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Theory of trap-controlled transient current injection
Journal of Non-CrystallineSolids 22 (1976) 215-218 © North-Holland PublishingCompany
LETTERS TO THE EDITOR THEORY OF TRAP-CONTROLLED TRANSIENT CURRENT INJECTION
A.I. RUDENKO * Physics Department, Brunel University, Uxbridge, Middlesex, UK
Received 6 April 1976
Transient current injection is a powerful instrument for studying such important properties and parameters of amorphous materials as mobility, band structure, trapping and release times, etc. (see e.g. ref. [1 ]). A theoretical study of transient current injection is also necessary to develop and analyse the operation of devices in present-day electronics (see e.g. ref. [2]). But theoretically the problem has been studied insufficiently. For the space-charge unperturbed current (the small signal case), which is considered in this paper, two limiting cases - the trap-free case and the fast trapping-release case - have been treated analytically under conditions of pulsed injection [3] and step-function injection [4]. Regarding the intermediate case of an arbitrary trapping-release rate, we may refer to the Monte-Carlo approach of Silver and coworkers [5] and to the recent stochastic model of Scher and Montroll [6]. But in spite of some successes to date we have to note that there has been no previous analytical solution of the basic equations governing trap-controlled carrier transport. In this paper we present the analytical solution of the space-charge unperturbed transient current injection problem for arbitrary correlations bet.ween transit, trapping and release times, We have obtained the solution for arbitrary distributions of trapping times and trapping level densities under various regimes of injection. We consider an insulator of thickness L across which a potential V0 is applied. From time t' = 0 carriers are injected into the bulk through the surface x' = 0. Two modes are usually used [4]. We may keep the voltage constant and measure the total current in the circuit Q-mode). We may also use open-circuit conditions and measure the voltage decay (V-mode). Under space-charge unperturbed conditions, the basic equations with boundary and initial conditions are as follows: a [n(x', t') + ~
ns(X', t')]/~t' + g E an(x', t ' ) / S x ' = O ,
S
ans(X' , t')/at' = (1/rs)n(x, ' ' t ' ) - (Os/rs)ns(X ' ' , t ') , elan(O,t')E =J0(t'),
n(x', O) = O ,
ns(X', O) = O.
* Permanent address: MoscowEngineering Physics Institute, Kashirskoeshosse,Moscow, USSR. 215
A.I. Rudenko / Theory o f trap-controlled transient current injection
216
i
: Om01 0.I
N
I
i
i
I0
0.s •r :
O.Oi 03
I
'I0
0.6 ~
0.4
g ~
0.2 I
0.0
I
I
I
I
0.2
O.tl
0.6
0.8
t.0
t.2
1.4
1.6
1.8
2.0
2.2
2A
T I M E (.,umber of tronsi/s)
Fig. I.
Here x' is the coordinate (0 ~
J(t') = (epE/L) / 0
dx' n(x', t ' ) .
For the V-mode we measure the voltage decay V(t') L
- d V ( t ' ) / d t ' = (47relaE/K ) / 0
dx' n(x', t') ,
where K is the dielectric constant. It is convenient to present the solution in terms of dimensionless variables:
x = x'/L,
t = laVot'/L2,
]0 = 47rJo L3/tIKV2 ,
Ts = laVo'[~/L2,
t9 = 47renL2/KV 0 ,
] = 47rJL3/pKV2 = -(dV/dt')/(IaV2/LZ) "
The principal measurable dimensionless variable is/(t) = f l dx p(x, t). For arbitrary distributions r s, 0 s, and under an arbitrary injection regime/o(t), the solution of the
A.L Rudenko I Theory of trap-controlled transient current injection
217
~r: 0.01 0.t I 0.010
o.oos
N E =-
0.006
F-Z ¢1:
0.004
0.002
20
40
60
80
t00
t20
t¢-I0
#60
t80
200
220
i 2zlO
TIME (number of transits) F i g . 2.
problem is given by p(x, t) = e x p ( - x / r ) J o ( t - x ) + e x p ( - x / r ) (s' ~ s)
×
Eexp{x g
f
t Os,lr 2 [(0siTs) -
(0,,/Ts,)l) f
dt'/o(t' - x)
X
× exp [-(Os[rs)(t - t ' ) ] I 1 [x/(4Os/r2)x(t - t')] [Osx/r2s(t - t')] 1/2 x
p(x,t) =0,
x > t.
Here, 1/r = Zs 1/rs and I 1 is a Bessel function of imaginary argument. Various models of trapping level distribution have been used, in particular: (1) the one single trapping level model s = 1 ; (2) the few trapping levels model s = 1 ... s O; (3) the "rectangular" trapping level density distribution 1
... ~ f s
de ....
l/~s--> l / r @ ) = 1/~,
0s ~ 0(e) ~ exp [ - ( e o / k T ) e ] ,
0
where e o is the edge of trapping level distribution; (4) the exponential trapping level density distribution
s
•, . ~ ? 0
de ....
l/rs~
1/r(e) -'-(1/r)e - e ,
0 s ~ O(e) o~ e x p [ _ ( e o / k T ) e ] ,
where e 0 is the characteristic "depth" of the distribution. Various regimes of injec-
218
A.L Rudenko / Theory of trap-controlled transient current in/ection
tion have been treated: (1) the pulsed regime ]o(t) = qfi(t); (2) the pulsed regime taking into account the capture of carriers by surface traps, where ]o(t) = cq ~(t) + (1 - c)] 1 exp(-t/ro), r 0 characterizes the surface trapping time (0 ~< c ~< 1); (3) the step-function injection regime ]0(t ) = 0, t < O,]o(t ) =/1, t > 0, etc. As an example we present the solution of the problem for the single level model under the pulsed regime o f injection. (Recently, Roberts has also considered this regime using the other approach [7].) /(t) =q(1 + 0 ) -1 (0 + e x p [ - ( 1 +O)t/r]},
0~
1
](t) = q exp(-Ot/r) f dx exp[--(1 - O)x/r] 0 X 11 [x/(40/r2)x(t - x)] [Ox/r2(t - x)] 1/2,
t> 1.
The curves plotted illustrate transient current after pulsed injection for various values of r and 0. We intend to report the more detailed data for other injection regimes and trapping models in a future full paper. Useful discussions with Dr. J. Hirsch, Professor C.A. Hogarth, Dr. A.E. Owen and Professor G.G. Roberts, are gratefully acknowledged.
References [1] J.M. Marshall, F.D. Fisher and A.E. Owen, Phys. Stat. Sol. 25(a)(1974)419; C. Main and A.E. Owen, in: Proc. 5th Int. Conf. on Amorphous and Liquid Semiconductors, Garmisch-Partenkirchen, ed. J. Stuke and W. Brenig (Taylor and Francis, London, 1974) p. 783. [2] J. Mort, in: Proc. 5th Int. Conf. on Amorphous and Liquid Semiconductors, GarmischPartenkirchen, ed. J. Stuke and W. Brenig (Taylor and Francis, London, 1974) p. 1361. [3] A. Many and G. Rakavy, Phys. Rev. 126 (1962) 1980. [4] J. Mort and H. Scher, J. Appl. Phys. 42 (1971) 3939. [5] M. Silver, K.S. Dy and I.L. Huang, Phys. Rev. Letters 27 (1971) 21. [6] H. Scher and E. Montroll, Phys. Rev. B12 (1975) 2455. [7 ] G.G. Roberts, private communication.
LETTERS TO THE EDITOR THEORY OF TRAP-CONTROLLED TRANSIENT CURRENT INJECTION
A.I. RUDENKO * Physics Department, Brunel University, Uxbridge, Middlesex, UK
Received 6 April 1976
Transient current injection is a powerful instrument for studying such important properties and parameters of amorphous materials as mobility, band structure, trapping and release times, etc. (see e.g. ref. [1 ]). A theoretical study of transient current injection is also necessary to develop and analyse the operation of devices in present-day electronics (see e.g. ref. [2]). But theoretically the problem has been studied insufficiently. For the space-charge unperturbed current (the small signal case), which is considered in this paper, two limiting cases - the trap-free case and the fast trapping-release case - have been treated analytically under conditions of pulsed injection [3] and step-function injection [4]. Regarding the intermediate case of an arbitrary trapping-release rate, we may refer to the Monte-Carlo approach of Silver and coworkers [5] and to the recent stochastic model of Scher and Montroll [6]. But in spite of some successes to date we have to note that there has been no previous analytical solution of the basic equations governing trap-controlled carrier transport. In this paper we present the analytical solution of the space-charge unperturbed transient current injection problem for arbitrary correlations bet.ween transit, trapping and release times, We have obtained the solution for arbitrary distributions of trapping times and trapping level densities under various regimes of injection. We consider an insulator of thickness L across which a potential V0 is applied. From time t' = 0 carriers are injected into the bulk through the surface x' = 0. Two modes are usually used [4]. We may keep the voltage constant and measure the total current in the circuit Q-mode). We may also use open-circuit conditions and measure the voltage decay (V-mode). Under space-charge unperturbed conditions, the basic equations with boundary and initial conditions are as follows: a [n(x', t') + ~
ns(X', t')]/~t' + g E an(x', t ' ) / S x ' = O ,
S
ans(X' , t')/at' = (1/rs)n(x, ' ' t ' ) - (Os/rs)ns(X ' ' , t ') , elan(O,t')E =J0(t'),
n(x', O) = O ,
ns(X', O) = O.
* Permanent address: MoscowEngineering Physics Institute, Kashirskoeshosse,Moscow, USSR. 215
A.I. Rudenko / Theory o f trap-controlled transient current injection
216
i
: Om01 0.I
N
I
i
i
I0
0.s •r :
O.Oi 03
I
'I0
0.6 ~
0.4
g ~
0.2 I
0.0
I
I
I
I
0.2
O.tl
0.6
0.8
t.0
t.2
1.4
1.6
1.8
2.0
2.2
2A
T I M E (.,umber of tronsi/s)
Fig. I.
Here x' is the coordinate (0 ~
J(t') = (epE/L) / 0
dx' n(x', t ' ) .
For the V-mode we measure the voltage decay V(t') L
- d V ( t ' ) / d t ' = (47relaE/K ) / 0
dx' n(x', t') ,
where K is the dielectric constant. It is convenient to present the solution in terms of dimensionless variables:
x = x'/L,
t = laVot'/L2,
]0 = 47rJo L3/tIKV2 ,
Ts = laVo'[~/L2,
t9 = 47renL2/KV 0 ,
] = 47rJL3/pKV2 = -(dV/dt')/(IaV2/LZ) "
The principal measurable dimensionless variable is/(t) = f l dx p(x, t). For arbitrary distributions r s, 0 s, and under an arbitrary injection regime/o(t), the solution of the
A.L Rudenko I Theory of trap-controlled transient current injection
217
~r: 0.01 0.t I 0.010
o.oos
N E =-
0.006
F-Z ¢1:
0.004
0.002
20
40
60
80
t00
t20
t¢-I0
#60
t80
200
220
i 2zlO
TIME (number of transits) F i g . 2.
problem is given by p(x, t) = e x p ( - x / r ) J o ( t - x ) + e x p ( - x / r ) (s' ~ s)
×
Eexp{x g
f
t Os,lr 2 [(0siTs) -
(0,,/Ts,)l) f
dt'/o(t' - x)
X
× exp [-(Os[rs)(t - t ' ) ] I 1 [x/(4Os/r2)x(t - t')] [Osx/r2s(t - t')] 1/2 x
p(x,t) =0,
x > t.
Here, 1/r = Zs 1/rs and I 1 is a Bessel function of imaginary argument. Various models of trapping level distribution have been used, in particular: (1) the one single trapping level model s = 1 ; (2) the few trapping levels model s = 1 ... s O; (3) the "rectangular" trapping level density distribution 1
... ~ f s
de ....
l/~s--> l / r @ ) = 1/~,
0s ~ 0(e) ~ exp [ - ( e o / k T ) e ] ,
0
where e o is the edge of trapping level distribution; (4) the exponential trapping level density distribution
s
•, . ~ ? 0
de ....
l/rs~
1/r(e) -'-(1/r)e - e ,
0 s ~ O(e) o~ e x p [ _ ( e o / k T ) e ] ,
where e 0 is the characteristic "depth" of the distribution. Various regimes of injec-
218
A.L Rudenko / Theory of trap-controlled transient current in/ection
tion have been treated: (1) the pulsed regime ]o(t) = qfi(t); (2) the pulsed regime taking into account the capture of carriers by surface traps, where ]o(t) = cq ~(t) + (1 - c)] 1 exp(-t/ro), r 0 characterizes the surface trapping time (0 ~< c ~< 1); (3) the step-function injection regime ]0(t ) = 0, t < O,]o(t ) =/1, t > 0, etc. As an example we present the solution of the problem for the single level model under the pulsed regime o f injection. (Recently, Roberts has also considered this regime using the other approach [7].) /(t) =q(1 + 0 ) -1 (0 + e x p [ - ( 1 +O)t/r]},
0~
1
](t) = q exp(-Ot/r) f dx exp[--(1 - O)x/r] 0 X 11 [x/(40/r2)x(t - x)] [Ox/r2(t - x)] 1/2,
t> 1.
The curves plotted illustrate transient current after pulsed injection for various values of r and 0. We intend to report the more detailed data for other injection regimes and trapping models in a future full paper. Useful discussions with Dr. J. Hirsch, Professor C.A. Hogarth, Dr. A.E. Owen and Professor G.G. Roberts, are gratefully acknowledged.
References [1] J.M. Marshall, F.D. Fisher and A.E. Owen, Phys. Stat. Sol. 25(a)(1974)419; C. Main and A.E. Owen, in: Proc. 5th Int. Conf. on Amorphous and Liquid Semiconductors, Garmisch-Partenkirchen, ed. J. Stuke and W. Brenig (Taylor and Francis, London, 1974) p. 783. [2] J. Mort, in: Proc. 5th Int. Conf. on Amorphous and Liquid Semiconductors, GarmischPartenkirchen, ed. J. Stuke and W. Brenig (Taylor and Francis, London, 1974) p. 1361. [3] A. Many and G. Rakavy, Phys. Rev. 126 (1962) 1980. [4] J. Mort and H. Scher, J. Appl. Phys. 42 (1971) 3939. [5] M. Silver, K.S. Dy and I.L. Huang, Phys. Rev. Letters 27 (1971) 21. [6] H. Scher and E. Montroll, Phys. Rev. B12 (1975) 2455. [7 ] G.G. Roberts, private communication.