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Simple calculation of the Landau levels of narrow-gap semiconductors in the Kane model
Volume 106A, number 4
PHYSICS LETTERS
3 I)ecember 1984
SIMPLE CALCULATION OF THE LANDAU LEVELS OF NARROW-GAP SEMICONDUCTORS IN THE KANE MODEL S. ASKENAZY a, P.R. WALLACE b, R.A. STRADLING c, j. GALIBERT a and P. PERRIER a a Service CNRS des Champs Magndtiques Intenses, Laboratoire de Physique des Solides, INSA. 31077 Toulouse Cedex, France b McGill University, Montreal, Canada c University o f St-Andrews, St-Andrews, Scotland, UK
Received 15 October 1984
By writing the equation for the Landau levels of a narrow-gap semiconductor in a new form, we present simple and rapid methods of calculation of these levels valid for arbitrary values of the gap, however small, provided that the energy is small relative to the spin-orbit coupling parameter a.
In the Kane model [1,2] for narrow-gap semiconductors, which takes account only of S- and P-bands, the heavy-hole band separates out and is completely flat. For the remaining three bands, the energy ~, is given by
k21
-/A/a12}.
furthermore pI2/MI 2 = h w c
defines the cyclotron frequency coc. It is also useful to define
X(eg + x)(Eg + A + x) =h2x2 {(Eg + ~A + X)l(2n + 1)/12 +
h2x2(Eg + ~A)/(Eg + A) = Eg fi2/2M,
(4)
U = ~A/(Eg + ] A ) ,
(1)
Eg is the energy gap, A the spin-orbit coupling parameter, X a momentum matrix element and I the cyclotron radius (ti/eB) 1/2. k z is the wave number parallel to the field./= +l distinguishes the two spin states. When the gap Eg is small, non-parabolicity is important, so we may not assume that X "~ Eg. A, however, is usually not small, so a first approximation to (1) may be found by assuming ~.,~ A, to give ~k(Eg + X) = [~2X2/(Eg + a ) l
in terms of which the spin term takes the form -/Eg × ~i~¢ocU. This gives a zero-energy&factor of (5)
g* = (U/M) m ,
m being the free electron mass. Finally, then we may write (2) in the form
X(Eg + x) =
l(n + ½ -
/tO O c + h2k
/2Ul.
Let us now look at the corrections to this lowest approximation. We may write (1) in the form X(Eg + ~k) = h 2 X2 { h(Eg + A)(Eg + A + X) -1
X {(Eg + ] a ) [ ( 2 n + 1)/12 + k21 - ] A / 3 1 2 } .
(2) × [(2n+ 1)/12 + k 21 + 11 - ~ , ( E g + A + ~ , ) - l l
We may define the zero-energy effective massM by M -1 = 2 (x2/Eg)(Eg + ]A)/(Eg + A).
So that
184
(3)
X ((Eg+ ~A)[(2n + 1)/l 2 + k 2 ] . - 1 A / 3 1 2 ) ) . The lowest approximation corresponds to keeping on the right-hand side only the terms which do not have X as a multiplier. The remaining terms may be 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 106A, number 4
PHYSICS LETTERS
(b) For numerical calculation, a convenient procedure is to solve (6) iteratively. We may write it in the form
grouped together as h2X 2 X(Eg + A + X) -1 {(Eg + A)[(2n + 1)]! 2 + kz2 ] - [(Eg + ~A)[(2n + l/l 2) + k 2]
),2 + XEg(1 - ~) -
-jA]312])
= ~h2X 2 ~k(Eg + A + ~k)-1 A[(2n + 1)/l 2 +k2+j/12]. Substituting again for ×2 from (3) and cancelling out the factor Eg + X which occurs on both sides of the equation we get the supplementary terms in the form
eg X(eg + ~ + x)-lu[(.
+~+}/)t,~c+t,2k2/~].
Eq. (1) may therefore be written exactly as X(/~.g + ),) = Eg {(n + ~ - ~]U)?/6o c +
+h2k2/2M
3 December 1984
EgE0 = O,
where
= [U/(Eg + A + •)-I ][(n + 1I-+ I/)h6Oc + pt2k2z/2M], (7) and
E 0 =(n + ½ - ½/U)•6o c
+n2k2/2M.
The solutions are 1 2 X= - } E g O - ~) + [;IE~(1 - ~)2 +EgE0 ] 1/2
(6)
xU(eg + a + x)-I [(. + } + }/)h~Oc + t,Zk2z/2M]}.
tl2k2/2m
The "free-electron" term which appears on the diagonal of the matrix hamiltonian of the Kane model cannot be taken into account in a simple way [3,4], since it does not commute with the rest of the hamiltonian; we shall therefore omit it from these considerations.
If ~(s) is calculated from (7), with a given approximation h (s) to X, eq. (8) will give the next approximation ~(s+l). (c) Another approximation, in general quite accurate, consists of developing ;k/(Eg+ A + X) in a series: 7,/ (~/~Eg+A) - X2/(Eg + A) 2 + .... where higher powers of + A) are neglected. To this approximation the eigenvalue equation remains quadratic: X2(1 + b) + XEg(1 - a) - EgE 0 = 0.
Further discussion. (a) We note that, f o r k z = 0, n = 0 a n d / = - 1 the "supplementary" terms in (6) are zero, and the lowest order of approximation gives the exact result. One might ask what has happened to the third root; in fact, it factors out and has the constant value h = --(Eg
(8)
Omitting
h2k2z/2M throughout
a = [U/(gg +
A)l(n + } + ½ / ) ~ w
for simplicity, e ,
b = aeg/(Eg + A).
+ zx).
(9)
(10) (l 1)
The solutions are then
Table 1 n
Itetative method
Formula (12)
s=O
s=l
s=2
spin ?
0 1 2 3 4
0.02 0.07 0.12 0.17 0.21
0.02644 0.08245 0.1247 0A602 0.1915
0.026428 0.082386 0.12472 0.16034 0.19176
0.02643 0.08233 0.1245 0.1598 0.1908
spin
0 I 2 3 4
0 0.09 0.14 0.19 0.22
0.045012 0.09452 0.1340 0.1679 0.1981
0.094509 0.13404 0.16804 0.19836
0.045012 0.09447 0.1339 0.1676 0.1975
185
Volume 106A, number 4
PtlYSICS LETTERS
= (1 + b) -1 (-½Eg( 1 - a) t 2 + [aE~(1 - a ) 2 + EgE0(1 + b ) ] l / 2 } .
(12)
Aside from the numerical iteration method (b), which can o f course be applied for any value of the parameters (though with a rate of convergence which improves as ~/(Eg + A) decreases), the approach outlined here is primarily useful (and accurate) when ~, ,~A. If this condition is satisfied, no additional condition on the gap energy Eg need be assumed; it may even be ze ro.
A numerical example. In order to test the accuracy of the formulae proposed, we have calculated, both the approximate formulae given in (c) and the iterative method outlined in (b), the levels n = 0, 1, 2, 3, 4 for each spin, in the conduction band, for the following parameters: Eg = 0.171 eV, A = 0.45 eV, hX = 8 eV A, B = 10 T, which are approximately those of Hg0.80CD0.20Se. The values obtained are shown in table 1. All energies are in electron volts. In the case of the iterative method we give for each level the initial guess, followed by the successive iterations. The result stabilizes at the last value given.
186
3 Dcccmber 1984
We note that the values given by the approximative formula (12) are always within less than 0.4% of the correct value. In the iterative method, the error after only one iteration starting from a fairly crude estimate, are always within 0.2% of the correct values. It will be noticed that (12) involves an error which increases with energy. Therefore, the level of accuracy shown in these calculations cannot be expected to persist at very high energy levels. It should be noted quite generally, however, that a very high level of accuracy can be attained using one iteration of (8) starting from the solution of (12), which is equivalent to giving a very accurate analytic expression for the energy.
References [1 ] E.O. Kane, J. Phys. Chem. Solids 1 (1957) 249. [2] W. Zawadski, in: New developments in semiconductors, eds. P.R. Wallace, R. Harris and M.J. Zuckermann (Noordhoff, Leyden, 1972) pp. 447 ff. [3] M. Singh and P.R. Wallace, Solid State Commun. 45 (1983) 9. [4] M. Singh, P.R. Wallace and S. Askenazy, J. Phys. C15 (1982) 6731.
PHYSICS LETTERS
3 I)ecember 1984
SIMPLE CALCULATION OF THE LANDAU LEVELS OF NARROW-GAP SEMICONDUCTORS IN THE KANE MODEL S. ASKENAZY a, P.R. WALLACE b, R.A. STRADLING c, j. GALIBERT a and P. PERRIER a a Service CNRS des Champs Magndtiques Intenses, Laboratoire de Physique des Solides, INSA. 31077 Toulouse Cedex, France b McGill University, Montreal, Canada c University o f St-Andrews, St-Andrews, Scotland, UK
Received 15 October 1984
By writing the equation for the Landau levels of a narrow-gap semiconductor in a new form, we present simple and rapid methods of calculation of these levels valid for arbitrary values of the gap, however small, provided that the energy is small relative to the spin-orbit coupling parameter a.
In the Kane model [1,2] for narrow-gap semiconductors, which takes account only of S- and P-bands, the heavy-hole band separates out and is completely flat. For the remaining three bands, the energy ~, is given by
k21
-/A/a12}.
furthermore pI2/MI 2 = h w c
defines the cyclotron frequency coc. It is also useful to define
X(eg + x)(Eg + A + x) =h2x2 {(Eg + ~A + X)l(2n + 1)/12 +
h2x2(Eg + ~A)/(Eg + A) = Eg fi2/2M,
(4)
U = ~A/(Eg + ] A ) ,
(1)
Eg is the energy gap, A the spin-orbit coupling parameter, X a momentum matrix element and I the cyclotron radius (ti/eB) 1/2. k z is the wave number parallel to the field./= +l distinguishes the two spin states. When the gap Eg is small, non-parabolicity is important, so we may not assume that X "~ Eg. A, however, is usually not small, so a first approximation to (1) may be found by assuming ~.,~ A, to give ~k(Eg + X) = [~2X2/(Eg + a ) l
in terms of which the spin term takes the form -/Eg × ~i~¢ocU. This gives a zero-energy&factor of (5)
g* = (U/M) m ,
m being the free electron mass. Finally, then we may write (2) in the form
X(Eg + x) =
l(n + ½ -
/tO O c + h2k
/2Ul.
Let us now look at the corrections to this lowest approximation. We may write (1) in the form X(Eg + ~k) = h 2 X2 { h(Eg + A)(Eg + A + X) -1
X {(Eg + ] a ) [ ( 2 n + 1)/12 + k21 - ] A / 3 1 2 } .
(2) × [(2n+ 1)/12 + k 21 + 11 - ~ , ( E g + A + ~ , ) - l l
We may define the zero-energy effective massM by M -1 = 2 (x2/Eg)(Eg + ]A)/(Eg + A).
So that
184
(3)
X ((Eg+ ~A)[(2n + 1)/l 2 + k 2 ] . - 1 A / 3 1 2 ) ) . The lowest approximation corresponds to keeping on the right-hand side only the terms which do not have X as a multiplier. The remaining terms may be 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 106A, number 4
PHYSICS LETTERS
(b) For numerical calculation, a convenient procedure is to solve (6) iteratively. We may write it in the form
grouped together as h2X 2 X(Eg + A + X) -1 {(Eg + A)[(2n + 1)]! 2 + kz2 ] - [(Eg + ~A)[(2n + l/l 2) + k 2]
),2 + XEg(1 - ~) -
-jA]312])
= ~h2X 2 ~k(Eg + A + ~k)-1 A[(2n + 1)/l 2 +k2+j/12]. Substituting again for ×2 from (3) and cancelling out the factor Eg + X which occurs on both sides of the equation we get the supplementary terms in the form
eg X(eg + ~ + x)-lu[(.
+~+}/)t,~c+t,2k2/~].
Eq. (1) may therefore be written exactly as X(/~.g + ),) = Eg {(n + ~ - ~]U)?/6o c +
+h2k2/2M
3 December 1984
EgE0 = O,
where
= [U/(Eg + A + •)-I ][(n + 1I-+ I/)h6Oc + pt2k2z/2M], (7) and
E 0 =(n + ½ - ½/U)•6o c
+n2k2/2M.
The solutions are 1 2 X= - } E g O - ~) + [;IE~(1 - ~)2 +EgE0 ] 1/2
(6)
xU(eg + a + x)-I [(. + } + }/)h~Oc + t,Zk2z/2M]}.
tl2k2/2m
The "free-electron" term which appears on the diagonal of the matrix hamiltonian of the Kane model cannot be taken into account in a simple way [3,4], since it does not commute with the rest of the hamiltonian; we shall therefore omit it from these considerations.
If ~(s) is calculated from (7), with a given approximation h (s) to X, eq. (8) will give the next approximation ~(s+l). (c) Another approximation, in general quite accurate, consists of developing ;k/(Eg+ A + X) in a series: 7,/ (~/~Eg+A) - X2/(Eg + A) 2 + .... where higher powers of + A) are neglected. To this approximation the eigenvalue equation remains quadratic: X2(1 + b) + XEg(1 - a) - EgE 0 = 0.
Further discussion. (a) We note that, f o r k z = 0, n = 0 a n d / = - 1 the "supplementary" terms in (6) are zero, and the lowest order of approximation gives the exact result. One might ask what has happened to the third root; in fact, it factors out and has the constant value h = --(Eg
(8)
Omitting
h2k2z/2M throughout
a = [U/(gg +
A)l(n + } + ½ / ) ~ w
for simplicity, e ,
b = aeg/(Eg + A).
+ zx).
(9)
(10) (l 1)
The solutions are then
Table 1 n
Itetative method
Formula (12)
s=O
s=l
s=2
spin ?
0 1 2 3 4
0.02 0.07 0.12 0.17 0.21
0.02644 0.08245 0.1247 0A602 0.1915
0.026428 0.082386 0.12472 0.16034 0.19176
0.02643 0.08233 0.1245 0.1598 0.1908
spin
0 I 2 3 4
0 0.09 0.14 0.19 0.22
0.045012 0.09452 0.1340 0.1679 0.1981
0.094509 0.13404 0.16804 0.19836
0.045012 0.09447 0.1339 0.1676 0.1975
185
Volume 106A, number 4
PtlYSICS LETTERS
= (1 + b) -1 (-½Eg( 1 - a) t 2 + [aE~(1 - a ) 2 + EgE0(1 + b ) ] l / 2 } .
(12)
Aside from the numerical iteration method (b), which can o f course be applied for any value of the parameters (though with a rate of convergence which improves as ~/(Eg + A) decreases), the approach outlined here is primarily useful (and accurate) when ~, ,~A. If this condition is satisfied, no additional condition on the gap energy Eg need be assumed; it may even be ze ro.
A numerical example. In order to test the accuracy of the formulae proposed, we have calculated, both the approximate formulae given in (c) and the iterative method outlined in (b), the levels n = 0, 1, 2, 3, 4 for each spin, in the conduction band, for the following parameters: Eg = 0.171 eV, A = 0.45 eV, hX = 8 eV A, B = 10 T, which are approximately those of Hg0.80CD0.20Se. The values obtained are shown in table 1. All energies are in electron volts. In the case of the iterative method we give for each level the initial guess, followed by the successive iterations. The result stabilizes at the last value given.
186
3 Dcccmber 1984
We note that the values given by the approximative formula (12) are always within less than 0.4% of the correct value. In the iterative method, the error after only one iteration starting from a fairly crude estimate, are always within 0.2% of the correct values. It will be noticed that (12) involves an error which increases with energy. Therefore, the level of accuracy shown in these calculations cannot be expected to persist at very high energy levels. It should be noted quite generally, however, that a very high level of accuracy can be attained using one iteration of (8) starting from the solution of (12), which is equivalent to giving a very accurate analytic expression for the energy.
References [1 ] E.O. Kane, J. Phys. Chem. Solids 1 (1957) 249. [2] W. Zawadski, in: New developments in semiconductors, eds. P.R. Wallace, R. Harris and M.J. Zuckermann (Noordhoff, Leyden, 1972) pp. 447 ff. [3] M. Singh and P.R. Wallace, Solid State Commun. 45 (1983) 9. [4] M. Singh, P.R. Wallace and S. Askenazy, J. Phys. C15 (1982) 6731.