Model calculation on the Meyer–Neldel rule for the field‐effect conductance of hydrogenated amorphous silicon

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Model calculation on the Meyer–Neldel rule for the field‐effect conductance of hydrogenated amorphous silicon

Byung‐Gook Yoon and Choochon Lee

Citation: Applied Physics Letters 51, 1248 (1987); doi: 10.1063/1.98694 View online: http://dx.doi.org/10.1063/1.98694

View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/51/16?ver=pdfcov Published by the AIP Publishing

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Model caiculation on the Meyer-Neldel rule for the fieidaeffect conductance of hydrogenated amorphous silicon

Byung-Gook Yoon^a⁾and Choochon Lee

Department of Physics, Korea Advanced Institute ojScience and Technology, P. O. Box 150. Chollgyang- ni, Seoul}3}, Korea

(Received 8 June 1987; accepted for publication 18 August 1987)

A model calculation was carried out to study the Meyer-Neldel rule of the field-effect

conductance of hydrogenated amorphous silicon (a-Si:H). It was found that the shift of Fermi level and the potential profile in the sample with temperature can explain the Meyer-Ne!del rule if a model density of states of a-Si:H is properly chosen. So it is of doubt to think that the conductivity prefactor varies in a single sample of band-bending case, as assumed by some authors.

It is well known that the pre-exponential factor of dark conductivity of hydrogenated amorphous silicon(a-Si:H) obeys the Meyer-Neldel rule (MNR)/,2 which states that

0'0

=

^(Tooexp(AE_{a ),} (1)

where 0'0 andE" are the experimentally determined conductivity prefactor and activation energy, respectively, and A is typically between 14 and 31 eV 1. The MNR holds when- ever the activation is altered by such methods as doping,3.4 light soaking,3,4 and field effect.⁵,6 The origin of the MNR is not yet clear and has been attributed to such mechanisms as the shift of mobility edge,7 surface band bending,s electron- phonon interaction,9 and statistical shift of the Fermi level

EF·IO.ll

Another related problem is whether the conductivity prefactor varies in a single sample when the activation energy is a function of position, or the MNR is not a local proper- ty but is apparently satisfied for the sample as a whole in the case of band bending. To study this problem, we assume that the conductivity prefactor is of a single value in a sample and work out a model calculation on the t.emperature dependence of the a-Si:H field-effect conductance.

The model sample in this work is a thin film throughout which the mode! density of states(DOS) in Fig. 1 is valid.

The electric potential varies along the x direction as shown in the insert of Fig, 1. The effects of external field and of surface states are represented by the boundary conditions of the potential at x

=

O. A single conductivity prefactor holds for O<,x<,d, where d is sample thickness, d

=

5000

A.

^The

boundary condition at x

=

d in field-effect analysis is

V~Oand

dV->Oasx-->d. (2) dx

The amount of band bending at x is determined by the Pois- son's equation

~:~ =~ roO',

dENCE){f[E- Vex)] -fCE)}, (3) where £s is the permittivity of a-Si:H, NeE) is the DOS,Jis the Fermi distribution function, and V(x) is measured in eV.

a) Permanent address: Department of Physics, Ulliversity of Ulsan, Ulsan, 690, Korea.

If a DOS is given, we can obtain Vex) satisfying conditions (2), by solving Eq. (3) numerically. In doing this it is useful to note that the asymptotic behaviors of Vand dV Idx are related to each other as x->d or V--O: as V-.O, Eq. (3) becomes approximately

d2~ =~NI\EF)V,

⁽⁴⁾

dx- !:s

where N () (E) would be the "zero temperature equivalent"

DOS of N(E) if we used the zero temperature statistics on the right-hand side of Eg. (3) and can be obtained by ¹²

N0(E_p

+

V)

= roc

dE N(E)( aleE - V) ). (5)

J-00 \

av ;

Equation (4) is easily solved as x - d:

Vex) cxexp( -Kx), dV = -KV, (6) dx

where K = [eN°(E_F)/£,] -lIZ. Thus, it is easier to inte- grate Eg. (3) starting at x = d, since then we need only one boundary value, say V(d).

When the temperature rises, the Fermi distribution

Id.

I

-1.8 -0.9

Energy (eV)

J

EG. 1. Model den~ity of states of a-Si:H used in this work. Enclosed in the box is the conduction-band diagram in real space of our model sample, where vex) is the electric potential measured in energy unit. The numerical values of density of states are (in eV ^--Iem -³and energy in eV): ^jQ22for E < - 1.6 and E> --0.15, !O22 exp[ 40(E + 0.15) I f o r - 0.45 < E

< -0.\5, lO'O exp [45.4(E +0.65)2] for - 0.8<E< -0.45, lO'8 exp [-17.9(R I-1.0)1 for -- l.O<E<O.8, lOIS for ~ 1.2<E

< -1.0, and 10²²exp[ -- 23.0(E +-1.6)J for -1.6<E< -1.2.

1248 Appl. Phys. Lett. 51 (Hi), 19 October 1987 0003-6951/87/421248-03$01.00 @ 1987 American Institute of Physics 1248 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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functionsfand EF change, and so does the potential profile.

But the total number of electrons is conserved, and we may write

f

dxdEN[E+ V(x) Jf(E,Er^,7)⁼constant, (7)

where r

=

k ^lJT is the fundamental temperature. In Eq. (7) as wen as Eq. (3), we neglected the correlation effect of dec- trow; and simply used the Fermi statistics. Ifwe differentiate Eq. (7) with respect to r, assuming that N(E) is temperature independent, we have

__ fdX

^dE

r

^N(E

+

V)JU --/)(E - E^{F )}

+

dN(E

+

V) fr\f

aV)]j

l r dE

ar

J

dXdE[NCE+ V)/(1-f)

+

dN(E+ V)

/r( a~)J.

dE aEp

(8)

With a given DOS we may proceed using Eq. (8) to calculate the temperature dependence of the conductance ofthe model sample as follows.

(i) Obtain Vex) at initial r.

(ii) Find

av lar

and

av laE

p from the calculations of the potential variations with small variations of l' and EF ,

respectively. In this step the following boundary c(lndition should be used, which represents the constancy of the gate voltage in the field-effect geometry:

d V

=

constant at x = 0 as "I andlor E F varies. (9 ) dx

(iii) Calculate dEl'I dr and obtain EF at a slightly higher temperature. At this higherT, calculate V(x) using condition (9). Now go to step (ii) to continue.

The conductance is calculated at each temperature by G

=

constant

i

^d^dx^{exp( -}

[Ec --

^{:(X) --}

Ed ).

( 10) In Fig. 2 the field-effect conductance of our model sample is calculated, assuming Ep

= -

0.85 eV at 7

=

0.026 eV, and plotted against liT, and in Fig. 3 the apparent conductance prefactor, against activation energy. The conductance curves are nearly straight lines and the MNR is satisfied with

FIG. 2. Calculated field-effect conductance ofthe model sample. Curves a-f correspond to different external fields at x = 0 and to a-f, respectively, in Figs. 3 and 4.

i249 Appl. Phys. Lett.. Vol. 51, No. 16, 19 October 1987

A = 23 eV -\ for 0.25 <E" < 0.6 eV.

The results can be interpreted as foHows. The conductance may be written empirically, letting Ee

=

0,

G = G(]() exp[

(V +

E_F)Ir] , (11) where

V

is the effective decrease of activation energy due to band bending and is temperature dependent. The apparent pre-exponential factor^{l l}becomes in this case

( dV dEF)

Ga=Gooexp

- + - - - .

dr dr ⁽¹²⁾

To estimate

(Iv

Idr, it is fairly satisfactory to consider the shift of V(O) with temperature. In Fig. 4, VW) is plotted againstT for the conductances in Fig. 2, and also written are AV(O)/a.randAE_F16.7, the mean shifts of V(O) andE}! for

"I between 0.026 and 0.03 eV. As the activation energy be-

comes smaller (or

V

larger), E F increases at a higher rate with temperature, but this causes

V

to decrease still more rapidly, which results in smaner dV Idr

+

dEj.ldr.

In summary, we have shown by a numerical method that the MNR of the field-effect conductance of a-Si:H can be explained by the temperature shift of E1' and the potential profile. This, together with our previous work, II tens us that the stati.stical shift of Ej. is at least one of the major reasons for the empirical MNR of a-Si:H. Furthermore, it is unnec-

0

*

c a

::J

I

.ci -4

.... I

3

"

(!)

Z ....l.

-!2 0.2 0.4 0.6

FIG, 3. Conductance prefactor vs activation energy of the conductance curves in Fig. 2.

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(eV)

VI x =rO"}---:~,.----,----:-:-?"",,,~...., 0.5

~ ~V,~O:4T

~~~f ~~

o.41~e

~ 4.5 -4.5 d

f

----'c I

26 - 13

I

0.3

--.----=--

b~

02f ,.

^-025

,oj

0.026 0.028 0.030 7: ^(eVl

FIG. 4. Variations of the potentia! at the sample surface, VeOl, with tern- per!fture. Also written are the mean shifts of Fermi level Ep and V(Ol between T == 0.026 eV and T ,=, 0.030 eV.

essary and probably incorrect to assume that the conductivity prefactor varies locally in a sample whose bands are bent.³•5

1250 Appl. Phys. Lett., Vol. 51, No. 16, 19 October 1987

This work was supported by the Korea Science and En- gineering Foundation and Ministry of Science and Technol- ogy of Korean Government.

'w.

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