponential band tails was found to be W = 0.027 eV for the tail c below the electron mobility edge and W v = 0.043 eV for the tail above the hole mobillty edge. No temperature dependence of these values was discussed in Ref. l, and its analysis implies that any temperature dependence must be weak. Detailed measurements of the optical absorption spectrum of a-Si:H have been reported even more recently. 2 Because this work was done at the same laboratory as that of Ref. I, the same materials were used, and comparison of the two types of results is made later in this paper. The portion of such optical data of relevance for this kind of comparison is that having an exponential energy dependence of the
absorption coefficient ~ which is often observed over a range of photon energy ~w near, but below, that corresponding to the pseudo gap G. In Ref. 2 this spectral region was studied in detail, with particular emphasis on the characteristic width E of the exponential part of ~(~w). The object of the present work is to assess how such observed values of E might o
relate to the individual widths of the energy band tails W and W whose values are inferred v e from other measurements. This type of quantitative comparison is possible for the first time now because of the availability of sufficient experimental data. One of the widespread beliefs about exponential optical absorption edges is that they reflect the joint density of states of the two energy band tails. 3'4 In fact, the two kinds of tails--one in ~(hw) and the other in g(E)-are sometimes referred to interehangeably. 5 The implication in such usage, of course, is that any energy dependence of the matrix element of the optical transitions is negligible. That approximation can be justified for transitions involving extended states (i.e., normal bandband transitions), but it has been shown that for hw
1347
1348
A v
-
-
~c
e,0[
the conduction band. The solutions of the first and third integrals simplify further by the use of the very., good approximations [Iw>>W and c . flw>>W . When the various parts are comblned and V expressed in terms of the proper variable (G-~w), we find the general result
v
~ ~ k
I
Ev
I.
ENERGY (orb. units)
Av
Av exp[-(E-Ev)/W v]
I gf(E) =
v expI-(C-fiw)/Wv]
- W2 c exp [- (G-~w)/Wc]
~
I
Ac Ac exp[_(Ec_E)/Wc ]
,
EJE
,
Ev
,
E>_Ec
,
Ev
A V
~m
C
(2W+G-fiw) exp [- (G-15m)/W],
(5)
whose width is just W, not 2W as might be thought. If one of the tail widths is zero, only one of the three integrals survives. For example, if W = 0 C
~
A A v c ~ W v expI-(G-flw)/Wvl.
(6)
(1)
(2)
a ~
gi (E)gf(E+~w) dE
(4) When
One other result that will be of use here is the absorption spectrum that would result if the only transitions to be included had both initial and final states in a band tail. This is equivalent to setting g(E)=0 in both bands, thus leaving only transitions of type (2) in the discussion preceeding Eq. (4). In this case
where A and A are constants. Because con. c v , slderatlon is restrxcted here to photon energies less than the gap energy, all optical transitions to be included in this analysis require either the initial or final state, or both, to be tail states. Ignoring some constants of no interest here, the absorption coefficient is given by
~w
1"
Ec
Model densities of states for the initial (left) and final (right) states of optical transitions. For photon energy ~w less than (E -E ), • C V one state or both must be in a band tail.
I
gi(E) =
W2
Some special cases are of interest. W =W =W c v A
Fig.
c
5w~w~w c)
A ex [(Ec-E}]~ [
t 0,
~
Vol. 44, No. 9
THE CASE OF a-Si:H*
(3)
where conservation of energy is already incorporated, and the crystalline requirement of momentum conservation has been removed because 7 of the amorphous nature of these materials. Note that the variable of integration is the energy of the initial state of the optical transition. The implication that initial states are occupied and final states are empty is appropriate for intrinsic materials. The simple forms used here for all the portions of g(E) make the evaluation of the integral in Eq.(3) a simple matter. It is essential, however, to count correctly all possible transitions consistent with 5w
Av A c W r ~w
lexp [- (G-nm)/W v ]
- exp[-(G-Nm)/Wc] }
(7)
where W -I=w -1-W -1. This becomes z e r o a t ~ = G r c v o n l y b e c a u s e t h e r e a r e no s t a t e s t h a t a r e permitted to contribute to ft. Some features of these results to be noted are: (I) Although Eq. (4) has two exponential terms, the one with the larger width always dominates. This conclusion is not the same as that for the general sum of two exponential terms which can exhibit 2-segment behavior; the nature of the pre-exponential factors in Eq. (4) preclude such behavior. (These factors can only be obtained by correct inclusion of transitions involving the bands.) Therefore the width of the absorption edge reflects that of the wider tail only, not both of them. The strength of this absorption, however, is proportional to both A and A . (2) Provided that the spectral V. range is restrlcted to energles well below that of the pseudo gap, the same conclusions are reached from Eq. (7). Therefore, to this extent, the presence or absence in the calculation of extended states in the "bands" does not matter, and the crude approximation used for g(E) in the bands also does not matter. (3) The similarity of the extreme cases of Eqs. (5,6) indicates the validity of these conclusions for all cases. For quantitative purposes, of course, the general solution, Eq. (4), must be used. Further confirmation of these conclusions has been found in numerical evaluations with squareroot g(E) in the bands. The conclusion concerning the edge width is not universal; it is due to the exponential shapes of the tails. For
THE CASE OF a-Si:H*
Vol. 44, No. 9
linear tails -- which are more complicated to treat -- a different conclusion is reached. These ideas have been applied to the data of Refs. (1,2) with the results shown in Fig. 2.
!
I l"- Ec- Ev = 1 7 e V
/
Wc= 0 O 2 7 e V Wv = 0 . 0 4 3 eV
x
data of Ref. 2, we note that the values of W c and W found in Ref. I had no temperature d e p e n d e n c e . In fact,the values obtained there came from measurements between 125-350 K. The values of E reported in Ref. 2, however, are o temperature dependent, wlth E ~0.065eV at 250 K oand E ~0.070eV at room temperature. Since these otemperatures are well within the range used in Ref. 1, it is appropriate to compare our Wef f
-10in
o IX LIJ
,v Io-; LL
/
LU O LO Z O QnO O9
/
Wef f = 0 . 0 4 5 eV
,/ ,0-3
10- 4
1.4
1.5
1.6
1.7
1.8
ENERGY ( E v )
Fig. 2.
Values of Eq. (4), (points), and the fit to them by a single exponential (solid line). The width parameter of the single exponential is W ~ = 0.045 eV. The dashed line ~ the width parameter of 0.070eV reported as the slope of the optical absorption edge at 300 K in Ref. 2.
The points are computed by Eq. A
c
W =0.027eV, 1 and W =0.043eV. I It is obvious c v that in the spectral range covering the exponential absorption edge (l.3-1.7eV) the computed values of ~ can be fitted very well by a single exponential whose width is Weff=0.045eV , a value very close to that of the valence band tail. A similar conclusion was reached in Ref. 4 by means of a less general analysis and an incomplete representation. Before comparing this Wef f with the optical
/
/
1349
(4) (omitting A v 2
and o t h e r c o n s t a n t s ) w i t h G=I.7eV,
with a value of E in this range. In Fig. 2, the dashed line s~ows the room-temperature edge for comparison with the edge calculated by Eq.(4). Regardless of which temperature in this range is used, however, the optical value of E o is substantially larger than the calculated W ~ , a discrepancy that is too large to be ell consistent with the uncertainties in measurement. The neglect of a possible energy dependence of the optical matrix element appears have the wrong sense to be responsible for this discrepancy. That is, optical absorption edges include electron-hole localization and correlation effects which have been shown to produce steeper edges (i.e., to have smaller widths) than the 8 density of states. A different possible source of the discrepancy may be found in recent Monte Carlo simulations of dispersive transport by hopping among states of an exponential tail^ (instead of activation to a mobility edge). 9 If, as Ref. 9 suggests, the correct conduction mechanism is not yet known, the values obtained in Ref. 1 for W and W may not be correct. • v This dlscrepancy ~etween E and W :~ has a • .0 . . elI practical impact as well as sclentzflc xnterest because of the recent linkage between band tails and the maximum possible voltage that can be expected in an a-Si:H solar cell. ]0 If the width of the valence-band tail were equal to the value of E =0.07eV, then the formulas of Ref. I0 would lead°to a maximum voltage (0.90V) that is less than some observed values (0.93V).
References Research reported herein was supported by SERI under contract No. XG-0-9372 and by RCA Laboratories, Princeton, NJ 08540.
5.
6. I.
2. 3. 4.
T. Tiedje, J. Cebulka, D. Morel, and B. Abeles, Phys, Rev. Lett., 46, 1425
(1981). G.D. Cody, T. T i e d j e , B. A b e l e s , B. Brooks, and Y. G o l d s t e i n , Phys. Rev. L e t t . 47, 1480 (1981). G. Moddel, D. Anderson, and W. P a u l , Phys, Rev. B 22, 1918 (1980). D. J . D u n s t a n , S o l i d S t a t e Commun. 43, 341 (1982). R. S. C r a n d a l l , S o l a r C e l l s 2, 319 (1980).
7.
8. 9. I0.
W. Jackson and N. Amer, J. de Physique, Colloque C4, Supplement I0, Vol. 42, P. C4-293 (Oct. 1981). D. Redfield, Phys, Rev. 130, 916 (1963). J. Dow and D. Redfield, Phys. Rev. B 5, 594 (1972). N.F. Mort and E.A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd ed., (Clarendon, Oxford, 1979) p. 288. J.D. Dow and J.J. Hopfield, J. Non-Cryst. Solids 8-10, 664 (1972). M. Silver, G. Schoenherr, and Baessler, Phys. Rev. Lett. 48, 352 (1982). T. Tiedje, Appl. Phys. Lett. 40, 627(1982).