Electronic States In An Equilateral Triangular Quantum Wire Rajesh Kumar1, S.C. Mukherjee2 and S.N.Singh3 1 St. Xavier’s College, Ranchi 834001 2 Yogada Satsang Mahavidayalaya, Ranchi 834004 3 University Department of Physics, Ranchi University, Ranchi 834008, Jharkhand Email:
[email protected]
Abstract: The states of an electron in a quantum wire with equilateral triangular cross-section have been discussed under the effective mass approximation. The similarities between the propagation of electron wave in quantum wires and electromagnetic waves in dielectric and metallic waveguides have been exploited to obtain an analytical solution leading to energy eigenvalues and wavefunctions representing the possible states of the electron under hard wall and massive wall confinements. The computed results are in qualitative agreement with the results obtained for right isosceles triangular and V-groove quantum wires under soft wall confinement. Keywords: Quantum waveguides, electronic states, Semiconductor Hetrostructures, electromagnetic waveguides, TE and TM modes, hard wall confinement, massive wall confinement. PACS No.: 73.21.-b, 73.21.Hb, 73.61.-r.
1. Introduction Many features of the propagation of electron wave in quantum wires can be studied on the basis of similarities with the propagation of electromagnetic waves in dielectric and metallic waveguides. The similarities have been investigated/used by a number of workers [1-4] under different contexts. It has been observed [5] that the consideration of massive wall confinement(MWC) [6-8], a concept introduced under the effective mass theory for solving problems involving semiconductor Hetrostructures , in theoretical investigation of electron/hole states in quantum wires of different geometries together with the conventional hard well confinement (HWC) establishes an interesting correspondence
between electron propagation in quantum wires and propagation of electromagnetic waves in metallic waveguides with perfect metallic boundaries. This correspondence is actually the consequence of similarities in the boundary conditions satisfied by the free electron wavefunction in quantum waveguides under MWC and HWC and longitudinal field components in case of metallic waveguides under TE and TM modes of propagation, respectively. The aim of this article is to use and further strengthen this correspondence exploiting many of the concepts used in the studies of the classical electromagnetic waveguides to the case of quantum waveguides /wires as well. 1
Recent advances in micro fabrication techniques have resulted into the fabrication of quantum wires of different configuration. In any such realistic quantum wire structure the confinement potential is bound to be finite giving rise to what is known as the soft wall confinement (SFC). Many such structures with cylindrical geometry have been investigated [913] using these modes of confinement. However, when the wire posses non- circular cross-sectional geometry as in case triangular or V- groove quantum wires, the application of SWC boundary conditions, requiring the continuity of the wavefunction and the quantity “wavefunction divided by the effective mass of the electron in the respective region” across the well and barrier interfaces, become difficult. Hence, some approximate/numerical method has to be used as has been done in case of right angled isosceles triangular quantum wire [14] and v-groove quantum wire [15]. In this article, we investigate the states of an electron in a triangular quantum waveguide with equilateral cross-section under HWC and MWC making use of the concepts employed in studies of classical electromagnetic waveguides with perfect metallic boundaries [16] and incorporating quantum mechanical features similar to that used for solving barrier problems in quantum mechanics. Thus, the combined approach leads to analytical expressions for energy spectrum and wavefunction, in form of a suitable combination of rectangular harmonics, for the electron. These results can be used for further investigation of the system of the stated geometry under SWC using the approach employed under in Ref. [14] or by using any other numerical /approximate method for comparison with experimental data. However, this has not been attempted here. The ground state energy has been computed and plotted
against the side of the triangle showing the same qualitative behavior as observed in case of isosceles right triangular quantum wire [14] and V-groove in our case are bound to be higher as we have only considered complete confinement situations whereas in Refs. [14] and [15], the electron has higher spatial spread on account of finite well-barrier conduction band offset leading to lower energy values. The energy of excited states can also be computed and information pertaining to these states can be obtained. We have calculated wavefunction at each point inside the triangle and plotted it as three dimensional plots. These plots are consistent with their respective boundary conditions.
2. Theory The cross-sectional plane of electron wave guiding structure under consideration is depicted in Figure -1 with z- axis taken to be perpendicular to plane of the figure. Each side of the equilateral triangle is ‘a’ and the equations representing the three sides are: for OP, = =
√ .
√
, for OQ, = −
Also
√
and for PQ,
∠ = 30 ̊ = ∠ . The
electron is free to move in the well region assumed with the electron would be the solution of the free particle effective mass Schrödinger equation and may be represented by r r
r
ψ 0 ( r ) = e − ik 0 . r ,
(1)
r where k 0 is the wave vector associated with the
r
r
electronic motion and is given by k0 = k0 kˆ0 . Here kˆ0 is a unit vector in the direction of propagation of the electron and in, general, it has the form 2
kˆ0 = α Xˆ + βYˆ + γ Zˆ
(2)
with , as direction cosines relative to the Cartesian coordinate system as shown in
Xˆ , Yˆ , Zˆ are unit vectors along r three orthogonal axes. k 0 is related to total
Fig. (1) and
of the electron can be constructed using the fact that in case of complete electron confinement the possible states would be the result of superposition of all possible, but independent, reflections of the electron wave Table: 1 Set of reflected unit vectors for different directions of incidence
energy of the electron as
Etotal =
h2 2 h2 2 k = E + kz , 0 2m* 2m*
(3) 1
α
β
Reflected – unit – vectors
(+)
(+)
−α Xˆ + β Yˆ 1 α + β 3 Xˆ + α 3 − β Yˆ 2 1 α − β 3 Xˆ + −α 3 − β Yˆ 2 α Xˆ + β Yˆ 1 −α + β 3 Xˆ + −α 3 − β Yˆ 2 1 −α − β 3 Xˆ + α 3 − β Yˆ 2
where E is the energy due to confined motion in the transverse plane , m* is the effective mass of the electron in the well material and
ħ = with h as Planck’s constant. KZ is equal to k0 as indicated in Eq. (2). E will be
2
(–)
(+)
Y
P 3
a O
(+)
( )̶
( (
) ( ) (
( (
) ( ) (
X
4
(–)
(–)
determined using the boundary conditions depending on type of confinement. The second term in right hand side of Eq. (3) is the kinetic energy of free motion along z-axis. The wavefunction representing the possible states
)
(
)
) (
)
α Xˆ − β Yˆ 1 −α − β 3 Xˆ + −α 3 + β Yˆ 2
(
Figure 1: An equilateral triangular quantum wire with z axis perpendicular to the plane of figure.
) )
1 α + β 3 Xˆ + −α 3 + β Yˆ 2
(
Q
)
−α Xˆ − β Yˆ 1 α − β 3 Xˆ + α 3 + β Yˆ 2
(
GaAs
)
) (
)
1 −α + β 3 Xˆ + α 3 + β Yˆ 2
(
) (
)
from the walls of the barrier. The idea is similar to the total internal reflection in case of dielectric waveguides or reflection of electromagnetic waves appropriate boundaries 3
in metallic waveguides. The reflected wavevectors can be constructed using the relation [17]
kˆr = kˆi − 2nˆ j (kˆi .nˆ j ),
ψ ( x , y ) = c11 .e ik α x e − ik β y + c12 .e
1 − ik . x α + 3 β 2
)
1 − ik . x α − 3 β 2
)
+ c13 .e
(4)
(
(
+ c 2 2 .e
reflection, kˆi is a unit vector in direction of
(
1 − ik . x − α − 3 β 2
(
+ c 23 .e
vectors normal to walls PQ, OP and OQ of the well- barrier interfaces, respectively. These have not been shown in figure but are represented mathematically as
+ c 31 .e ik α x e ik β y + c 32 .e + c 3 3 .e
(5)
kˆ = α Xˆ + β Yˆ ,
+ c 43 .e
r
r
but k
r
r will be two dimensional counterpart of k 0 . The set of unit vectors obtained in the directions of reflection are given in Table-1. The complete wavefunction obtained after the superposition of all the independent reflected wave vectors given in the fourth column of Table -1 is given by:
3α −β
)
1 − iky − 3 α − β 2
(
(
(
)
)
)
)
.e
.e
(
1 − iky 2
1
.e 2
.e
1 iky 2
)
iky
(
(
3α +β
)
3α −β
)
3α +β
)
1 − iky − 3 α + β 2
(
1 − ik . x − α − 3 β 2
(
1 − ik . x − α + 3 β 2
(
)
)
.e
)
.e
1 − iky − 3 α + β 2
1 − iky 2
(
(
3α +β
(7) where the coefficients are such that 4
4
∑∑ c
2
ij
assumed to be total, hence k0 = k
1 − ik . x α − 3 β 2
1 − ik . x α + 3 β 2
+ c 4 2 .e
(6)
omitting z-part which will remain constant. Furthermore, the magnitude after reflection will be unchanged as the reflections have been
(
+ c 41 .e − ik α x e ik β y
where δ m , n is the Kronecker delta symbol. It can be seen that if we substitute Eqs. (2) and (5) in Eq. (4), all reflected unit vectors will have the same z- component, hence, it is easier to take the unit vectors in the direction of incidence as
.e
1 − ik . x − α + 3 β 2
incidence and nˆ j with j=1,2 and 3 are unit
1 ˆ j +1 X (1 + δ 1, j ) + ( − 1 ) Yˆ 3 (1 − δ 1, j ) , 2
1 − iky 2
+ c 2 1 .e − ik α x e − ik β y
where kˆr is unit vector in the direction of
nˆ j =
.e
=1 .
(8)
i =1 j =1
The boundary conditions corresponding to the HWC require that the wavefunction must vanish at =
√
, = −
√
and x=
√
for all y. On
applying these conditions one by one and making simplifications, the constants appearing in Eq.(7) can be eliminated. The final solution leads to the even parity wavefunction [16] given by ψ ( x , y ) = A[sin 2 k1 x.cos 2 k 2 y − sin k 3 x.cos k 4 y − sin k 5 x.cos k 6 y ],
(9)
4
)
,
)
where A is an arbitrary constant, which may be complex, and for odd parity the wavefunction would be
The boundary conditions corresponding to the MWC require that [17]
ψ ( x, y ) = B[sin 2k1 x.sin 2k2 y + sin k3 x.sin k4 y
∂ψ
+ sin k5 x.sin k6 y],
and
where B is another constant. In Eqs. (9) and (10), k1, k2,- - -, k6 are given by
k2 = k β =
2nπ with n=0,1,2,3,-----, , 3a
( x, y ) =
− 3
∂x
(10)
2mπ , with m=0,1,2,3,-----, 3a
3
∂x ∂ψ
k1 = kα =
( x, y ) =
∂ψ
∂ψ
∂y ∂ψ
( x, y ) = 0
∂x
( x, y ) at
y =
( x, y ) at
∂y
at x =
x , 3
y = −
x , 3
3a fo r a ll y . 2
(13) When these conditions are applied, one by one, to the wavefunction in Eq. (7) and on some algebraic simplifications, we get the even parity wavefunction as ψ
(x, y ) =
C [c o s 2 k1 x c o s 2 k 2 y + c o s k 3 x × cos k4 y + cos k5 x cos k6 y ]
k3 =
k4 =
k5 =
2π ( m + n ) 3a
,
(14) and the odd parity wavefunction as
2π ( n − 3m )
and k6 =
,
2π ( n + 3m ) 3a
(11)
.
For even parity states m ≠ 0 and m ≠ n. Similarly, for odd parity states m ≠ 0, n ≠ 0, m ≠ n and n ≠ 3m. These conditions are due to the requirements that wavefunction must be complete and should not be zero except as required by the boundary conditions. The
energy 2
E=
2
h 4π 2 m* 3 a 2
(x, y ) =
D [c o s 2 k 1 x s in 2 k 2 y − c o s k 3 x s in k 4 y − c o s k 5 x s in k 6 y ],
2π ( m − n ) 3a
ψ
,
3a
spectrum
is
given
2 1 2 m + 3 n = Em , n ( say )
by (12)
with m and n satisfying the conditions stated above for even and odd parity states.
(15) where k1, k2,….., k6 are given in Eq. (11) and for the even function in Eq.(14) all integral values of m and n are allowed whereas for odd function of Eq.(15) n≠0 and n≠3m. C and D are arbitrary constants to be determined by the normalization condition. The energy eigenvalues are given by Eq. (12) but now m and n will satisfy the requirements of wavefunctions in Eqs. (14) and (15). Furthermore, the parity consideration of all wavefunctions is about the axis of symmetry, that is, about x-axis and has been tested through computations of the wavefunctions. It is to be noted that solutions corresponding to the HWC boundary condition, which is nothing but the Dirichlet boundary condition, are available in different, but mathematically equivalent, forms in some standard texts[18-21] 5
MWC:Even and Odd Parity 450 400 Ground state energy in meV
and similar expressions have been obtained earlier during modal analysis under TM and TE modes in case of classical electromagnetic waveguides with perfect metallic boundaries[16] but no attempt has been made to deal with kind of quantum system discussed in this article based on the analogy mentioned earlier.
350 300 250
even
200 150 100 odd 50
3. Numerical Results
0 5
10 15 sidelength of triangle in nm
HWC:Even and Odd Parity
20
800
Figure 3: Plot of GS energy as a function of the side of the triangle
Ground state energy in meV
700
600
odd
500
400
300
even 200
0
5
10
15
20
sidelength of triangle in nm
GS even wavefunction
4 100
10
2
8
0
6
-2 5
4
Figure 2: Plot of GS energy as a function of the side of triangle.
-5
y
x
0
Figure 4: Plot of GS even wavefunction in HWC for side length 10 nm .
2
GS odd wavefunction
The eigenvalues have been computed considering the well material to be made up of GaAs in which the effective mass of electron is 0.0665 times the free electron mass. In Fig. (2), the variation of the ground state (GS) energy, corresponding to HWC , as a function of the side of the triangle for even and odd parity solutions have been shown. For MWC the same plots are shown in Fig.(3). The results that energy values should increase with decreasing spatial confinement [11-13] is true in this situation also. The curves have the same qualitative behavior as observed under SWC incase of isosceles right triangular [14] and Vgroove [15] quantum wires. It is to be noted here that our results cannot be compared with
2
0
1 0 -1 10
-2 5 5 0
y
-5
0
x
Figure 5: Plot of GS odd wavefunction in HWC for side length 10 nm.
6
experimental data in view of the fact that HWC and MWC are two theoretical concepts treated on equal footing based on the assumption that in the former case barrier potential is very high in comparison to well potential and in the latter case the effective mass of the carrier is very high, tending to infinity, in the barrier region in comparison to the well region. It has
3
GS odd wavefunction
2 1 0 -1 -2
10
-3 5
5 0 -5
0
x
y
Figure 6: Plot of GS odd wavefunction in MWC for
side length 10 nm
GS even wavefunction
of y, varying from =
x=
4 2 10 0
8 6
-2 5
4 2
0
y
MWC. This is a consequence of the different nature of boundary conditions, the former satisfying the Dirichlet type boundary condition whereas the latter satisfies the Neumann type condition. This feature was also observed in case of quantum wire with circular boundaries. We have also computed the energy eigenvalues and corresponding wavefunctions pertaining to excited states and it has been observed that for all the situations Em,n ≠ En,m. Furthermore, for HWC and for both the even and odd parity, only those states exist for which (m+n) is odd but no such condition holds for MWC in which case (m+n) can be odd or even. The nonnormalized wavefunction for the ground state has been computed for both modes of confinement and for even and odd parity states. The three dimensional plots in Figs.(4-6) are the plots of the probability density(PD) as a function
-5
0
x
Figure 7: Plot of GS even wavefunction in MWC for side length 10 nm
been shown [5,8,12,13] that HWC and MWC are the special cases of SWC under appropriate limiting conditions. It is to be noted that HWC even parity ground state has lower energy in comparison to odd parity state but the reverse is the case with
√ ,
√
to = −
√
and
i.e. , along a line perpendicular to X-axis
and lying within the well region. These plots are as expected and are consistent with their respective boundary conditions. The plot of PD for even parity state under MWC has not been shown here because it is almost similar to that in Figure-4 except at the end points where it is not zero in view of the fact that under MWC the normal derivative of the wavefunction is zero and not the function itself. In all these figures, Figs. (4-7), y –axis is in arbitrary units because the wavefunctions are not normalized. 4. Conclusion The states of an electron in a quantum wave guiding structure have been discussed making use of the effective mass theory, some of the quantum mechanical concepts and the similarities between the free electron wave propagation of electromagnetic waves in 7
classical metallic waveguides. Expressions pertaining to energy eigenvalues and the eigenfunctions of the electron have been obtained explicitly under HWC and MWC and results have been discussed through numerical computations. The results obtained here cannot compared directly with experimental data because in any realistic system of this nature the electron confinement is achieved though a finite confining potential arising due to finite conduction band offset between the well and the barrier materials, however , some of our results show the same qualitative behavior as reported for isosceles right triangular quantum wire [14] and V-groove quantum wire [15], which have been fabricated and studied extensively. Moreover, the wavefunctions obtained here can form the basis for further studies of the system under the SWC similar to that in Ref. [14]. References [1] J.P. Carini, J.T. Londergan, Kieran Mullen and D.P. Murdock, Phys. Rev. B46,15538 (1992); 48,4503(1993); J.P. Carini, J.T. Londergan, D.P. Murdock, D. Trincle and C.S. Young , ibid, 55, 9842(1997). [2] P.K. Tien, Rev. Mod. Phys. 49,361(1977). [3] R.J. Black and A. Ankiewicz, Am. J. Phys. 53, 554 (1985) [4] J.D. Maynard, Rev. Mod. Phys. 73 , 401 (2001). [5] S.K.Dey, S.N.Singh, A. Kapoor and G.S. Singh, Phys. Rev. B67, 113304 (2003). [6] G. Bastard, Wave Mechanics Applied to Semiconductor Hetrostructures (Les Editions de Physique, Les Ulis, France, 1988). [7] M.G. Burt, J.Phys. Condens. Matter 4 , 6651 (1992).
[8] P.C. Hammer and D.T. Wang, Phys. Rev. B47, 6603(1993) [9] N.C. Constantinou and B.K. Ridley, J. Phys. Condens. Matter 1 ,2283 (1989) [10] N.C. Constantinou , M. Masale and D.R. Tilley, J.Phys. Condens 4, 4499 (1992) [11] M.Masale, N.C. Constantinou and D.R. Tilley, Phys. REV. B46, 15432 (1992) [12] S.N. Singh , S.K.Dey and G.S. Singh , Ind. J. Phys. 70A, 715 (1996) [13] S.N. Singh, Achint Kapoor, S.K.Dey and G. S. Singh, in Physics of Semiconductor Nanostrucures: Proceedings of a National Symposium, edited by K.P. Jain (Narosa Publisihng House, New Delhi, 1997), pp.131-138. [14] Samita Gango Padhyay and B.R. Nag, J.Appl. Phys. 81 (12),7885 (1997) [15] Geraido Crecl and Gerald Weber, Semicond. Sci. Technol. 14, 690 (1999). [16] P.L. O verfelt and D.J. White , IEEE Transactions on MTT MTT34 ,161 (1986) [17]] Jaroslaw Klos,Physica, E36,154 (2007) [18] M.Siemoni and M. Jofre, IET Microw. Antennas Propag., 04(3), 297 (2010)
[19] S.A. Schelkunoff, Electromagnetic Waves, ( New York, Van Nostrand,1943) [20] Mathews and Walker, Mathematical Methods of Physics, ( W. A. Benjamin, INC. New York 1970). [21] A. V. Rozokov and Franco Nori Phys. Rev.B (81) , 155401 (2010)
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