Solid State Communications 151 (2011) 730–733
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Solid State Communications
journal homepage: www.elsevier.com/locate/ssc
The role that conduction band tail states play in determining the optical response
of hydrogenated amorphous silicon
Jasmin J. Thevaril a , Stephen K. O’Leary b,∗
a
Department of Electrical and Computer Engineering, University of Windsor, 401 Sunset Avenue, Windsor, Ontario, Canada N9B 3P4
b
School of Engineering, The University of British Columbia, 3333 University Way, Kelowna, British Columbia, Canada V1V 1V7
article
info
Article history:
Received 21 January 2011
Accepted 11 February 2011
by M. Grynberg
Available online 19 February 2011
Keywords:
A. Semiconductors
A. Hydrogenated amorphous silicon
D. Optical response
D. Conduction band tail breadth
abstract
We aim to explore the role that conduction band tail states play in shaping the optical response of
hydrogenated amorphous silicon. We do so within the framework of an empirical model for the valence
band and conduction band density of states functions, one that considers valence band band, valence
band tail, conduction band band, and conduction band tail states. We examine the sensitivity of the joint
density of states function to variations in the conduction band tail breadth, all other parameters being held
fixed at their nominal hydrogenated amorphous silicon values. We find that when the conduction band
tail is narrower than the valence band tail, its role in shaping the corresponding spectral dependence
of the joint density of states function is relatively minor. This justifies the use of a simplified empirical
model for the density of states functions that neglects the presence of the conduction band tail states
in the characterization of the optical response of this material. Experimental data corresponding to
hydrogenated amorphous silicon, demonstrating that the conduction band tail breadth is always less than
the valence band tail breadth for this material, is then presented. Finally, fundamental reasons for the
observed asymmetry in the band tail breadths are reviewed.
© 2011 Published by Elsevier Ltd
The optical response of hydrogenated amorphous silicon (aSi:H) has been a focus for intensive investigation for many years
[1–10]. Insights into this response have been gleaned through
the development of models for the spectral dependence of the
imaginary part of dielectric function, ϵ2 (h̄ω). Such models serve
two useful functions: (1) they provide a theoretical framework
for the purposes of materials characterization, and (2) they allow
for the quantitative prediction of device performance [11]. For the
specific case of a-Si:H, Jackson et al. [12] suggest that
ϵ2 (h̄ω) = 4.3 × 10−45 R2 (h̄ω)J (h̄ω),
(1)
where R2 (h̄ω), the normalized dipole matrix element squared
average, is in units of Å2 , and J (h̄ω), the joint density of states
(JDOS) function, is in units of cm−6 eV−1 . At zero temperature,
J (h̄ω) ≡
∫
∞
Nv (E )Nc (E + h̄ω) dE ,
(2)
−∞
where Nv (E ) and Nc (E ) denote the valence band and conduction
band density of states (DOS) functions, respectively, Nv (E ) 1E
and Nc (E ) 1E representing the number of one-electron valence
∗
Corresponding author. Tel.: +1 250 807 8091; fax: +1 250 807 9850.
E-mail address: stephen.oleary@ubc.ca (S.K. O’Leary).
0038-1098/$ – see front matter © 2011 Published by Elsevier Ltd
doi:10.1016/j.ssc.2011.02.008
band and one-electron conduction band electronic states, between
energies [E , E + 1E ], per unit volume.
Many of the empirical models that have been proposed for
the spectral dependence of ϵ2 (h̄ω) are themselves built upon
empirical models for Nv (E ) and Nc (E ) [13]. Such models include
that proposed by Tauc et al. [14] in 1966, Chen et al. [15] in 1981,
Redfield [16] in 1982, Cody [17] in 1984, O’Leary et al. [18] in
1997, Jiao et al. [19] in 1998, O’Leary and Malik [20] in 2002,
and O’Leary [21] in 2004. Most of these models assume squareroot distributions of band states and exponential distributions
of tail states. For the specific case of a-Si:H, there is general
consensus that the valence band tail is the dominant tail. As
a consequence, the tail exhibited by ϵ2 (h̄ω) is assumed to
be principally determined by that associated with the valence
band.
In this Communication, we aim to explore the role that
conduction band tail (CBT) states play in shaping the optical
response of a-Si:H. We do so within the framework of an empirical
model for Nv (E ) and Nc (E ), one that includes both valence band
tail (VBT) and CBT states. The spectral dependence of the resultant
JDOS function, J (h̄ω), will be the focus of our analysis, this function
determining, in large measure, the optical response of a-Si:H.
We begin by examining how variations in the conduction band
tail breadth impact upon the spectral dependence of the JDOS
function, all other parameters being held at their nominal a-Si:H
values. How the breadth of the conduction band tail influences
J.J. Thevaril, S.K. O’Leary / Solid State Communications 151 (2011) 730–733
the breadth of the optical tail associated with this JDOS function
is then assessed. A review of experimental measurements of the
valence band and conduction band tail breadths associated with aSi:H, with the aim of assessing how variations in the conduction
band tail breadth associated with this material impact upon the
corresponding optical tail, will then be presented, this review
drawing upon the insights gleaned from our JDOS analysis. Finally,
we discuss some recent theoretical analyses of the VBT and CBT
states, and provide reasons for the observed asymmetry in the
breadths of such states.
We cast our analysis within the framework of a general
empirical model for the DOS functions, Nv (E ) and Nc (E ), that
captures the basic expected features. For the case of a-Si:H, there
is general consensus that Nv (E ) and Nc (E ) exhibit square-root
functional dependencies in the band regions and exponential
functional dependencies in the tail regions. Following O’Leary [21],
we thus set






EvT − Ev
Ev − E


exp
,
 Ev − EvT exp
γv
γv
Nv (E ) = Nvo
E > E vT



Ev − E , E ≤ EvT ,
(3)
and


E − Ec , E ≥

 EcT




Ec − EcT
E − Ec
Nc (E ) = Nco
EcT − Ec exp
exp
,

γc
γc


E < EcT ,
(4)
where Nvo and Nco denote the valence band and conduction band
DOS prefactors, respectively, Ev and Ec represent the valence band
and conduction band band edges, γv and γc are the breadths
of the valence band and conduction band tails, EvT and EcT
being the critical energies at which the exponential and squareroot distributions interface; it should be noted that this model
implicitly requires that Ev − EvT ≥ 0 and EcT − Ec ≥ 0. It is
noted that this general empirical model for the DOS functions
includes valence band band (VBB) states, VBT states, conduction
band band (CBB) states, and CBT states. We further note that there
are eight independent modeling parameters in this model, i.e.,
Nvo , Nco , Ev , Ec , γv , γc , EvT , and EcT [22]. It is clear, from Eqs. (3) and
(4), that both Nv (E ) and Nc (E ) are continuous functions of energy,
i.e., Nv (Ev−T ) = Nv (Ev+T ) and Nc (Ec−T ) = Nc (Ec+T ).
In order to further narrow the scope of our analysis, we set EvT
to Ev − γv /2 and set EcT to Ec + γc /2. With these settings, Eq. (3)
reduces to





γv
1
Ev − E


exp
−
exp
,


 2
2
γv
γ
v
Nv (E ) = Nvo
E > Ev −


2



 Ev − E , E ≤ Ev − γv ,
(5)
2
and Eq. (4) reduces to

γc

 E − Ec , E ≥ Ec +

2







γc
1
E − Ec
Nc (E ) = Nco
exp −
exp
,

2
2
γc


γ

c
 E < Ec + ,
(6)
2
where now the number of independent modeling parameters
has been reduced to six, i.e., Nvo , Nco , Ev , Ec , γv , and γc [23]. We
note that now the DOS functions, Nv (E ) and Nc (E ), in addition to
being continuous functions of energy, have derivatives that are
731
Fig. 1. The valence band and conduction band DOS functions associated with aSi:H. The valence band DOS function, Nv (E ), specified in Eq. (5), is determined assuming the nominal a-Si:H parameter selections Nvo = 2 × 1022 cm−3 eV−3/2 , Ev =
0.0 eV, and γv = 50 meV. The conduction band DOS function, Nc (E ), specified in
Eq. (6), is determined assuming the nominal a-Si:H parameter selections Nco =
2 × 1022 cm−3 eV−3/2 , Ec = 1.7 eV, and γc = 27 meV. Only differences between the
energies Ev and Ec will impact upon the obtained JDOS results. The critical points at
which the band states and tail states interface, EvT and EcT , are clearly marked with
the dotted lines and the arrows.
Table 1
The nominal a-Si:H modeling parameter selections employed for the purposes of
this analysis. These modeling parameters relate to Eqs. (5) and (6).
Parameter (units)
Value
Nvo (cm−3 eV−3/2 )
Nco (cm−3 eV−3/2 )
Ev (eV)
Ec (eV)
γv (meV)
γc (meV)
2 × 1022
2 × 1022
0.0
1.7
50
27
continuous functions of energy when γv and γc are non-zero. We
also note that for the special case that γc → 0,
Nc (E ) → Nco

E − Ec ,
0,
E ≥ Ec
E < Ec ,
(7)
which is in accord with the simplified empirical DOS model
of O’Leary and Malik [20], i.e., the CBT states are neglected.
This empirical model for the DOS functions, Nv (E ) and Nc (E ),
i.e., Eqs. (5) and (6), forms the framework for our analysis.
For the nominal a-Si:H modeling parameters, Nvo = Nco = 2 ×
1022 cm−3 eV−3/2 , Ev = 0.0 eV, Ec = 1.7 eV, γv = 50 meV, and
γc = 27 meV, the corresponding DOS functions are as depicted in
Fig. 1, these modeling parameter selections being representative of
a-Si:H [13,19–21,24–29]. These nominal a-Si:H modeling parameter selections, employed for the purposes of this analysis, are tabulated in Table 1.
We now evaluate the JDOS function, J (h̄ω), corresponding to
our empirical model for the DOS functions, i.e., with Nv (E ) and
Nc (E ) as set in Eqs. (5) and (6), respectively. For the purposes of
this analysis, we examine the sensitivity of this JDOS function to
variations in the conduction band tail breadth, γc , the valence band
tail breadth, γv , being fixed at its nominal a-Si:H value, i.e., 50 meV.
The other modeling parameters are also set to their nominal a-Si:H
values, i.e., Nvo = Nco = 2 × 1022 cm−3 eV−3/2 and Eg ≡ Ec − Ev =
1.7 eV; Eg may be referred to as the energy gap [21]. In Fig. 2,
we plot the spectral dependencies of the resultant JDOS functions,
J (h̄ω), for the γc selections 0, 35, and 50 meV. We note that in all
cases, two regions of behavior are observed: (1) for h̄ω less than
the band gap, the JDOS function exhibits an exponential functional
dependence, and (2) for h̄ω greater than the band gap, the JDOS
732
J.J. Thevaril, S.K. O’Leary / Solid State Communications 151 (2011) 730–733
Fig. 2. The JDOS function, J (h̄ω), associated with a-Si:H, determined through an
evaluation of Eq. (2), for various selections of γc . For all cases, Nvo , Nco , Ev , Ec , and γv
are held at their nominal a-Si:H values, i.e., Nvo = Nco = 2×1022 cm−3 eV−3/2 , Eg ≡
Ec − Ev = 1.7 eV, and γv = 50 meV.
Fig. 3a. The dependence of E0 , as defined in Eq. (8), on the photon energy, h̄ω, for
a number of selections of γc . For all cases, Nvo , Nco , Ev , Ec , and γv are held at their
nominal a-Si:H values, i.e., Nvo = Nco = 2 × 1022 cm−3 eV−3/2 , Eg ≡ Ec − Ev =
1.7 eV, and γv = 50 meV.
function exhibits an algebraic functional dependence. We also note
that while the result corresponding to the γc = 0 meV case, which
corresponds to the simplified empirical DOS model of O’Leary and
Malik [20], forms a lower bound, the result corresponding to the
γc = 50 meV case forms an upper bound, the result corresponding
to the γc = 35 meV case being sandwiched in between these
bounds.
In order to further quantitatively examine the spectral dependence of the JDOS function, we follow the spirit of the analysis of
Orapunt and O’Leary [30] and define the tail breadth as being the
reciprocal of the logarithmic derivative of the JDOS function, i.e.,
[
E0 ≡
d ln [J (h̄ω)]
dh̄ω
] −1
,
(8)
where we note that E0 , as defined, is a continuous function of the
photon energy, h̄ω; for an exact exponential tail, this tail breadth,
as defined, corresponds to a single number, the exponential
breadth, but any deviations from an exact exponential dependence
will lead to a spectral dependence of E0 . For the same nominal aSi:H modeling parameter selections employed in Fig. 2, i.e., Nvo =
Nco = 2 × 1022 cm−3 eV−3/2 , Eg ≡ Ec − Ev = 1.7 eV, and
γv = 50 meV, in Fig. 3a we plot the spectral dependence of E0 on
the photon energy, h̄ω, for the three selections of γc considered in
Fig. 2, i.e., 0, 35, and 50 meV. We note that for the case of γc set to
0 meV, that E0 remains fixed at 50 meV, i.e., γv , until h̄ω exceeds
Ec − EvT , beyond which it monotonically increases, this monotonic
increase corresponding to the transition between the exponential
and algebraic regions of the JDOS function, as was previously
observed by Orapunt and O’Leary [30]. For the case of γc set to
35 meV, it is seen that while E0 asymptotically approaches 50 meV
for low photon energies, it is also observed that for higher photon
energies it increases monotonically with the photon energy, the
transition between the exponential and algebraic regions being
less abrupt than for the case of γc set to 0 meV. An even gentler
transition between these regions is observed for the case of γc set
to 50 meV.
We continue our analysis by examining the dependence of E0
on γc for a fixed value of the photon energy, h̄ω. In particular,
for the nominal a-Si:H parameter selections, i.e., Nvo = Nco =
2 × 1022 cm−3 eV−3/2 and Eg ≡ Ec − Ev = 1.7 eV, setting γv
to 50 meV and h̄ω to 1 eV, we examine the dependence of E0 on γc .
We plot this dependence in Fig. 3b. We note that E0 is essentially
equal to 50 meV, i.e., γv , until γc exceeds γv , beyond which E0
seems to asymptotically approach γc . This observation, and the
Fig. 3b. The dependence of E0 , as defined in Eq. (8), on the conduction band tail
breadth, γc , for the photon energy, h̄ω, set to 1 eV. For all cases, Nvo , Nco , Ev , Ec ,
and γv are held at their nominal a-Si:H values, i.e., Nvo = Nco = 2 ×
1022 cm−3 eV−3/2 , Eg ≡ Ec − Ev = 1.7 eV, and γv = 50 meV. The approximate
analytical expression for E0 , i.e., Eq. (9), is also depicted with the dotted line.
symmetry in the JDOS function, suggests that an approximate
analytical expression for E0 may be obtained by setting
E0 ≃ max.(γv , γc ).
(9)
We plot this approximate analytical expression for E0 in Fig. 3b
along with the corresponding exact result, noting that only when
γv and γc are of comparable values is there any noticeable deviation
between the approximate and exact results. This suggests that
when γc is less than γv the use of a model for the DOS functions
that neglects the presence of the CBT states, such as that devised
by O’Leary and Malik [20], is justified in the characterization of the
optical response.
We now review some experimental measurements of the
valence band and conduction band tail breadths associated with
a-Si:H. In Fig. 4, we plot experimentally determined values of
γv and γc corresponding to a-Si:H. Experimental results from
Rerbal et al. [31], Sherman et al. [32], Tiedje et al. [33], and Winer
and Ley [34], are employed for the purposes of this analysis; details
of the means used to perform these experiments are adequately
discussed in the literature. We also plot the γv and γc values
obtained by O’Leary [13] through a fit with some a-Si:H ϵ2 (h̄ω)
experimental data. We note that in all cases the valence band
tail breadth, γv , exceeds the corresponding conduction band tail
breadth, γc , and in most cases by a considerably margin. This
J.J. Thevaril, S.K. O’Leary / Solid State Communications 151 (2011) 730–733
733
that neglects the presence of the CBT states, such as that devised
by O’Leary and Malik [20], in the characterization of the optical
response of this material. Experimental data corresponding to aSi:H, demonstrating that γc is always less than γv for this material,
was then presented. Finally, fundamental reasons for the observed
asymmetry in the band tail breadths were reviewed.
Acknowledgement
The authors gratefully acknowledge financial support from the
Natural Sciences and Engineering Research Council of Canada.
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Fig. 4. The dependence of γv on γc . Results from the experiments of Rerbal
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shown. Experimental results are depicted with the solid symbols. Modeling results
are depicted with the open symbols.
[1]
[2]
[3]
[4]
[5]
[6]
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to variations in the conduction band tail breadth, γc , all other
parameters being held fixed at their nominal a-Si:H values. We
found that when the conduction band tail is narrower than the
valence band tail, its role in shaping the corresponding spectral
dependence of the JDOS function is relatively minor. This justifies
the use of a simplified empirical model for the DOS functions
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