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Schottky barrier profiles in amorphous silicon-based materials
Journal of Non-Crystalline Solids 35 & 36 (1980) 731-736 ©North-Holland Publishing Company
SCHOTTKY BARRIER PROFILES IN AMORPHOUSSILICON-BASED MATERIALS *
.+
M. Shur , W. Czubaty3 , and A. Madan+ *University of Minnesota Minneapolis, Minnesota 55455 U.S.A. +Energy Conversion Devices, inc. 1675 West Maple Road Troy, Michigan 48084 U.S.A.
The localized state density for a-Si:H and a-Si:H:F can be reasonably approximated by a hyperbolic function. Solution of Poisson's equation leads to potential profiles in the depletion region. The predicted width of the depletion region is in reasonable agreement with the results obtained from C-V experiments. INTRODUCTION Recent studies, 1'2 have shown that amorphous Si:H:F alloys can be fabricated using radio frequency glow discharge in SiF4 and H2. These materials have been shown to possess a low density of localized states and high photoconductivity. They are devoid of undesirable photostructural changes, and can readily be made p type or n type. Becauseof their properties, these alloys could be useful in photovoltaic device applications. In the photovoltaic action of amorphous silicon-based alloys, the main component of the short-circuit current is due to d r i f t within the depletion region rather than diffusion because of the small diffusion lengths of holes.3 Therefore, the collection efficiency of this type of cell is linked to the width of the depletion region which is determined by the barrier height, by the density of states, and by the Fermi level position which is controlled by the doping. In this paper we would l i k e to present a simple analytical model which relates Fermi level position to the doping density and which allows us to calculate the width of the depletion region and the C-V characteristics exhibited by a-Si:H:F and a-Si:H devices. Our results are in good agreementwith the computer calculations performed for Si:H by Spear, Le Comber, and Snell,4 with their experimental data for a-Si:H devices, and also with our own experimental data for a-Si:H:F samples. EXPERIMENTAL DETAILS We have used the radio frequency glow discharge apparatus. The premixed SiF4 and H2 gases were fed into the system at a constant rate, and the pressure was maintained at - I t o r r during deposition. The plasma was generated between plates using an r . f . discharge operated at 13.56 MHz. The doping was achieved by admitting a controlled amount of PH3 into the reaction chamber. The samples for the studies of the Fermi level position versus doping were deposited on heated glass substrates with SiF4/H2 = 10/1. The samples were furnished with evaporated Al contacts in a coplanar configuration with a I mmgap. Linear I-V characteristics for fields up to 104 V cm-1 were observed. The dark conductivity (oD) as a function of temperature was measured in a vacuum better than 10-5 t o r r . The activation energy, AE, was then associated with Ecb-EF for T = OOK, where Ecb is the bottom of the conduction band and EF is the position of the Fermi level. 731
732
m. Shut et al. / Schottky Barrier Profiles
For the C-V measurements, samples in a sandwich configuration were used. Using 500 ppm of PHR in the premix of SiF4/H2=IO/I, a heavily doped n+ layer (approximately 200~t~ick) was f i r s t deposited onto a molybdenum, stainless steel, or nichrome substrate to provide an ohmic contact. This deposition was followed by the deposition of the active material (0.5 to lum thick). F i n a l l y , a 20% transmissive Au contact was evaporated on top. DESCRIPTION OF THE MODEL We assume that the density of states can be approximated by g(E) - groin 2 exp ( E_E ~_ ) + groin 2 expC~
)
(1)
Here, the energy E is measured from the minimum of the density of states, Ech is a characteristic energy, and gmin is the minimum density of states. As shown below, the general agreement between the analytical approximation and precise curve is s u f f i c i e n t to account for the properties of the depletion layer. The depletion width, C-V characteristics, etc. calculated in the frame of the analyt i c a l model agree well with the computer calculations of Spear et al. 3 who used the actual experimental density of states curves. In eq. ( i ) we assume that localized states are composed of donor states given by the f i r s t term and acceptor states 9iven by the second term. The positive and negative charge densities n- and pt can be then deduced from Fermi integrals. These densities depend weakly on temperature since the characteristic energy Ech (~ 70-100 meV) is considerably larger than kT. The calculation of the net charge density in the zero temperature l i m i t yields + p - n = -gmin Ech Sinh XF (2) where the dimensionless s h i f t of the Fermi level position is defined by XF -
EF - EFo Ech
(3)
I f the sample is doped with donor concentration ND, the Fermi level s h i f t XF found from the charge n e u t r a l i t y condition is given by XF =
Sinh - I
where
ND Ech gmin
Sinh-l(n)
n _
ND Ech gmin represents the dimensionless doping density.
(4) (5)
~F versus q is shown in Figure 1 where i t is compared with the results presented In reference 4 and with our experimental measurements. THE SHAPE AND WIDTH OF THE SCHOTTKYBARRIER The shape and the width of the barrier can be analyzed by solving Poisson's equation. The solution of the Poisson's equation describing the v a r i a t i o n of the bot tom of the conduction band with distance in a dimensionless form is given by V
I ~
dXc
(6)
W - 3 / ~ ~P [nX c + Cosh (X F - Xc) - Cosh (XF~½ XT where W = } / Z o is the dimensionless coordinate and Zo = (~os/92gmin)2, Xc =EGh/ Ech, XT
ET/Ech, ET = kT.
For the i n t r i n s i c
material (~ = 0), the inzegral in
733
M~ Shut et al. / Schottky Barrier Profiles
eq, (6) can be c a l c u l a t e d a n a l y t i c a l l y , The shape of the b a r r i e r p r o f i l e f o r d i f f e r e n t doping d e n s i t i e s i s shown in Figure 2 where we p l o t Xc versus W.
,
/
4
4-
+
XF ~
2
++ !
0
t I
0.1
--
l 10
. . . .
I 100
. 1000
Ti
Fig.I--Dimensionless Fermi l e v e l XF versus dimensionless doping d e n s i t y n. XF is zero f o r i n t r i n s i c m a t e r i a l s . + is data of Spear et a l . 3 f o r a-Si:H. 0 i s experimental data f o r a-Si:H:F.
! 10
,ci .5
4
3
2
I
_
_
L
-
~ 0.5
I
I
]
I
1,5
2
W
Fig.2--Dimensionless Schottky b a r r i e r p r o f i l e .
21, ¸
734
M. Shur et al. / Schottkg Barrier Profiles
As can be seen from eq. (6), the barrler wldth is slmply proportional to (g . ) 2 because• Z0 o<(g i~_)-½ For a-Si'H ~aterial. g-<~ = 1" 5x1017cm-3eV-1 and Zq - -~SR" n. " " ~ H • ' for a-SI:H:F ma~erlal, gmin = 2.1016cm-3eV-~ leading to Zo=1710~. We estlmate the depletion width to be of the order of 1500~and 4000~ for a-Si:H and a-Si:H:F materials, respectively (see Figure 4). This estimate indicates a potential superiority of the a-Si:H:F based devices. C-V CALCULATIONS C-V measurements can provide valuable information about the density of states and the width of the depletion layer. As in reference 4, we w i l l distinguish between two different situations, the l i m i t i n g case of low frequencies and the high frequency situation. In the former case, the space charge density within the depletion region related to the deep levels has sufficient time to respond to the voltage variations; in the l a t t e r case, the space charge distribution is frozen and the response to the applied voltage comes from the mobile carriers in the semiconductor at the boundary of the depletion layer. At low frequencies the capacitance CLF can be expressed as follows: n - Sinh (XF - Xs) CLF = Cch
(7)
o
# T [ n X s + Cosh (X F - Xs) - Cosh XJ
where
Es Eso qV ~ ~ ½ . . . u, u = : (8) Ech Ech Ech ' Cch (q o gmin ) Using the values of g~in above, we find Cch = 1.52xlO-3F/m2 for the a-Si:H and Cch : O.558xlO-3F/m 2 ~or the a-Si:H:F materials. xs .
In Figure 3 the calculation in the frame of our model (for ~ = O) is compared with the results of the computer calculations and low frequency measurements presented in reference 4. Such a comparison shows a reasonable agreement.
C (1()~ F/m = )
I
V (volts)
Fig. 3--Low frequency C-V charact e r i s t i c s for the silane device• Solid line is computer calculation4 for Ec-EF = .65eV. + is experimental results.4 Dashed line is analytical calculation.
735
M. Shut et alo / Schottky Barrier Profiles
As mentioned above, at high frequencies the space charge density in the depletion region does not respond to the ac voltage v a r i a t i o n . Thus, the depletion region behaves l i k e a charged d i e l e c t r i c of thickness WB sandwiched between two p a r a l l e l plates. Therefore, the capacitance per u n i t of cross sectional area at high f r e quency CHF is given by CHF :
CoE 1 ~ Z B - ~ ~ Cch WBB
(9)
where ~ is a numerical constant of the order of 1. Deviation from ~ =I s i g n i f i e s a nonparabolic potential p r o f i l e . Our c a l c u l a t i o n shows that for low and i n t e r mediate doping l e v e l s , the capacitance is much lower at high frequency than at low frequency. Also, the capacitance at high frequency is almost independent of the bias. At large doping d e n s i t i e s , CHF and CLF approach each other because the c o n t r i b u t i o n of the l o c a l i z e d states to the space charge becomes less important and is masked by the charge density of the ionized donors. C-V MEASUREMENTS C-V measurements corresponding to the high frequency l i m i t were performed on various samples whose a c t i v a t i o n energy ranged from O.7eV to O.39eV. In these experiments the in-phase (R) and the out-of-phase (X) components of the impedance of the samples were measured. The results were i n t e r p r e t e d using the equivalent c i r c u i t diagram of the device. Using eq. (9), we can determine the depletion width from the C-V measurements at high frequency. In Figure 4 we show the v a r i a t i o n of WR with the p o s i t i o n of EF. Included in that f i g u r e are some experimental points f o r the depletion width determined from the high frequency C-V measurements which provide a reasonable agreement between the theory and experiment. In a d d i t i o n , we show the points taken from Spear et a l . 4 which show a reasonable agreement with the a n a l y t i c a l model for a-Si:H a l l o y . z
,.< 0 0 0
I
i
i
3
)<
° ~ a - S i : H
=:2 I-G
:F o
I¢
N 1 Ig
a-Si : H-~
~"...~
~
m 0
.7
I
.6
I
.5 Ecb- E r
I
.4
[
.3
(eV)
Fig. 4--The b a r r i e r width versus the Fermi level for a-Si:H:F and a-Si:H a l l o y s . The c i r c l e s are experimental data points for a-Si:H:F; the crosses represent experimental data of Spear et a l . 4
736
M. Shut et al. / Schottkg Barrier Profiles
CONCLUSION In the above we have suggested a simple a n a l y t i c a l model which assumes a simple density of states spectrum. The model predicts the p o s i t i o n of the Fermi level as a function of doping, the shape and the width of the depletion l a y e r , and the C-V c h a r a c t e r i s t i c s at low and high frequencies in amorphous materials. This anal y t i c a l theory is in reasonable agreement with the computer c a l c u l a t i o n s performed by Spear et a l . 4 with C-V measurements at low and high frequencies, and measurements of the Fermi level p o s i t i o n as a function of doping in a-Si:H and a-Si:H:F materials. Accordlng to t h l s model the depletlon wldth Is proportlonal to gmin 2. Our results suggest that the depletion width can reach~1500~and -4000~ f o r a-Si:H and a-Si:H:F m a t e r i a l s , r e s p e c t i v e l y . The s i m p l i c i t y of t h i s a n a l y t i c a l model makes i t s u i t a b l e f o r c a l c u l a t i o n of c o l l e c t i o n e f f i c i e n c y and other device parameters and f o r a computer-aided design of amorphous solar c e l l s . REFERENCES [i]
Ovshinsky, S.R. and Madan, A., Nature 276 (1978) 482-484.
[2]
Madan, A., Ovshinsky, S.R., and Benn, E. - Phil. Mag., In Press.
[3]
Carlson, D.E. and Wronski, C.R., J. Electr. Mater. 6 (1977) 95406.
[4]
Spear, W.E., Le Comber, P.G., and Snell, A . J . , Phil. Mag., In Press.
SCHOTTKY BARRIER PROFILES IN AMORPHOUSSILICON-BASED MATERIALS *
.+
M. Shur , W. Czubaty3 , and A. Madan+ *University of Minnesota Minneapolis, Minnesota 55455 U.S.A. +Energy Conversion Devices, inc. 1675 West Maple Road Troy, Michigan 48084 U.S.A.
The localized state density for a-Si:H and a-Si:H:F can be reasonably approximated by a hyperbolic function. Solution of Poisson's equation leads to potential profiles in the depletion region. The predicted width of the depletion region is in reasonable agreement with the results obtained from C-V experiments. INTRODUCTION Recent studies, 1'2 have shown that amorphous Si:H:F alloys can be fabricated using radio frequency glow discharge in SiF4 and H2. These materials have been shown to possess a low density of localized states and high photoconductivity. They are devoid of undesirable photostructural changes, and can readily be made p type or n type. Becauseof their properties, these alloys could be useful in photovoltaic device applications. In the photovoltaic action of amorphous silicon-based alloys, the main component of the short-circuit current is due to d r i f t within the depletion region rather than diffusion because of the small diffusion lengths of holes.3 Therefore, the collection efficiency of this type of cell is linked to the width of the depletion region which is determined by the barrier height, by the density of states, and by the Fermi level position which is controlled by the doping. In this paper we would l i k e to present a simple analytical model which relates Fermi level position to the doping density and which allows us to calculate the width of the depletion region and the C-V characteristics exhibited by a-Si:H:F and a-Si:H devices. Our results are in good agreementwith the computer calculations performed for Si:H by Spear, Le Comber, and Snell,4 with their experimental data for a-Si:H devices, and also with our own experimental data for a-Si:H:F samples. EXPERIMENTAL DETAILS We have used the radio frequency glow discharge apparatus. The premixed SiF4 and H2 gases were fed into the system at a constant rate, and the pressure was maintained at - I t o r r during deposition. The plasma was generated between plates using an r . f . discharge operated at 13.56 MHz. The doping was achieved by admitting a controlled amount of PH3 into the reaction chamber. The samples for the studies of the Fermi level position versus doping were deposited on heated glass substrates with SiF4/H2 = 10/1. The samples were furnished with evaporated Al contacts in a coplanar configuration with a I mmgap. Linear I-V characteristics for fields up to 104 V cm-1 were observed. The dark conductivity (oD) as a function of temperature was measured in a vacuum better than 10-5 t o r r . The activation energy, AE, was then associated with Ecb-EF for T = OOK, where Ecb is the bottom of the conduction band and EF is the position of the Fermi level. 731
732
m. Shut et al. / Schottky Barrier Profiles
For the C-V measurements, samples in a sandwich configuration were used. Using 500 ppm of PHR in the premix of SiF4/H2=IO/I, a heavily doped n+ layer (approximately 200~t~ick) was f i r s t deposited onto a molybdenum, stainless steel, or nichrome substrate to provide an ohmic contact. This deposition was followed by the deposition of the active material (0.5 to lum thick). F i n a l l y , a 20% transmissive Au contact was evaporated on top. DESCRIPTION OF THE MODEL We assume that the density of states can be approximated by g(E) - groin 2 exp ( E_E ~_ ) + groin 2 expC~
)
(1)
Here, the energy E is measured from the minimum of the density of states, Ech is a characteristic energy, and gmin is the minimum density of states. As shown below, the general agreement between the analytical approximation and precise curve is s u f f i c i e n t to account for the properties of the depletion layer. The depletion width, C-V characteristics, etc. calculated in the frame of the analyt i c a l model agree well with the computer calculations of Spear et al. 3 who used the actual experimental density of states curves. In eq. ( i ) we assume that localized states are composed of donor states given by the f i r s t term and acceptor states 9iven by the second term. The positive and negative charge densities n- and pt can be then deduced from Fermi integrals. These densities depend weakly on temperature since the characteristic energy Ech (~ 70-100 meV) is considerably larger than kT. The calculation of the net charge density in the zero temperature l i m i t yields + p - n = -gmin Ech Sinh XF (2) where the dimensionless s h i f t of the Fermi level position is defined by XF -
EF - EFo Ech
(3)
I f the sample is doped with donor concentration ND, the Fermi level s h i f t XF found from the charge n e u t r a l i t y condition is given by XF =
Sinh - I
where
ND Ech gmin
Sinh-l(n)
n _
ND Ech gmin represents the dimensionless doping density.
(4) (5)
~F versus q is shown in Figure 1 where i t is compared with the results presented In reference 4 and with our experimental measurements. THE SHAPE AND WIDTH OF THE SCHOTTKYBARRIER The shape and the width of the barrier can be analyzed by solving Poisson's equation. The solution of the Poisson's equation describing the v a r i a t i o n of the bot tom of the conduction band with distance in a dimensionless form is given by V
I ~
dXc
(6)
W - 3 / ~ ~P [nX c + Cosh (X F - Xc) - Cosh (XF~½ XT where W = } / Z o is the dimensionless coordinate and Zo = (~os/92gmin)2, Xc =EGh/ Ech, XT
ET/Ech, ET = kT.
For the i n t r i n s i c
material (~ = 0), the inzegral in
733
M~ Shut et al. / Schottky Barrier Profiles
eq, (6) can be c a l c u l a t e d a n a l y t i c a l l y , The shape of the b a r r i e r p r o f i l e f o r d i f f e r e n t doping d e n s i t i e s i s shown in Figure 2 where we p l o t Xc versus W.
,
/
4
4-
+
XF ~
2
++ !
0
t I
0.1
--
l 10
. . . .
I 100
. 1000
Ti
Fig.I--Dimensionless Fermi l e v e l XF versus dimensionless doping d e n s i t y n. XF is zero f o r i n t r i n s i c m a t e r i a l s . + is data of Spear et a l . 3 f o r a-Si:H. 0 i s experimental data f o r a-Si:H:F.
! 10
,ci .5
4
3
2
I
_
_
L
-
~ 0.5
I
I
]
I
1,5
2
W
Fig.2--Dimensionless Schottky b a r r i e r p r o f i l e .
21, ¸
734
M. Shur et al. / Schottkg Barrier Profiles
As can be seen from eq. (6), the barrler wldth is slmply proportional to (g . ) 2 because• Z0 o<(g i~_)-½ For a-Si'H ~aterial. g-<~ = 1" 5x1017cm-3eV-1 and Zq - -~SR" n. " " ~ H • ' for a-SI:H:F ma~erlal, gmin = 2.1016cm-3eV-~ leading to Zo=1710~. We estlmate the depletion width to be of the order of 1500~and 4000~ for a-Si:H and a-Si:H:F materials, respectively (see Figure 4). This estimate indicates a potential superiority of the a-Si:H:F based devices. C-V CALCULATIONS C-V measurements can provide valuable information about the density of states and the width of the depletion layer. As in reference 4, we w i l l distinguish between two different situations, the l i m i t i n g case of low frequencies and the high frequency situation. In the former case, the space charge density within the depletion region related to the deep levels has sufficient time to respond to the voltage variations; in the l a t t e r case, the space charge distribution is frozen and the response to the applied voltage comes from the mobile carriers in the semiconductor at the boundary of the depletion layer. At low frequencies the capacitance CLF can be expressed as follows: n - Sinh (XF - Xs) CLF = Cch
(7)
o
# T [ n X s + Cosh (X F - Xs) - Cosh XJ
where
Es Eso qV ~ ~ ½ . . . u, u = : (8) Ech Ech Ech ' Cch (q o gmin ) Using the values of g~in above, we find Cch = 1.52xlO-3F/m2 for the a-Si:H and Cch : O.558xlO-3F/m 2 ~or the a-Si:H:F materials. xs .
In Figure 3 the calculation in the frame of our model (for ~ = O) is compared with the results of the computer calculations and low frequency measurements presented in reference 4. Such a comparison shows a reasonable agreement.
C (1()~ F/m = )
I
V (volts)
Fig. 3--Low frequency C-V charact e r i s t i c s for the silane device• Solid line is computer calculation4 for Ec-EF = .65eV. + is experimental results.4 Dashed line is analytical calculation.
735
M. Shut et alo / Schottky Barrier Profiles
As mentioned above, at high frequencies the space charge density in the depletion region does not respond to the ac voltage v a r i a t i o n . Thus, the depletion region behaves l i k e a charged d i e l e c t r i c of thickness WB sandwiched between two p a r a l l e l plates. Therefore, the capacitance per u n i t of cross sectional area at high f r e quency CHF is given by CHF :
CoE 1 ~ Z B - ~ ~ Cch WBB
(9)
where ~ is a numerical constant of the order of 1. Deviation from ~ =I s i g n i f i e s a nonparabolic potential p r o f i l e . Our c a l c u l a t i o n shows that for low and i n t e r mediate doping l e v e l s , the capacitance is much lower at high frequency than at low frequency. Also, the capacitance at high frequency is almost independent of the bias. At large doping d e n s i t i e s , CHF and CLF approach each other because the c o n t r i b u t i o n of the l o c a l i z e d states to the space charge becomes less important and is masked by the charge density of the ionized donors. C-V MEASUREMENTS C-V measurements corresponding to the high frequency l i m i t were performed on various samples whose a c t i v a t i o n energy ranged from O.7eV to O.39eV. In these experiments the in-phase (R) and the out-of-phase (X) components of the impedance of the samples were measured. The results were i n t e r p r e t e d using the equivalent c i r c u i t diagram of the device. Using eq. (9), we can determine the depletion width from the C-V measurements at high frequency. In Figure 4 we show the v a r i a t i o n of WR with the p o s i t i o n of EF. Included in that f i g u r e are some experimental points f o r the depletion width determined from the high frequency C-V measurements which provide a reasonable agreement between the theory and experiment. In a d d i t i o n , we show the points taken from Spear et a l . 4 which show a reasonable agreement with the a n a l y t i c a l model for a-Si:H a l l o y . z
,.< 0 0 0
I
i
i
3
)<
° ~ a - S i : H
=:2 I-G
:F o
I¢
N 1 Ig
a-Si : H-~
~"...~
~
m 0
.7
I
.6
I
.5 Ecb- E r
I
.4
[
.3
(eV)
Fig. 4--The b a r r i e r width versus the Fermi level for a-Si:H:F and a-Si:H a l l o y s . The c i r c l e s are experimental data points for a-Si:H:F; the crosses represent experimental data of Spear et a l . 4
736
M. Shut et al. / Schottkg Barrier Profiles
CONCLUSION In the above we have suggested a simple a n a l y t i c a l model which assumes a simple density of states spectrum. The model predicts the p o s i t i o n of the Fermi level as a function of doping, the shape and the width of the depletion l a y e r , and the C-V c h a r a c t e r i s t i c s at low and high frequencies in amorphous materials. This anal y t i c a l theory is in reasonable agreement with the computer c a l c u l a t i o n s performed by Spear et a l . 4 with C-V measurements at low and high frequencies, and measurements of the Fermi level p o s i t i o n as a function of doping in a-Si:H and a-Si:H:F materials. Accordlng to t h l s model the depletlon wldth Is proportlonal to gmin 2. Our results suggest that the depletion width can reach~1500~and -4000~ f o r a-Si:H and a-Si:H:F m a t e r i a l s , r e s p e c t i v e l y . The s i m p l i c i t y of t h i s a n a l y t i c a l model makes i t s u i t a b l e f o r c a l c u l a t i o n of c o l l e c t i o n e f f i c i e n c y and other device parameters and f o r a computer-aided design of amorphous solar c e l l s . REFERENCES [i]
Ovshinsky, S.R. and Madan, A., Nature 276 (1978) 482-484.
[2]
Madan, A., Ovshinsky, S.R., and Benn, E. - Phil. Mag., In Press.
[3]
Carlson, D.E. and Wronski, C.R., J. Electr. Mater. 6 (1977) 95406.
[4]
Spear, W.E., Le Comber, P.G., and Snell, A . J . , Phil. Mag., In Press.