Thin Solid Films, 148 (1987) 121-125 ELECTRONICS AND OPTICS
121
SOME PROBLEMS ENCOUNTERED DURING THE MEASUREMENT OF THE ACTIVATION ENERGY OF DARK CONDUCTIVITY OF UNDOPED HYDROGENATED AMORPHOUS SILICON FILMS R. MEAUDRE D~partement de Physique des Mat~riaux, Unit~ Associ~e 172 au Centre National de la Recherche Scientifique, Universit( Claude Bernard, Lyon 1, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne C~dex (France) (Received June 17, 1986; accepted September 8, 1986)
The effect of band bending at the interfaces in structures of planar configuration can be overcome by using a sandwich configuration for the transport measurement. However, other difficulties arise owing to the presence of ohmic or injecting contacts. In the case of n +/n/n ÷ or a-Si/a-Si: H/a-Si configurations it was shown that if the film thickness L is smaller than the Debye length L D associated with localized states at the Fermi level the activation energy of conductivity measured in the ohmic region of the J(V) characteristi c is W' = W-2eVo(1-L/2LD) where W is the true activation energy and V0 is the surface potential corresponding to the case where L/2LD >> 1. In the injection regime, easily obtained for low values of applied voltage V (typically 1 0 - ~ - 1 0 - 2 V ) , the measured activation energy is given by W' = W-(2cte/eNL2)V where ct ~ 0.6 and N is the density of states at the Fermi level.
1. INTRODUCTION The conductivity a of undoped hydrogenated amorphous silicon (a-Si:H) films at and above room temperature is generally assumed to be due to transport in extended states above the mobility edge E c in the conduction band. a is given by a = ao exp{--(Ec--Ev)/kT} where ao is the pre-exponential factor *'2 and EF is the Fermi level. In this temperature range where the conductivity is thermally activated, one has cr = aobs e x p ( - W / k T ) where aobs and W are the experimentally observed pre-exponential factor and the activation energy respectively. Conductivity measurements are often performed on structures of planar configuration with two metal electrodes deposited onto the film and the conductivity is measured in a direction parallel to the substrate. It is then well known that, fairly often, because of the bending of the bands on the free surface of the film or at the film-substrate interface, the measured activation energy and pre-exponential factor do not represent the true value expressing the bulk properties of the films 3 5. The effect of band bending at the interfaces can be overcome by using a sandwich configuration for the transport measurement. However, other difficulties
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122
R. MEAUDRE
arise such as a non-negligible contact resistance which may be n o n - o h m i c or the possibility of injecting contacts. The aim of this paper is to consider only the second effect. 2.
ACTIVATION ENERGY A N D PRE-EXPONENTIAL FACTOR OF DARK C O N D U C T I V I T Y IN
S A N D W I C H STRUCTURES
Let us consider the b a n d diagram shown in Fig. 1 for an a-Si:H film at equilibrium in the absence of an external applied voltage. The u n d o p e d film of thickness L is provided with two ohmic or injecting contacts for electrons in the form of n + - a - S i : H (Fig. l(a)) 6'7 or u n h y d r o g e n a t e d a-Si (Fig. l(b)) 8 lo. Because of the bending of the bands the conductivity is a function o f x (Fig. 1):
E~(x)-- E v
~(x) = aoex p
; 77
Ec(x) -- Ev = (E~ -- Ev) b
--
J
(1)
e V(x)
(2)
where ( E c - E F ) b is the value of E ~ - E F in the bulk region of a film of infinite thickness. The bending of the bands is given by the potential V(x) which can be calculated by solving Poisson's equation: d 2 V(x) -
-
q(x) + - -
dx 2
(3)
= 0 ~;
where q(x) = --e2N V(x) is the space charge density and N is the density of states in the gap which is assumed to be spatially and energetically uniform. Equation (3) can be written as d 2 V(x)
V(x)
dx 2
LD 2
-
0
(4)
Current f l o w (
-----
f
EC
eVo / ~rl ÷
EF
eV(x)
~ V o f ~
Ec
n
------
Ev
"
d(x)dx
t
EF
l~-~i " ~-~i:~e~lxl I
- L/2
i
0
%
f
Ev
(a) (b) Fig. I. Band d i a g r a m of o h m i c or injecting contacts for electrons at e q u i l i b r i u m in the absence of an external applied voltage: (a) n +/n (undoped)/n + structure; (b) a-Si/a-Si:H/a-Si structure.
ACTIVATION ENERGY OF C O N D U C T I V I T Y IN
a-Si:H
FILMS
123
where the Debye length LD associated with localized states is given by // /3 "~1/2
LD = ~2e2 J
(5)
Taking the middle of the film as the origin of the x axis the solution ofeqn. (4) is V(x)= Voexp(--2--~D){exp(~DD)+ exp(--~DD) }
(6)
where eVo is the total band bending when the film is of infinite thickness. Using eqns. (1), (2) and (6) the resistivity of the film in a direction perpendicular to the surface electrodes is then given by -1
p(x)= ao
f(Ec--Ev)b] V eVo f L x x exp,- ~ .;exPL--~-exp~--~L--~D){exp(~DD)+exp(--~DD)}I
(7) 2.1. Ohmic regime When the applied voltage is not too high and does not lead to injection of electrons into the bulk (see below) an ohmic behaviour is obtained and the resistance between two identical contacts of unit area is given by R =2
fo
p(x) dx
(8)
- L/2
and the measured conductivity is derived from am = L R - 1. For L/2LD < 1. using eqns. (7) and (8) a m can be expressed as
am = a° exp[-- {(E¢-- EF)b-- 2eV°(1-T L/2LD)}
(9)
and for L/2L o >> 1 we obtain a m = a 0 exp
kT
J
(lO)
7T
(11)
As usual11 (E c -- EF) b =
(Ec -
EF) o --
where (Ec - E v ) 0 is the energy obtained when (E~- E F ) b is extrapolated to T = 0 K. Equations (10) and (11) show that when the influence of the contact is negligible (L ~ 2LD) the measured pre-exponential factor and activation energy of conductivity are aom = aoexp(y/k) and W = ( E e - - E F ) o respectively, whereas eqn. (9), obtained for the case of L/2LD < 1 shows that the pre-exponential factor is unchanged whereas the activation energy becomes
We then see that the measured activation energy can be different from the true
124
R. MEAUDRE
value expressing the bulk properties of the film. Let us take a numerical example. If (E c - - E F ) b takes the two values 0.2 eV and 0.8 eV in the n + and u n d o p e d regions of
the sample respectively, then eVo = 0.6 eV (Fig. l(a)). F o r N = 1021 m -3 eV ~ (see for instance Tiedje et al?2), e = 1 0 - 1 ° F m 1 and L = 1 0 - 6 m one obtains L/2L o ----=0.63 and 2eVo(1-L/2LD) = 0.44eV. F o r N = 5 xl021 m - 3 e V -1, e=10-~°Fm -1 and L = 5 × 1 0 7 m one obtains L/2LD=0.71 and 2eVo ( 1 - L / 2 L D ) = 0 . 3 4 e V . These two examples show that the true value of the activation energy can easily be decreased by about 0.4 eV. It can be seen that the case shown in Fig. l(b) corresponds to lower values o f e V0 than the case shown in Fig. l(a) and leads to a less disturbed situation. However, in case (a) the Fermi level at the n + - u n d o p e d a-Si:H interface is near the conduction b a n d edge of the u n d o p e d film where the density of states can be high. Then according to eqns. (5) and (12) the lowering of the measured activation energy could be reduced.
2.2 Space-charge-limited conduction regime Following L a m p e r t and M a r k 13 and den Boer ~4, S o l o m o n et al. 7 have shown that, as soon as the applied voltage V is higher than 10 2 V~, space-charge-limited c o n d u c t i o n occurs and for L>> LD the current density-voltage characteristic J = f(V) is given by
J = noep~expk~)
(13)
where Vc = eNkT1L2/e, VE = eNkTL2/2e and ~ is a factor varying little from 0.52 to 0.66. no and ~ are the equilibrium density and the mobility of carriers. The density of states below the mobility edge Ec varies as e x p { - ( E c - E ) / k T I } and has the value N at the Fermi level. T is the temperature. F o r N = 5 x l021 m - 3 e V -a, k T a = 0.125 eV (Solomon et al.7), L = 1 0 - 6 m ande=10 l ° F m - l w e o b t a i n V ~ = l V . Then as soon as V > 1 0 - 2 V ~ = 1 0 - 2 V , deviation from ohmic behaviour can be seen. Let us now suppose that we measure the activation energy W' of the conductivity in that part of the characteristic given by eqn. (13). We obtain W'-
k A(ln a) 2~e A(T 1~ - W - e N L ~ V
(14)
where W is the true activation energy of conductivity noe #. Taking a = 0.6, N = 5.1021 m - ~ e V - 1 and L = 10 -6 m we obtain W'(eV) = W ( e V ) - 0.15 V(eV) If V = 2 V then the measured activation energy is 0.3 eV lower than the true activation energy. It is then seen that measurement of the activation energy of conductivity without taking precautions, i.e. without ensuring that we are in the ohmic region of the J(V) characteristic, leads to completely erroneous values. According to eqn. (14), N and W can be derived from the slope of a plot of W' vs. V and the intercept on the W' axis respectively.
ACTIVATION ENERGY OF CONDUCTIVITY IN a - S i : H FILMS
125
3. CONCLUSION
Measurement of the activation energy of conductivity by using ohmic contacts in a sandwich configuration may lead to completely erroneous values, more especially as the films have good properties (low density of states in the gap). Accurate values can be obtained in the ohmic regime if the separation of the electrodes is greater than the Debye length associated with localized states at the Fermi level. Before measurement of the activation energies one must ensure that the ohmic regime is obtained, otherwise the values of activation energies depend strongly on the applied voltage and may be considerably lower than the true values. However, in the case where current measurement in the ohmic regime cannot be performed (if the value of the current is too low for instance) we have shown (cf eqn. 14) that a correct value for the activation energy can be derived from a study of this quantity as a function of the applied voltage. REFERENCES
1 N.F. Mott, Philos. Mag. B, 49(1984)L75;52(1985)19. R. Meaudre, Philos. Mag. B, 51 (1985) L57. M. Tanielan, H. Fritzsche, C. C. Tsai and E. Symbalisty, Appl. Phys. Lett., 33 (1978) 353. I. Solomon, T. Dietl and D. Kaplan, J. Phys. (Paris), 39 (1978) 1241. J. Tardy and R. Meaudre, Solid State Commun., 39 (1981) 1031. J.I. Pankove and D. E. Carlson, Appl. Phys. Lett., 31 (1977) 450. I. Solomon, R. Benferhat and H. Tran-Quoc, Phys. Rev. B, 30 (1984) 3422. J.C. Bruy6re and A. Deneuville, J. Phys. (Paris), 41 (1980) L27. P. Viktorovitch and G. Moddel, J. Appl. Phys., 51 (1980) 4847. M. Meaudre and R. Meaudre, Philos. Mag. B, in the press. N . F . Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials, Oxford University Press, Oxford, 1979. 12 T. Tiedje, T. D. Moustakas and J. M. Cebulka, Phys. Rev. B, 23 (1981) 5634. 13 M . A . Lampert and P. Mark, in H. G. Booker and N. Declaris (eds.), Current Injection in Solids, Academic Press, New York, 1970. 14 W. den Boer, J. Phys. (Paris), Colloq. C4, 42 (1981) 451. 2 3 4 5 6 7 8 9 10 11