Download Extraction Current Transients: New Method of Study of Charge

VOLUME 84, NUMBER 21
PHYSICAL REVIEW LETTERS
22 MAY 2000
Extraction Current Transients: New Method of Study of Charge Transport
in Microcrystalline Silicon
G. Juška, K. Arlauskas, and M. Viliūnas
Department of Solid State Electronics, Vilnius University, Sauletekio 9, III K, 2040 Vilnius, Lithuania
J. Kočka
Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 162 53 Prague 6, Czech Republic
(Received 3 August 1999)
The transport properties of microcrystalline silicon, namely, mobility and conductivity, are investigated
by a new method, for which the simple theory as well as numerical modeling is presented. The basic
idea of the new method is verified on amorphous hydrogenated silicon by comparison with the widely
used time-of-flight method. Contrary to time of flight, the new method can be used even for relatively
conductive materials. Preliminary results on microcrystalline silicon clearly indicate the critical role of
amorphouslike tissue in transport in microcrystalline silicon.
PACS numbers: 72.20.Fr, 71.55.Jv, 72.20.Jv
Time of flight (TOF) is a basic method for evaluation
of the charge carrier drift mobility [1,2] in low mobility
materials like organic semiconductors and amorphous hydrogenated silicon (a-Si:H). While in a-Si:H the measurement and interpretation of TOF are straightforward, for
microcrystalline silicon (mc-Si:H) the TOF measurement
encounters many problems related to its heterogeneous
structure and mainly relatively high dark conductivity.
One TOF limitation (necessary for the prevention of
electric field redistribution) is that the material dielectric
relaxation time (ts ) should be longer than the delay
time between the application of voltage and light pulse
(tD ). Relatively high room temperature bulk conductivity
(s ⬵ 1025 V 21 cm21 ) and so short ts indicate violation
of this TOF limitation. The capacitance higher than the
geometrical one even in the MHz range represents the
proof of redistribution of the applied electric field and its
concentration to the contact regions [3,4] (so that TOF
evaluated drift mobility is expected to be overestimated).
This is also evident from the increasing charge collection
with the decrease of the delay time from usually tD 艐
20 ms to tD 艐 50 ns [3]. There is a second, even more
important limitation of TOF, which is also difficult to
fulfill for conductive materials—ts should be larger than
the small signal transit time (ttr ). If this limitation is not
fulfilled the number of equilibrium carriers is sufficient
for significant redistribution of electric field within time
shorter than ttr . Moreover, the package of drifting charge
disappears before its arrival to the opposite electrode.
In this Letter, we present the basic idea of the new
method, based on equilibrium charge carrier extraction,
together with the simple theory for evaluation of the drift
mobility and bulk conductivity, applicable also to high conductivity materials. We have used device grade a-Si:H,
the transport properties of which are well known for the
verification test and present first experimental results on
mc-Si:H. These results together with the study of ac conductivity can substantially contribute to the understanding
4946
0031-9007兾00兾84(21)兾4946(4)$15.00
of mechanism of transport in mc-Si:H, complicated by heterogeneity and even anisotropy [4].
The a-Si:H sample used in this study was 2 mm thick
p-i-n junction, based on device-grade a-Si:H prepared at
250 ±C by 13 MHz glow discharge decomposition of pure
SiH4 . The 5.6 mm thick mc-Si:H was prepared at 220 ±C
by 130 MHz decomposition of 5% SiH4 in a SiH4 1 H2
mixture on ZnO covered glass, with top NiCr contact.
The basic idea of the new method is illustrated in Fig. 1.
We apply two consecutive pulses of linearly increasing
FIG. 1. Schematic illustration of the CELIV method. U is the
form of applied voltage to the sample, j is the corresponding
current transient calculated for ts 苷 ttr (see text). Full line (a)
corresponds to ideal contact blocking and no recovery; dashed
line (b) is calculated for partial equilibrium recovery (up to 13
of normal). The inset illustrates schematically the band diagram
of the typical device.
© 2000 The American Physical Society
VOLUME 84, NUMBER 21
PHYSICAL REVIEW LETTERS
voltage and follow current transients related to extraction of equilibrium carriers. We call this method “carrier
extraction by linearly increasing voltage” (CELIV). Extracted charge is equal to the difference of these two transients for sufficiently small delay 共td 兲. Increase of the
delay between the pulses allows the study of the recovery of equilibrium. The second pulse is also important for
identification of nonideal conditions (for example, nonperfectly blocking contact).
Contrary to TOF we investigate extraction of the equilibrium carriers instead of drift of the photogenerated package of charge (and so many TOF problems are excluded).
The measuring setup is extremely simple—just an oscilloscope, the sample, and a function generator. Moreover, for
our linearly increasing voltage there is no big initial spike
of displacement current (typical for TOF as a result of application of steplike voltage before the light pulse), which
usually complicates the measurement of small charge transients in conductive materials.
For calculation of extraction current transients we combine the continuity, current, and Poisson equations. To
simplify the theory, let us assume that we have a sample
of thickness d (see Fig. 1 inset) with a completely blocking electrode at position x 苷 0 and that the equilibrium
free carrier density is n. As a result of the application of
linearly increasing voltage carriers are at time t extracted
up to the “extraction depth” l共t兲, where 0 , l共t兲 , d.
Integration of the Poisson equation through interelectrode distance 共x兲 gives us the extracted charge Q共t兲 as
Q共t兲
enl共t兲
苷
苷 E共0, t兲 2 E共d, t兲 ,
(1)
´´0
´´0
where ´ and ´0 are relative and absolute permittivity, respectively; E共0, t兲 and E共d, t兲 are electric field at position 0
(blocking electrode) and d (back electrode), respectively.
By integration of the continuity equation through x
we have
dQ共t兲
dl共t兲
苷 en
苷 jd 苷 sE共d, t兲 ,
(2)
dt
dt
where conductivity s 苷 enm (m is the mobility).
The electric field linearly decreases in the depletion (carrier extracted) region [0 , x , l共t兲] as
enx
E共x, t兲 苷 E共0, t兲 2
,
(3)
´´0
and it is coordinate independent and equal to E共d, t兲
at l共t兲 , x , d. Then for linearly increasing voltage
22 MAY 2000
U共t兲 苷 At
Z d
E共x, t兲 dx 苷 At
0
苷 E共d, t兲d 1
E共0, t兲 2 E共d, t兲
l共t兲 .
2
(4)
By combination of Eqs. (1), (2), and (4) we obtain the
Riccati equation for l共t兲 in the form
s
dl共t兲
mAt
1
l 2 共t兲 苷
.
dt
2´´0 d
d
(5)
Then by averaging through x of the general expression for
current
j共t兲 苷 ´´0
dE共x, t兲
1 s共x, t兲E共x, t兲 ,
dt
(6)
we obtain current transient
j共t兲 苷
苷
´´0 A
s Z d
1
E共d, t兲 dx
d
d l共t兲
sE共d, t兲 关d 2 l共t兲兴
´´0 A
1
.
d
d
(7)
Substitution for E共d, t兲 from Eq. (2) and use of Eq. (5)
give us the expression for transient
µ
∂µ
∂
´´0 A
s
l共t兲 mAt
s
j共t兲 苷
1
12
2
l 2 共t兲 .
d
m
d
d
2´´0 d
(8)
Current transient calculated following Eqs. (5) and (8) is
shown by the full line in Fig. 1. For the nonideal case,
when blocking is not perfect or partial recovery of equilibrium appears, the expected transients are illustrated by the
dashed line in Fig. 1.
From the initial slope of the transient we can evaluate
bulk conductivity
Ç
d关 j共t兲兾j共0兲兴
s 苷 ´´0
.
(9)
dt
t苷0
From the initial step—j共0兲 苷 共´´0 A兲兾d —also permittivity can be evaluated.
For low conductivity material, when ts 苷 ´´0 兾s ¿
ttr (where ttr corresponds to the full interelectrode distance
transit time of small charge in the case of linearly increasing voltage) the extracted charge negligibly changes the
electric field distribution and we can simplify Eq. (8) with
the help of Eq. (5) as [5]
s
∑
∂∏
µ
2
A
mAt 2
´´0 1 st 1 2
苷 ttr ,
for t , d
j共t兲 苷
2
d
2d
mA
(10)
A
´´0 苷 j共0兲
j共t兲 苷
for t . ttr .
d
Thus, the mobility can be evaluated from the maximum current s
time
2
ttr
tmax 苷 p 苷 d
(11)
3mA
3
4947
VOLUME 84, NUMBER 21
PHYSICAL REVIEW LETTERS
and Dj 苷 j共tmax 兲 2 j共0兲 is
2 ttr
Dj
苷 p
, 1.
j共0兲
3 3 ts
(12)
For high conductivity or slow voltage increase (small
A), when ts ø ttr , s
s
2
2
3 ts ttr
3 ts d
tmax 苷
苷
.
(13)
2
mA
In this case Dj . j共0兲 and evaluation of m on the basis of
Eq. (13) is easy, while by TOF it is impossible.
Up to now we have neglected the existence of traps,
typical for a-Si:H and mc-Si:H. There is a large number of
traps in these materials; fortunately the shallow and deep
traps are relatively well separated. As a consequence of
inclusion of traps instead of microscopic mobility the m
means so called drift mobility and n is the concentration
of the carriers thermalized within shallow traps.
An important condition for correct TOF measurement is
that ttr should be smaller than carrier deep capture time
(tC ). For comparison of m evaluated from tmax of our
CELIV method we have to use enough high A to fulfill the
same condition ttr , tC .
In Fig. 2(a) there is the current transient corresponding
to the first CELIV pulse, measured on a-Si:H, for which
22 MAY 2000
we have used homogeneous illumination to increase its
relatively low conductivity. For mc-Si:H which is enough
conductive the CELIV measurements have been done in
the dark, without illumination. We have used undoped
mc-Si:H (it means lightly n-type) and so electrons dominate transport. Since in a-Si:H the mobility of holes is at
least 100 times smaller than mobility of electrons, holes are
considered immobile during the extraction of electrons and
their charge is included into the calculated redistribution
of the electric field. For a-Si:H the CELIV evaluated m 苷
0.7 cm2 兾V s of electrons agree very well with TOF results.
Surprisingly when we decreased A [see Fig. 2(b)] the second peak also appeared in addition to the first.
For understanding the origin of this peak it is important to emphasize the differences between TOF and our
method. In TOF the package of nonequilibrium carriers
thermalized in shallow traps is extracted. In our method
the first peak is related to the extraction of the equilibrium
free (shallow trapped) carriers and with increasing time of
experiment the carriers from deeper and deeper states are
extracted (as well as in post-transit TOF), and this leads to
the second CELIV peak.
While for two extreme cases (ts ø ttr and ts ¿ ttr )
a simple analytical solution was found, for the intermediate case and inclusion of traps the numerical modeling is
necessary.
In Fig. 3 there are the results of modeling without
trapping and with single trap level. For high and low A
the modeling very well reproduces tmax 共A兲 艐 A20.5 and
A20.33 , predicted by Eqs. (11) and (13). When trapping is
included then in both limiting cases the same expressions
for tmax and Dj like without trapping are obtained if
we substitute m by 共mf兲, where f is the trapping factor
102
∆ j, j(0), tmax
j(0)
101
tmax
A-1/3
100 1/f = 100
10-1
no trapping
j
10-2
10-3
10-2
10-1
100
A-1/2
101
102
A
FIG. 2. The current transient corresponding to the first pulse of
the CELIV method, measured under homogeneous illumination
at room temperature on a 2 mm thick a-Si:H sample in case (a)
with A 苷 2 V兾300 ns and in case ( b) with A 苷 5 V兾2.5 ms.
4948
FIG. 3. Numerical modeling of Dj, j共0兲, and tmax as a function
of A (voltage slope). A 苷 1 corresponds to the case ttr 苷 ts .
Bold lines correspond to the case with the included trapping
(tC 苷 1, tR 苷 100); full lines represent the case without trapping. Current is normalized to j共0兲 at A 苷 1, and time is normalized to ts .
VOLUME 84, NUMBER 21
PHYSICAL REVIEW LETTERS
f 苷 共nshallow 兲兾共nshallow 1 ndeep 兲, or for single trap f 苷
tC 兾共tC 1 tR 兲, where tR is the release time.
When trapping is included two maximums appear at
ttr 艐 tC (see Fig. 3). In standard TOF charge collection
this corresponds to a transition from the linear part of
Hecht’s formula to the saturation.
In Fig. 4 there are tmax measured for a-Si:H and
mc-Si:H, plotted for a wide range of A. The features,
predicted by the modeling are clearly expressed. Certain
deviations (of the slope, for example) are due to the inclusion of only the single trap level, which cannot precisely
account for the influence of a rather complex tail and deep
states distribution.
In the same Fig. 4 the m共A兲 is plotted for a-Si:H and
mc-Si:H. In the high A region for the same mc-Si:H sample
m共CELIV兲 ⬵ 0.8 cm2 兾V s is lower than m共TOF兲 ⬵
2 3 cm2 兾V s [3], overestimated even for tD , 100 ns
[3]. In both materials with decreasing A the m共CELIV兲
decreases. This demonstrates that drift mobility is generally time dependent [6] and only in some cases and in
a limited time window (for example, in TOF at t , tC )
the m is constant. With the help of our method we can
evaluate not only TOF equivalent m but also “quasistatic”
drift mobility, influenced by deep trapping factor.
An important task for potentially anisotropic mc-Si:H
[4] is the evaluation of bulk conductivity perpendicular
to the substrate [s共⬜兲], not influenced by contact barriers. For sufficiently high A the initial slope of the current
transient [see Eq. (9)] gives us for mc-Si:H the room temperature value s共⬜兲 艐 3 3 1025 V 21 cm21 . This value
corresponds well to the high frequency ac conductivity
“plateau” value [4] used as an alternative way how to exclude the contact barriers and find s共⬜兲.
In conclusion, we have presented a new method, the so
called carrier extraction by linearly increasing voltage, for
evaluation of bulk transport properties of materials in the
sandwich geometry. The basic advantages of the CELIV
method are as follows: (i) the possibility to evaluate drift
mobility and bulk conductivity (not influenced by contact barriers of sandwich sample) of relatively highly conductive materials, (ii) great simplicity and applicability to
a wide class of materials (heterogeneous, organic, etc.),
and (iii) the possibility to evaluate from time of current
maximum(s) not only TOF equivalent drift mobility but
also quasistatic drift mobility, influenced by the trapping
factor f共t兲.
There is one disadvantage in comparison with
TOF—only majority carriers can be studied.
We have presented the simple theory as well as results
of numerical modeling, which takes in the simplest form
into account trapping and very well describe the preliminary results on a-Si:H and mc-Si:H. One important conclusion, evident from Fig. 4, is that transport in mc-Si:H is
very similar to a-Si:H and so we can say that a-Si:H-like
“tissue,” surrounding c-Si grains controls the transport
properties of mc-Si:H.
22 MAY 2000
FIG. 4. The current maximum time 共tmax 兲 and drift mobility
共m兲 as a function of voltage slope (A) evaluated from the measured current transients by the CELIV method at room temperature on a-Si:H (duration of pulses was changed while Umax was
fixed at 1.5 V), and mc-Si:H (duration of pulses was changed
while Umax was fixed at 5 V) samples (see text). The dashed
lines mark the value of A, at which two maximums of current
transient were equal.
Recently we have tested the CELIV method also on
crystalline Si, for example, for lightly p-type Si we have
obtained mobility of holes 290 cm2 兾V s. This value
is close to the expected value (hole mobility around
400 cm2 兾V s). More results and a detailed discussion will
be published elsewhere.
We thank Professor A. Shah’s group from Neuchatel
University for the mc-Si:H sample, P. Šnajdrová for
manuscript preparation, and Czech Academic Grant
No. A1010809, Lithuanian VMSF Grant No. 375, and
NEDO contract for partial financial support.
[1] W. E. Spear, J. Non-Cryst. Solids 59 & 60, 1 (1983).
[2] J. Kočka, in Proceedings of International School on Condensed Matter Physics Varna ’92, edited by J. M. Marshall,
N. Kirov, and A. Vavrek (World Scientific, Singapore,
1992), p. 129.
[3] G. Juška, K. Arlauskas, K. Genevičius, and J. Kočka,
Mater. Sci. Forum 297 – 298, 327 (1999).
[4] J. Kočka, A. Fejfar, B. Rezek, A. Poruba, M. Vaněček,
P. Torres, J. Meier, N. Wyrsch, A. Shah, and A. Matsuda, in
Proceedings of the 2nd World Conference on Photovoltaic
Solar Energy Conversion, Vienna, 1998 (EC, Ispra, 1998),
p. 785.
[5] A. Petravičius, G. Juška, and R. Baubinas, Sov. Phys.
Semicond. 9, 1530 (1976).
[6] J. Kočka, C. E. Nebel, and C. D. Abel, Philos. Mag. B 63,
221 (1991).
4949