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Surface states in the one-dimensional model of a crystal with an arbitrary potential at the boundary
Volume 60A, number 5
PHYSICS LETTERS
21 March 1977
SURFACE STATES IN THE ONE-DIMENSIONAL MODEL OF A CRYSTAL WITH AN ARBITRARY POTENTIAL AT THE BOUNDARY J. PEISERT Instytut Matematyki, Politechnika Wroclawska, Wybrze~eWyspian’skiego 27, 50-370 Wroehiw, Poland Received 8 February 1977 The approximate formula for the number of surface states and the approximate equation for its energy values has been obtained for the model of a one-dimensional semi-infinite crystal with an arbitrary potential at the boundary and the Kronig-Penney-type potential inside the crystal.
The boundary potential dependence of energy value and the number of surface states have been studied recently [1—6]in a few particular models by the use of numerical calculations. We present the general approximate formulas connected with this problem. We assume that in the one-dimensional model it is sufficient to choose the Kronig-Penney potential to describe an interior of the crystal (x < 0). Then it is easy to state that the surface state energy value equation has the form: 2a2 2~2~E~ ka cotg ka k = af(E), (1) —
—
with the “existence condition”
<—-f—
2
where E = fl2k2/2m, is energy, a the lattice constant, P power of Kronig-Penney’s delta and f(E) is a function describing the reflection of an electron at the boundary of the crystal. It is the logarithmic derivative of the solution of the Schradinger equation, taken at the boundary of the crystal (x = 0). For a rectangular barrier model it is equal to [7} r2
We find an approximation of f(E) with the aid of the WKB method. For the smooth boundary potential we can easily find that [8] r2m
[-~-(E
—
x
Xtan
f 0
0(E)
11/2 V(0))J
r2m 11/2 ~ dxl—(E—V(x))I —-i2 1 Lh
f
—
0
and it is of course the number of f(E) singularities. Each singularity appearing inside the energy would produce in general two surface states but thegap condition (2) admits only one of them. Now we can write the expression for the surface states number in the successive nth energy gap: mn~N(xo(En))_N@o(~)) where E~,E~are the lower and upper limits of the nth energy gap, respectively (and of course they do not depend on the V(x)). Eq. (1) with the functionf(E) (3) allows to describe all the models [1—6] and to compare their results in the uncomplicated and physically clear way. Finally, we have to bear in mind that the WKB method does not deal properly with the region beyond the classically available one.
-11/2
fo(E)=_[~~!~(Vo_E)]
f(E)
when x0(E) is the classical turning point of a particle of energy E moving into the potential V(x). The number of zeros of the solution u(x, if) of the Schrodinger equation is CE) 1 (2m\’I~ XO 1/2 N(x0(E)) ~ [E V(x)] dx,
The author would like to thank Professor K.F. Wojciechowski for constant encouragement and advice.
References .
Lll (3)
.
-
F. Flores, E. Louis and 3. Rubio, J. Phys. C. Solid State Phys. 5 (1972) 3469.
443
Volume 60A, numberS [2] [3] [4] [5]
444
PHYSICS LETTERS
N. Garcia and J. Solana, Surface Sci. 36 (1973) 262. M. Kolai and I. Bartol, Czech. J. Phys. B23 (1973) 179. M. Stçilicka, Phys. Lett. 57A (1976) 255. A. Modinos, Surface Sci. 5 (1975)1 57.
21 March 1977
[6] S.G. 1)avison and K.P. Tan, Surface Sd. 27 (1971) 297. [7] S.G. Davison and il). Levine, Surface states (Academic Press, New York and London. 1970). [8) A.S. Davydov. Kvantovaya mechanika (Moskva 1963).
PHYSICS LETTERS
21 March 1977
SURFACE STATES IN THE ONE-DIMENSIONAL MODEL OF A CRYSTAL WITH AN ARBITRARY POTENTIAL AT THE BOUNDARY J. PEISERT Instytut Matematyki, Politechnika Wroclawska, Wybrze~eWyspian’skiego 27, 50-370 Wroehiw, Poland Received 8 February 1977 The approximate formula for the number of surface states and the approximate equation for its energy values has been obtained for the model of a one-dimensional semi-infinite crystal with an arbitrary potential at the boundary and the Kronig-Penney-type potential inside the crystal.
The boundary potential dependence of energy value and the number of surface states have been studied recently [1—6]in a few particular models by the use of numerical calculations. We present the general approximate formulas connected with this problem. We assume that in the one-dimensional model it is sufficient to choose the Kronig-Penney potential to describe an interior of the crystal (x < 0). Then it is easy to state that the surface state energy value equation has the form: 2a2 2~2~E~ ka cotg ka k = af(E), (1) —
—
with the “existence condition”
<—-f—
2
where E = fl2k2/2m, is energy, a the lattice constant, P power of Kronig-Penney’s delta and f(E) is a function describing the reflection of an electron at the boundary of the crystal. It is the logarithmic derivative of the solution of the Schradinger equation, taken at the boundary of the crystal (x = 0). For a rectangular barrier model it is equal to [7} r2
We find an approximation of f(E) with the aid of the WKB method. For the smooth boundary potential we can easily find that [8] r2m
[-~-(E
—
x
Xtan
f 0
0(E)
11/2 V(0))J
r2m 11/2 ~ dxl—(E—V(x))I —-i2 1 Lh
f
—
0
and it is of course the number of f(E) singularities. Each singularity appearing inside the energy would produce in general two surface states but thegap condition (2) admits only one of them. Now we can write the expression for the surface states number in the successive nth energy gap: mn~N(xo(En))_N@o(~)) where E~,E~are the lower and upper limits of the nth energy gap, respectively (and of course they do not depend on the V(x)). Eq. (1) with the functionf(E) (3) allows to describe all the models [1—6] and to compare their results in the uncomplicated and physically clear way. Finally, we have to bear in mind that the WKB method does not deal properly with the region beyond the classically available one.
-11/2
fo(E)=_[~~!~(Vo_E)]
f(E)
when x0(E) is the classical turning point of a particle of energy E moving into the potential V(x). The number of zeros of the solution u(x, if) of the Schrodinger equation is CE) 1 (2m\’I~ XO 1/2 N(x0(E)) ~ [E V(x)] dx,
The author would like to thank Professor K.F. Wojciechowski for constant encouragement and advice.
References .
Lll (3)
.
-
F. Flores, E. Louis and 3. Rubio, J. Phys. C. Solid State Phys. 5 (1972) 3469.
443
Volume 60A, numberS [2] [3] [4] [5]
444
PHYSICS LETTERS
N. Garcia and J. Solana, Surface Sci. 36 (1973) 262. M. Kolai and I. Bartol, Czech. J. Phys. B23 (1973) 179. M. Stçilicka, Phys. Lett. 57A (1976) 255. A. Modinos, Surface Sci. 5 (1975)1 57.
21 March 1977
[6] S.G. 1)avison and K.P. Tan, Surface Sd. 27 (1971) 297. [7] S.G. Davison and il). Levine, Surface states (Academic Press, New York and London. 1970). [8) A.S. Davydov. Kvantovaya mechanika (Moskva 1963).