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The effect of surface states on the transport coefficients of thin semiconductor films
SURFACE
SCIENCE 30 (1972) 692-696 0 North-Holland
THE
EFFECT
OF SURFACE
TRANSPORT THIN
Publishing Co.
STATES
COEFFICIENTS
SEMICONDUCTOR
ON THE OF
FILMS
Received 18 January 1972
Although a considerable amount of work has been directed towards understanding the conduction processes in semiconductor thin films, analysis of transport coefficients, such as the Hall, Seebeck and magnetoresistance, have been restricted to considering the effects of surface scatteringi-3) or polycrystallinity’r). In this paper we examine how the Hall and Seebeck coefficients and the conductivity are affected by the presence of trapping states on the film surface; a similar analysis, but for conductivity alone, has been performed by Goodwin and Marks). The model which has been adopted is necessarily extremely simple, but it will serve to demonstrate the difficulty of interpreting transport coefficient measurements when a high density of surface states is present. Fig. 1 shows the band structure assumed for these calculations, for the two cases of n-type and p-type conductivity. The model adopted is based on the results of field-effect measurements made on PbTe and PbSe films by Egerton and Juhaszs). This showed that there was a high density of both donors and acceptors (-2 x 10” cmp2 of each) at the film-substrate interface, and that in low carrier concentration material these states tend to clamp the Fermi level at the interface at about 1 kT below the intrinsic level (i.e. that position of the Fermi level which would give exactly intrinsic material). In addition we assume that: (a) the Fermi level at this interface is clamped 1 kT below the intrinsic level at all carrier densities of the film; (b) the same situation exists on the film-air interface [there is evidence that, if anything, the donor and acceptor densities on this surface are even higher than on the film-substrate interface’)]; (c) the crystallite size is large enough to neglect polycrystalline effects; (d) the surface scattering is specular [this has been confirmed for PbSe2)]; (e) the energy bands are parabolic. The Hall and Seebeck coefficients and the conductivity of this model are found by considering it as a set of elemental planes, parallel to the film sur692
EFFECT
Air
2d
693
STATES
Air
Substrate
Film -
OF SURFACE
Film c.b.
h
--
\ \
Substrate 4
\ \
\
2d
__
\
\
\
n - type
P - type
Fig. 1. Model of band-bending in thin semiconductor films used for the calculations presented here. (-+ indicates position of intrinsic level in flat-band case, i. e. no surface states).
face, each of thickness is thenz)
dz. For a film of thickness
2d, the Hall coefficient,
R,,
d
2d
R (z) [r~(z)]” dz s
the Seebeck coefficient,
CL,becomes d
(2)
J
(T(Z) dz
-d
and the conductivity,
6, is d
1 (J
=
--
2d s
a(z)dz.
(3)
-d
The position-dependent Hall coefficient, R(z), Seebeck coefficient, IX(Z), and conductivity, O(Z), are determined by the electron and hole densities, n(z) andp(z) at the position z. These can in turn be related to the band-bending in
694
J. J. HARRIS
AND
A. J.CROCKER
the space charge region by introducing the parameter g, defined as the energy separation in kT units between the Fermi level and the intrinsic level at the position z. Then if ni is intrinsic carrier concentration, the electron and hole densities are given by n(Z)
=:
FZi ecr,
p(z)
=
n,
e-‘,
(4)
assuming that Boltzmann statistics are applicable. Substituting (4) into the standard formulae for the Hall and Seebeck coefficients and conductivity of ambipolar material enables (l), (2) and (3) to be rewritten as
(e-’ -
c2e’) dz
--..-. (e-’ + c e”) dz
I
d
1
[(A f urv i- u) eerr - c(A+w,,-u)eU]dz
t( = -kb.-I_.. .
~-~-
d
-----
li
GW
i? (e-’
+
c e’)
dz
s
d
0
where e is the electronic charge; r is the Hall factor; c is the ratio of electron mobility, p, to hole mobility &; A is the scattering parameter; and wIv and oIc are the energy separations in kT units between the intrinsic level and the valence and conduction bands respectively. These equations have also made use of the symmetry of the model about z=O. The reIationship between u and z is found by solution of the one-dimensional Poisson equation. This has been done for our model by Frankl*), who obtained du -S, = 7(2 [cash u - cash u0 -I- (ZQ,- U) sinh z+J>*, (3 dz I) where u. is the value of u at z = 0; us is the value of u in the absence of surface states; S,, is the sign of y=tr--u,; and L, is the “intrinsic Debye length”, (eokT/8rre2ni)*, e0 being the static dielectric constant, Eq. (5) enables the integrals over z in (la), (2a) and (3a) to be replaced by ones over U, the limits being changed from 0 and d to u. and y (the latter is the value of u at z = t_ d).
EFFECT
OF SURFACE
695
STATES
These integrals have been performed numerically using values of the parameters corresponding to PbTe at room temperature, i.e. ni = 8.5 x 10” cme3, X&=17OOA, r= 1, pi,=900 cm’ V-’ set-“, ,q= 1600 cm2 V-r set-’ (so that c= 1.8), w,v=&O, wrc= 5.6 and A=2 (corresponding to acoustic phonon scattering), This last value is probably incorrect, as recent analysisg) has shown that polar optical phonon scattering is also important in PbTe, and that the relative contributions of this and acoustic phonon scattering are strongly dependent on carrier concentration; however, including this effect is not justified in view of the other approximations already made. The results of these integrations are dispIayed in fig. 2 as the variations of
a
I
-2ooor -6
= -
-4
' P
-
. -2 type
'
' 0 %
'
. 2
fi
n-type
* 4
. -
Fig. 2. Variation of Hall coefficient, RH, Seebeck coefficient, a and Hall mobility, pi, with carrier concentration parameter, URfor PbTe films of thicknesses (a) co, (b) 5000 A and (c) 1700 A.
696
J. J.
HARRIS
AND A. J. CROCKER
R,, a and the Hall mobility, pn( = R,a) with the parameter uB [this is related to the carrier concentrations in the absence of surface states by IZ=q exp uB and p =ni exp (- Q)] for three values of the film thickness, 2d= 1700 A, 5000 A, and co, the latter corresponding to bulk material. These graphs show that as the film thickness is reduced, the positive and negative peaks of the R, and c1 curves move towards higher ]~a] values, as do the points at which the Hall mobility begins to deviate from the limiting values, ,LL~ and p,,. The effect of a large density of surface states clamping the Fermi level near mid-gap at the film surfaces is thus seen to extend the range of bulk dopings (i.e. uB values) for which the film behaves as if it were ambipolar. This makes it extremely difficult to interpret the observed transport coefficient measurements on thin films in terms of the concentrations of donors or acceptors within the film, or in the case of semiconductors like the lead chalcogenides, in terms of deviations from stoichiometry. A more sophisticated model than the one adopted here would be needed for such an interpretation to be satisfactorily achieved. J. J. HARRIS and A. J. CROCKER Zenith Radio Research Corporation (U.K.) Ltd., 6 Da&on Gardens, Stanmore, Middlesex, England References 1) 2) 3) 4) 5) 6) 7) 8)
A. Amith, J. Phys. Chem. Solids 14 (1960) 271. M. H. Brodsky and J. N. Zemel, Phys. Rev. 155 (1967) 780. R. L. Petritz, Phys. Rev. 110 (1958) 1254. G. H. Blount, R. H. Bube and R. L. Robinson, J. Appl. Phys. 41 (1970) 2190. T. A. Goodwin and P. Mark, Progr. Surface Sci. 1 (1971) 1. R. F. Egerton and C. Juhasz, Brit. J. Appl. Phys. (J. Phys. D) [2] 2 (1969) 975. M. J. Lee, private communication. D. R. Frank], Electrical Properties of Semiconductor Surfaces (Pergamon Press, 1967) p. 42. 9) Yu. I. Ravich, B. A. Efimova and V. I. Tamarchenko, Phys. Status Solidi (b) 43 1971) 453.
SCIENCE 30 (1972) 692-696 0 North-Holland
THE
EFFECT
OF SURFACE
TRANSPORT THIN
Publishing Co.
STATES
COEFFICIENTS
SEMICONDUCTOR
ON THE OF
FILMS
Received 18 January 1972
Although a considerable amount of work has been directed towards understanding the conduction processes in semiconductor thin films, analysis of transport coefficients, such as the Hall, Seebeck and magnetoresistance, have been restricted to considering the effects of surface scatteringi-3) or polycrystallinity’r). In this paper we examine how the Hall and Seebeck coefficients and the conductivity are affected by the presence of trapping states on the film surface; a similar analysis, but for conductivity alone, has been performed by Goodwin and Marks). The model which has been adopted is necessarily extremely simple, but it will serve to demonstrate the difficulty of interpreting transport coefficient measurements when a high density of surface states is present. Fig. 1 shows the band structure assumed for these calculations, for the two cases of n-type and p-type conductivity. The model adopted is based on the results of field-effect measurements made on PbTe and PbSe films by Egerton and Juhaszs). This showed that there was a high density of both donors and acceptors (-2 x 10” cmp2 of each) at the film-substrate interface, and that in low carrier concentration material these states tend to clamp the Fermi level at the interface at about 1 kT below the intrinsic level (i.e. that position of the Fermi level which would give exactly intrinsic material). In addition we assume that: (a) the Fermi level at this interface is clamped 1 kT below the intrinsic level at all carrier densities of the film; (b) the same situation exists on the film-air interface [there is evidence that, if anything, the donor and acceptor densities on this surface are even higher than on the film-substrate interface’)]; (c) the crystallite size is large enough to neglect polycrystalline effects; (d) the surface scattering is specular [this has been confirmed for PbSe2)]; (e) the energy bands are parabolic. The Hall and Seebeck coefficients and the conductivity of this model are found by considering it as a set of elemental planes, parallel to the film sur692
EFFECT
Air
2d
693
STATES
Air
Substrate
Film -
OF SURFACE
Film c.b.
h
--
\ \
Substrate 4
\ \
\
2d
__
\
\
\
n - type
P - type
Fig. 1. Model of band-bending in thin semiconductor films used for the calculations presented here. (-+ indicates position of intrinsic level in flat-band case, i. e. no surface states).
face, each of thickness is thenz)
dz. For a film of thickness
2d, the Hall coefficient,
R,,
d
2d
R (z) [r~(z)]” dz s
the Seebeck coefficient,
CL,becomes d
(2)
J
(T(Z) dz
-d
and the conductivity,
6, is d
1 (J
=
--
2d s
a(z)dz.
(3)
-d
The position-dependent Hall coefficient, R(z), Seebeck coefficient, IX(Z), and conductivity, O(Z), are determined by the electron and hole densities, n(z) andp(z) at the position z. These can in turn be related to the band-bending in
694
J. J. HARRIS
AND
A. J.CROCKER
the space charge region by introducing the parameter g, defined as the energy separation in kT units between the Fermi level and the intrinsic level at the position z. Then if ni is intrinsic carrier concentration, the electron and hole densities are given by n(Z)
=:
FZi ecr,
p(z)
=
n,
e-‘,
(4)
assuming that Boltzmann statistics are applicable. Substituting (4) into the standard formulae for the Hall and Seebeck coefficients and conductivity of ambipolar material enables (l), (2) and (3) to be rewritten as
(e-’ -
c2e’) dz
--..-. (e-’ + c e”) dz
I
d
1
[(A f urv i- u) eerr - c(A+w,,-u)eU]dz
t( = -kb.-I_.. .
~-~-
d
-----
li
GW
i? (e-’
+
c e’)
dz
s
d
0
where e is the electronic charge; r is the Hall factor; c is the ratio of electron mobility, p, to hole mobility &; A is the scattering parameter; and wIv and oIc are the energy separations in kT units between the intrinsic level and the valence and conduction bands respectively. These equations have also made use of the symmetry of the model about z=O. The reIationship between u and z is found by solution of the one-dimensional Poisson equation. This has been done for our model by Frankl*), who obtained du -S, = 7(2 [cash u - cash u0 -I- (ZQ,- U) sinh z+J>*, (3 dz I) where u. is the value of u at z = 0; us is the value of u in the absence of surface states; S,, is the sign of y=tr--u,; and L, is the “intrinsic Debye length”, (eokT/8rre2ni)*, e0 being the static dielectric constant, Eq. (5) enables the integrals over z in (la), (2a) and (3a) to be replaced by ones over U, the limits being changed from 0 and d to u. and y (the latter is the value of u at z = t_ d).
EFFECT
OF SURFACE
695
STATES
These integrals have been performed numerically using values of the parameters corresponding to PbTe at room temperature, i.e. ni = 8.5 x 10” cme3, X&=17OOA, r= 1, pi,=900 cm’ V-’ set-“, ,q= 1600 cm2 V-r set-’ (so that c= 1.8), w,v=&O, wrc= 5.6 and A=2 (corresponding to acoustic phonon scattering), This last value is probably incorrect, as recent analysisg) has shown that polar optical phonon scattering is also important in PbTe, and that the relative contributions of this and acoustic phonon scattering are strongly dependent on carrier concentration; however, including this effect is not justified in view of the other approximations already made. The results of these integrations are dispIayed in fig. 2 as the variations of
a
I
-2ooor -6
= -
-4
' P
-
. -2 type
'
' 0 %
'
. 2
fi
n-type
* 4
. -
Fig. 2. Variation of Hall coefficient, RH, Seebeck coefficient, a and Hall mobility, pi, with carrier concentration parameter, URfor PbTe films of thicknesses (a) co, (b) 5000 A and (c) 1700 A.
696
J. J.
HARRIS
AND A. J. CROCKER
R,, a and the Hall mobility, pn( = R,a) with the parameter uB [this is related to the carrier concentrations in the absence of surface states by IZ=q exp uB and p =ni exp (- Q)] for three values of the film thickness, 2d= 1700 A, 5000 A, and co, the latter corresponding to bulk material. These graphs show that as the film thickness is reduced, the positive and negative peaks of the R, and c1 curves move towards higher ]~a] values, as do the points at which the Hall mobility begins to deviate from the limiting values, ,LL~ and p,,. The effect of a large density of surface states clamping the Fermi level near mid-gap at the film surfaces is thus seen to extend the range of bulk dopings (i.e. uB values) for which the film behaves as if it were ambipolar. This makes it extremely difficult to interpret the observed transport coefficient measurements on thin films in terms of the concentrations of donors or acceptors within the film, or in the case of semiconductors like the lead chalcogenides, in terms of deviations from stoichiometry. A more sophisticated model than the one adopted here would be needed for such an interpretation to be satisfactorily achieved. J. J. HARRIS and A. J. CROCKER Zenith Radio Research Corporation (U.K.) Ltd., 6 Da&on Gardens, Stanmore, Middlesex, England References 1) 2) 3) 4) 5) 6) 7) 8)
A. Amith, J. Phys. Chem. Solids 14 (1960) 271. M. H. Brodsky and J. N. Zemel, Phys. Rev. 155 (1967) 780. R. L. Petritz, Phys. Rev. 110 (1958) 1254. G. H. Blount, R. H. Bube and R. L. Robinson, J. Appl. Phys. 41 (1970) 2190. T. A. Goodwin and P. Mark, Progr. Surface Sci. 1 (1971) 1. R. F. Egerton and C. Juhasz, Brit. J. Appl. Phys. (J. Phys. D) [2] 2 (1969) 975. M. J. Lee, private communication. D. R. Frank], Electrical Properties of Semiconductor Surfaces (Pergamon Press, 1967) p. 42. 9) Yu. I. Ravich, B. A. Efimova and V. I. Tamarchenko, Phys. Status Solidi (b) 43 1971) 453.