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Electronic density-of-states near a rough surface
Solid State Communications, Printed in Great Britain.
Vol. 66, No. 2, pp. 241-243,
ELECTRONIC
0038-1098188 $3.00 + .OO Pergamon Press plc
1988.
DENSITY-OF-STATES
NEAR
A ROUGH
SURFACE
K.M. Hong Department
of Electrical
and Electronic Engineering, University Pokfulam Road, Hong Kong
of Hong Kong,
(Received 9 December 1987 by H. Kamimura)
We use the infinite-barrier model in conjunction with the coherent potential approximation to treat the problem of a rough surface. The local densities of states near the surface are calculated.
SURFACE asperities at the Si-SiO, interface have long been recognized to have a major influence on the transport properties of electrons in the inversion layer of metaloxide-silicon structures [IA]. Previous theoretical treatments of such surface roughness effects are all based on a continuum description of the surface. However, experimental results [3-51 indicate that the asperities are of dimensions of 4-5 A, i.e., the surface roughness is typically just one or two atomic layers thick, so that a continuum model may not be an appropriate description of the situation. In this Communication we give an alternative approach to the problem and consider surface roughness in the atomic limit. Furthermore, we apply the coherent potential approximation [6, 71 to this problem and obtain the electronic density-of-states (DOS) near the interface. For simplicity, we consider a semiconductor with a simple cubic structure with lattice constant a. We describe the electrons in the bulk crystal by the tightbinding-hamiltonian H where
function in a mixed Bloch-Wannier a semi-infinite crystal: G,(Z, I’, k,,; w)
-J,
if R and R’ are nearest
V,ifR = R’andI o othenvise 3
(3)
Here, A
=
h sgn(Re <)/(2JJm),
(4)
q
=
-
(5)
5 + sgn(Re <)dn,
(6)
l3w - 4k,,M2J),
t; =
4k,,) = -
2J(cosk,a
+ cosk,a).
(7)
It may be noted this infinite-barrier model of a semiinfinite crystal gives a Green’s function which is identical to the one obtained by Kalkstein and Soven [8] using a severed-bond model. However, as can be seen in the following, the present model is more readily extended to give a description of a rough surface. From here onwards, we shall use a system of units whereh = 25 = a = 1.
(1)
‘“n-&
Here R = (Z, m, n) denotes a lattice site and IR) a Wannier wave funciton. We create a semi-infinite crystal with a (10 0) surface by introducing a potential barrier V, in the I = 0 plane, i.e. =
q’+“),
neighbours,
i 0, otherwise.
(RI VIIR’)
A (t+-r’ -
for
for I, I’ 3 1.
(RIHIR’) =
=
representation
P =
I
1
‘13
0.2
1
\
= 0, (2)
By doing calculations with a general V and eventually letting V + 00, we immediately obtain the Green’s
-2Dl -4
I -3
I -2
I _,
I 0
I 1
I 2
\I 3
4
w
* Work supported Research Grant.
in part by a Hong Kong University
Fig. 1. The real and imaginary parts of the surface coherent potential oR(m) of a crystal with l/3 of surface atoms randomly removed. 241
ELECTRONIC
242
DENSITY-OF-STATES
NEAR
SURFACE
P = 'I3
1.0 P;2/3
A ROUGH
Vol. 66, No. 2
I
. -0.5
_‘ -
-I
-3
-2
-1
0 lJ
1
2
3
L
Fig. 2. The real and imaginary parrts of the surface coherent potential bR(m) of a crystal with 2/3 of surface atoms randomly removed. Note the difference in scales between Figs. 1 and 2. We roughen the I = 1 surface layer by introducing the random perturbation VZ, given by
Fig. 4. The local DOS N(1; w) in the various atomic layers (l) of a crystal with a rough (10 0) surface. l/3 of the sites in the 1 = I surface layer are inaccessible to the electrons.
a(o),ifR
(RI I’,lR’) V, with probability
=
P,if R = R’ and I = 1
i 0, otherwise. (8) Thus in the limit of V -+ co, we have a description of a rough surface in which a fraction P of the surface atoms have been randomly removed, or made unavailable to the electrons. This problem can now be treated within the coherent potential approximation. In essence, our treatment is the same as that developed by Tsukada [9] in the study of chemisorbed layers. We approximate the effect of V, by a surface coherent potential r~ localized in 1 = 1 layer:
(R’a’R’)
W,
Green’s
I’, k,,; 0) +
=
function
a(@)
(9) is then given by the
G,,(L I’, k,,; 01
a(o)Go(A1, k,,; o)G,(1, I’, k,,; 0) ’ 1 - a(o)Go(L1, k,,; 0)
where a(o) is determined tion [6, 7, 91 P -=
= 1,
0 2 otherwise,
=
The effective equation
= R’andI
- +;
by the self-consistency
G(l, 1, 4; 0).
(10) rela-
(11)
P=O
Fig. 3. The local DOS N(/; o) in the various atomic layers (I) of a crystal with a perfect (10 0) surface. Minor ripples in the curves are not shown. The I = 1 layer is the surface layer.
Fig. 5. The local DOS N(I; w) in the various atomic layers (I) of a crystal with a rough (10 0) surface. 2/3 of the sites in the I = 1 surface layer are inaccessible to the electrons.
Vol. 66, No. 2
ELECTRONIC
DENSITY-OF-STATES
Here N,, is the number of sites per plane parallel to the surface and we have taken the limit v + co. Using the fact that, for any functionf(x),
NEAR A ROUGH SURFACE
243
case P = 2/3. It may be seen either from the figures or directly from equation (15) that, as may be expected, as P increase (i.e. as the first layer is increasingly eroded away) the DOS of the I + 1st layer gradually assumes the shape of the DOS of the Ith layer of the (12) f[Etk,,)l drK’tO_ft- 2t), -1 perfect semi-infinite crystal. Furthermore the total where K’(k) is the complementary complete elliptic number of states in the first layer is reduced from 1 to integral with modulus k, we can rewrite equation (11) 1 - P, as a fraction P of the sites have become unavailable. as Thus we have demonstrated how the problem of 4 ’ P K’(t) a rough surface can be treated in the coherent potendt (13) -=7 20(o) - 25 - q * tial approximation and we have used it to calculate the a(o) -I( local density-of-states. More importantly, it should be Here q and 5 are defined as in equations (5) and (6) noted that such a formulation allows one to use the with .s(k,,)replaced by - 2t. We have solved equation powerful machinery of the Green’s funcitons ap(13) numerically by the Newton-Raphson method for proach to tackle many other problems related to a”(o) corresponding to the retarded Green’s function rough surfaces. (o + o + i0). The results for P = l/3 and 2/3 are shown in Figs. 1 and 2. From a”(o) we can calculate the DOS per site in the lth layer: REFERENCES N(I; o) = - --& c Im GR(Z, 1, k,i; co) (14) 1. Y.C. Cheng, Surf. Sci. 27, 663 (1971). 11kll 2. Y.C. Cheng, Proc. of the 3rd Conf. on Solid State4 ’ Devices, Tokyo, 1971 [J. Japan Sot. Appl. Phys. = dtK’(t) i 41, Suppl., 173 (1972)]. 7 -I 3. T. Ando, J. Phys. Sot. Japan 43, 1616 (1977). 21-I 4. T. Ando, A.B. Fowler & F. Stern, Rev. Mod. VI . (15) Phys. 54 437 (1982). 1 + 2+(o) > A. Hartstein, T.H. Ning & A.B. Fowler, Surf. Sci. 58, 178 (1976). Figures 3 to 5 illustrate the results of this calculation. P. Soven, Phys. Rev. 156, 809 (1967). Figure 3 shows the DOS of a perfect semi-infinite R.J. Elliot, J.A. Krumhansl & P.L. Leath, Rev. crystal. Figure 4 shows the DOS of the crystal with Mod. Phys. 46, 465 (1974). one third of the atoms in the first layer randomly D. Kalkstein & P. Soven, Surf. Sci. 26,85 (1971). removed (P = l/3), while Fig. 5 corresponds to the M. Tsukada, J. Phys. Sot. Japan 41,899 (1976).
&;
= f 1‘
Vol. 66, No. 2, pp. 241-243,
ELECTRONIC
0038-1098188 $3.00 + .OO Pergamon Press plc
1988.
DENSITY-OF-STATES
NEAR
A ROUGH
SURFACE
K.M. Hong Department
of Electrical
and Electronic Engineering, University Pokfulam Road, Hong Kong
of Hong Kong,
(Received 9 December 1987 by H. Kamimura)
We use the infinite-barrier model in conjunction with the coherent potential approximation to treat the problem of a rough surface. The local densities of states near the surface are calculated.
SURFACE asperities at the Si-SiO, interface have long been recognized to have a major influence on the transport properties of electrons in the inversion layer of metaloxide-silicon structures [IA]. Previous theoretical treatments of such surface roughness effects are all based on a continuum description of the surface. However, experimental results [3-51 indicate that the asperities are of dimensions of 4-5 A, i.e., the surface roughness is typically just one or two atomic layers thick, so that a continuum model may not be an appropriate description of the situation. In this Communication we give an alternative approach to the problem and consider surface roughness in the atomic limit. Furthermore, we apply the coherent potential approximation [6, 71 to this problem and obtain the electronic density-of-states (DOS) near the interface. For simplicity, we consider a semiconductor with a simple cubic structure with lattice constant a. We describe the electrons in the bulk crystal by the tightbinding-hamiltonian H where
function in a mixed Bloch-Wannier a semi-infinite crystal: G,(Z, I’, k,,; w)
-J,
if R and R’ are nearest
V,ifR = R’andI o othenvise 3
(3)
Here, A
=
h sgn(Re <)/(2JJm),
(4)
q
=
-
(5)
5 + sgn(Re <)dn,
(6)
l3w - 4k,,M2J),
t; =
4k,,) = -
2J(cosk,a
+ cosk,a).
(7)
It may be noted this infinite-barrier model of a semiinfinite crystal gives a Green’s function which is identical to the one obtained by Kalkstein and Soven [8] using a severed-bond model. However, as can be seen in the following, the present model is more readily extended to give a description of a rough surface. From here onwards, we shall use a system of units whereh = 25 = a = 1.
(1)
‘“n-&
Here R = (Z, m, n) denotes a lattice site and IR) a Wannier wave funciton. We create a semi-infinite crystal with a (10 0) surface by introducing a potential barrier V, in the I = 0 plane, i.e. =
q’+“),
neighbours,
i 0, otherwise.
(RI VIIR’)
A (t+-r’ -
for
for I, I’ 3 1.
(RIHIR’) =
=
representation
P =
I
1
‘13
0.2
1
\
= 0, (2)
By doing calculations with a general V and eventually letting V + 00, we immediately obtain the Green’s
-2Dl -4
I -3
I -2
I _,
I 0
I 1
I 2
\I 3
4
w
* Work supported Research Grant.
in part by a Hong Kong University
Fig. 1. The real and imaginary parts of the surface coherent potential oR(m) of a crystal with l/3 of surface atoms randomly removed. 241
ELECTRONIC
242
DENSITY-OF-STATES
NEAR
SURFACE
P = 'I3
1.0 P;2/3
A ROUGH
Vol. 66, No. 2
I
. -0.5
_‘ -
-I
-3
-2
-1
0 lJ
1
2
3
L
Fig. 2. The real and imaginary parrts of the surface coherent potential bR(m) of a crystal with 2/3 of surface atoms randomly removed. Note the difference in scales between Figs. 1 and 2. We roughen the I = 1 surface layer by introducing the random perturbation VZ, given by
Fig. 4. The local DOS N(1; w) in the various atomic layers (l) of a crystal with a rough (10 0) surface. l/3 of the sites in the 1 = I surface layer are inaccessible to the electrons.
a(o),ifR
(RI I’,lR’) V, with probability
=
P,if R = R’ and I = 1
i 0, otherwise. (8) Thus in the limit of V -+ co, we have a description of a rough surface in which a fraction P of the surface atoms have been randomly removed, or made unavailable to the electrons. This problem can now be treated within the coherent potential approximation. In essence, our treatment is the same as that developed by Tsukada [9] in the study of chemisorbed layers. We approximate the effect of V, by a surface coherent potential r~ localized in 1 = 1 layer:
(R’a’R’)
W,
Green’s
I’, k,,; 0) +
=
function
a(@)
(9) is then given by the
G,,(L I’, k,,; 01
a(o)Go(A1, k,,; o)G,(1, I’, k,,; 0) ’ 1 - a(o)Go(L1, k,,; 0)
where a(o) is determined tion [6, 7, 91 P -=
= 1,
0 2 otherwise,
=
The effective equation
= R’andI
- +;
by the self-consistency
G(l, 1, 4; 0).
(10) rela-
(11)
P=O
Fig. 3. The local DOS N(/; o) in the various atomic layers (I) of a crystal with a perfect (10 0) surface. Minor ripples in the curves are not shown. The I = 1 layer is the surface layer.
Fig. 5. The local DOS N(I; w) in the various atomic layers (I) of a crystal with a rough (10 0) surface. 2/3 of the sites in the I = 1 surface layer are inaccessible to the electrons.
Vol. 66, No. 2
ELECTRONIC
DENSITY-OF-STATES
Here N,, is the number of sites per plane parallel to the surface and we have taken the limit v + co. Using the fact that, for any functionf(x),
NEAR A ROUGH SURFACE
243
case P = 2/3. It may be seen either from the figures or directly from equation (15) that, as may be expected, as P increase (i.e. as the first layer is increasingly eroded away) the DOS of the I + 1st layer gradually assumes the shape of the DOS of the Ith layer of the (12) f[Etk,,)l drK’tO_ft- 2t), -1 perfect semi-infinite crystal. Furthermore the total where K’(k) is the complementary complete elliptic number of states in the first layer is reduced from 1 to integral with modulus k, we can rewrite equation (11) 1 - P, as a fraction P of the sites have become unavailable. as Thus we have demonstrated how the problem of 4 ’ P K’(t) a rough surface can be treated in the coherent potendt (13) -=7 20(o) - 25 - q * tial approximation and we have used it to calculate the a(o) -I( local density-of-states. More importantly, it should be Here q and 5 are defined as in equations (5) and (6) noted that such a formulation allows one to use the with .s(k,,)replaced by - 2t. We have solved equation powerful machinery of the Green’s funcitons ap(13) numerically by the Newton-Raphson method for proach to tackle many other problems related to a”(o) corresponding to the retarded Green’s function rough surfaces. (o + o + i0). The results for P = l/3 and 2/3 are shown in Figs. 1 and 2. From a”(o) we can calculate the DOS per site in the lth layer: REFERENCES N(I; o) = - --& c Im GR(Z, 1, k,i; co) (14) 1. Y.C. Cheng, Surf. Sci. 27, 663 (1971). 11kll 2. Y.C. Cheng, Proc. of the 3rd Conf. on Solid State4 ’ Devices, Tokyo, 1971 [J. Japan Sot. Appl. Phys. = dtK’(t) i 41, Suppl., 173 (1972)]. 7 -I 3. T. Ando, J. Phys. Sot. Japan 43, 1616 (1977). 21-I 4. T. Ando, A.B. Fowler & F. Stern, Rev. Mod. VI . (15) Phys. 54 437 (1982). 1 + 2+(o) > A. Hartstein, T.H. Ning & A.B. Fowler, Surf. Sci. 58, 178 (1976). Figures 3 to 5 illustrate the results of this calculation. P. Soven, Phys. Rev. 156, 809 (1967). Figure 3 shows the DOS of a perfect semi-infinite R.J. Elliot, J.A. Krumhansl & P.L. Leath, Rev. crystal. Figure 4 shows the DOS of the crystal with Mod. Phys. 46, 465 (1974). one third of the atoms in the first layer randomly D. Kalkstein & P. Soven, Surf. Sci. 26,85 (1971). removed (P = l/3), while Fig. 5 corresponds to the M. Tsukada, J. Phys. Sot. Japan 41,899 (1976).
&;
= f 1‘