Amplitudes of Multichannel Scattering on a Two-Dimensional Potential Barrier

with Constant Height D. M. Sedrakiana, L. R. Sedrakianb, and A. V. Margaryana

aYerevan State University, Yerevan, Armenia bRussian–Armenian (Slavonic) University, Yerevan, Armenia

Received November 13, 2010

Abstract⎯The immersing method is applied to solve the N-channel scattering problem for concrete potential. In particular, we consider the particle scattering on a two-dimensional potential barrier, which is constant in the scattering direction and arbitrary in the cross-section direction. For this case the scattering amplitudes mt and mr ( 1,2, , )m N= … are determined. A transition from the obtained formulas to the case of δ-potential is performed. For this case transmission amplitudes mt and reflection amplitudes mr are obtained. It is also shown that the product of transmission and reflection amplitudes along the channel m does not depend on the scattering channel.

DOI: 10.3103/S1068337211030017 Keywords: two-dimensional potential barrier, multichannel scattering, reflection, transmission

1. INTRODUCTION Recently, in connection with increasing possibilities of nanotechnology to create low-dimensional

structures, it becomes possible to fabricate structures with arbitrary given structural features. Intense investigations of physical properties of one- and quasi-one-dimensional structures began in the middle 70s, when it became possible to realize so-called superlattices, i.e., periodic systems formed by alteration of two or several structural elements. Nowadays such systems can be considered as have received the most study. Modern technologies allow manufacturing the systems with a more complex structure, namely, the so-called low-dimensional aperiodic structures, where the spatial periodicity of structural elements made of the same material, is disrupted. At present, the intense study of physical properties of low-dimensional systems with a complex structure takes place. It should be noted that theoretical study of low-dimensional structures and waves, propagating in them, is a complex mathematical problem. By now a number of exact and approximate methods have been developed for description of waves in one-dimensional and quasi-one-dimensional media. The best known among these methods are perturbation theory, transfer-matrix method, immersing method, phase function method, and method of Green’s functions [1].

One can state that up to now there are not general and exact theoretical approaches for consideration of these problems in two- and three-dimensional statements. Even within the framework of approximate methods, derivation of a final result requires a great number of numerical calculations. For such problems in [2] a method was developed, which is a generalization of the immersing method.

In this paper, by using this method we study the problem of multichannel scattering of a quantum particle on an arbitrary two-dimensional potential. This problem differ from the one-dimensional one: in our case the particle scattering can be accomplished by changing the energy level of the transverse motion. These levels are discrete because the particle motion is confined in this direction. In actuality the number of levels is infinite, but depending on the initial energy the energy levels, which are higher than some level, are not excited. Here we assume that only the lowest N energy states are excited. We consider the problem of particle scattering on a barrier, the height of which is constant in the scattering direction and is arbitrary in the perpendicular direction. As a result of investigation, the amplitudes of transmission

1 2, , , Nt t t… and reflection 1 2, , , Nr r r… will be obtained. Consider, in particular, multichannel scattering of an electron on the potential ( ) ( ) ( ), ,V x y V x V y=

where

ISSN 1068–3372, Journal of Contemporary Physics (Armenian Academy of Sciences), 2011, Vol. 46, No. 3, pp. 93–98. © Allerton Press, Inc., 2011. Original Russian Text © D.M. Sedrakian, L.R. Sedrakian, A.V. Margaryan, 2011, published in Izvestiya NAN Armenii, Fizika, 2011, Vol. 46, No. 3, pp. 147–154.

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( ), ,

0, .K a x b

V xa x b≤ ≤⎧

= ⎨ ≥ ≥⎩ (1)

Here K is a constant and ( )V y is an arbitrary function satisfying the condition ( ) ( )0 .V V c= =∞ According to [2–6], for deriving the multichannel scattering amplitudes one should solve the following

system of linear differential equations with respect to the functions ( ) ( ) ( )1 2, , , :NL x L x L x…

( ) ( ) ( )2

22 0, 1,2, ,

Nm

m m mi ii m

d L xq L x V L x m N

dx ≠

+ − = =∑ … (2)

where

( ) ( ) ( ) ( )

2 2 2 2 2

0

, , ,

.

m m mm m m m

a

mn m n

q k V k ma

V x K y V y y dy∗

π= − = χ − χ χ =

= Φ Φ∫ (3)

Here the functions ( )n yΦ are defined as ( )( )2 sin ,n a a ny⎡ ⎤Φ = π⎣ ⎦ where 1,2, , ,n N= … and the potentials ( )mnV x are nonzero in the region a x b≤ ≤ and vanish at x a≤ and .x b≥

The system of equations (2) is integrated up to the point x b= with the following boundary conditions at the point x a= [7]:

( )

( )

1 111 1, ,

0, 2,3, .

ik a ik a

x a

mm

x a

dLL a e ik edx

dLL a m Ndx

− −

=

=

= − =

= = = … (4)

Let us denote the values of the functions ( ) ( ) ( )1 2, , , NL x L x L x… at the point x b= by 1 2, , , NL L L… . The scattering amplitudes are determined by the quantities 1 2, , , ND D D… and 1 2, , , ND D D… which are connected with 1 2, , , NL L L… by the following relations [6]:

( ) ( )( )( ) ( )( )

1 2 , 1,2 , ,

1 2 , 1,2 , ,

m

m

ik bm m m

ik bm m m

D b M L e m N

D b M L e m N

−= + =

= − =

…

… (5)

where

1 .mm

m x b

dLMik dx =

= (6)

According to [6, 7], the scattering amplitudes mt and mr are defined by the following formulas:

( )21 1 1 1 1 1

2 *

, ,

, , 2,3, , ,m m m m

t D D r D D D D

t D D r D D m N

∗∗

∗

= =

= = = … (7)

where

2 2 21

2 1

Nm

mm

kD D Dk=

= +∑ . (8)

Thus, by integrating the system of equations (2) and determining the values of the functions ( ) ( ) ( )1 2, , , NL x L x L x… at the point x b= one can derive the scattering amplitudes mt and .mr

2. SOLUTION OF THE SYSTEM OF EQUATIONS FOR THE FUNCTIONS ( )mL x

We seek for the solution of the system of equations (2) in the following form:

( ) ( ) .ii x am mk

kL x A e χ −=∑ (9)

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Here iχ are yet indeterminate constants and the quantities miA are determined from algebraic equations which are obtained by the substitution of solution (9) into the system of equations (2):

( )( )

( )

2 21 1 12 2 1

2 212 1 2 2 2

2 21 1 2 2

0,

0,

0.

i i i N Ni

i i i N Ni

N i N i i N Ni

q A V A V A

V A q A V A

V A V A q A

−χ + − − − =

− + −χ + − − =

− − − + −χ + =

. . . . . . . . . . . . . . . . (10)

The constant iχ are derived from the condition that the determinant of system (10) is equal to zero

( )( )

( )

2 21 12 1

2 212 2 2

2 21 2

0.

i N

i N

N N i N

q V V

V q V

V V q

χ −

χ −=

χ −

. . . . (11)

Let us denote these constants by iQ± and take into account that their number is 2N. Here we assume that the derived roots differ from each other. If the roots are repeated, the solution is constructed in another way.

If equality (11) is fulfilled, then from the system of equations (10) one can obtain the quantities 1 .mi mi ic A A= (12)

If we write down solution (9) in the form

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( )

1 1 11

1 11 1

,

, 2,3, , ,

i i

i i i i

NiQ x a iQ x a

i ii

N NiQ x a iQ x a iQ x a iQ x a

m mi mi mi i ii i

L x A e B e

L x A e B e c A e B e m N

− − −

=

− − − − − −

= =

= +

= + = + =

∑

∑ ∑ … (13)

then

( ) ( )( )( ) ( )( )

11 1

1

1 11

,

, 2,3, , .

i i

i i

NiQ x a iQ x a

i i ii

NiQ x a iQ x am

i mi i ii

dL iQ A e B edxdL iQ c A e B e m Ndx

− − −

=

− − −

=

= −

= − =

∑

∑ … (14)

Substituting formulas (13) and (14) into boundary conditions (4), we get

1

1 1, 0, 2,3, , ,

N Nik a

i mi ii i

F e c F m N−+ +

= =

= − = =∑ ∑ … (15)

1

1 1, 0, 2,3, , ,

N Nik a

i mi ii i

F e c F m N−− −

= =

= − = =∑ ∑ … (16)

where

( )1 1 1 11

, .ii i i i i i

QF A B F A Bk

+ −= + = − − (17)

The quantities iF + and iF − are connected by the evident relation

( )1 1 1 11

,ii i i i i i

QF A B A B Fk

+ −= + = − − = (18)

i.e., it is sufficient to find, for instance, iF + from the system of equations (15). The determinant of this system coincides with the determinant of the matrix mic since, according to formula (12), 1 1ic = for all values of i. Solving the system of equations (15) and substituting the expressions

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1 11 1

1 11 , 12 2i i i i

i i

k kA F B FQ Q

+ +⎛ ⎞ ⎛ ⎞= − = +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ (19)

into formula (13), for the functions ( )mL x and ( )mM x we obtain

( ) ( ) ( )1

1cos sin

N

m mi i i ii i

kL x c F Q x a i Q x aQ

+

=

⎛ ⎞= − − −⎜ ⎟

⎝ ⎠∑ , (20)

( ) ( ) ( )1

1 1

cos sinN

im mi i i i

im

QkM x c F Q x a i Q x ak k

+

=

⎛ ⎞⎛ ⎞= − − − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

∑ . (21)

The desired quantities mD and mD ( )1,2, ,m N= … are determined by substitution of expressions (20) and (21) into formulas (5).

3. DETERMINATION OF THE QUANTITIES ( )mD b AND ( )mD b

By substituting the functions ( )mL x and ( )mM x at the point x b= into formula (5) we obtain

( ) ( ) ( )

( ) ( ) ( )

1 1

1 1

1 1 2

1

1 1 2

1

1 1 cos sin ,2

1 1 cos sin ,2

N

N

m

m m

m

dN i k k i k k xim mi i i i

i m m i

dN i k k i k k xim mi i i i

i m m i

Qk kD b c Q d i Q d e ek k Q

Qk kD b c Q d i Q d e ek k Q

− − ++

=

+ − −+

=

⎛ ⎞⎛ ⎞ ⎛ ⎞= Φ − + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞ ⎛ ⎞= Φ − + + +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

∑

∑ (22)

where 1ik a

i iF e−+ +Φ = , (23)

Nx is the coordinate of the center, and d is the barrier width. Let us find the scattering amplitudes for a δ-shaped potential

( ) ( ) ( )0 0, .V x y P x x y y= δ − δ − (24)

The quantities mD and mD for this problem can be determined from formulas (22), if the potentials ikV in them are chosen so that they are finite at 0.d → Other quantities proportional to d, naturally, will

tend to zero. With this aim it is necessary to replace ( )V y by ( )0y yδ − in formulas (3) and require that

0lim .dK

dK P→→∞

= (25)

In this case the expression ikdV is replaced by a finite quantity vik ( ) ( )0 0v .ik i kP y y= Φ Φ (26) It is also necessary to take cos 1,iQ d ≈ sin i iQ d Q d≈ and replace 2

iQ d by 2.ix Then, taking into account the relations

1 1

1, 0, 1N N

i mi ii i

c m+ +

= =

Φ = − Φ = ≠∑ ∑ , (27)

following from expressions (15) and (23), for 1 1, , ,mD D D and mD we get the expressions

( ) ( )

1

1 1

211 111 1

1 1

1 1

1 , ,2 2

, ,2 2

N

m N m N

ik x

i k k x i k k xm mm m

m m

iu iuD D ek k

iu iuD e D ek k

−

− − − +

= + =

= = (28)

where

21

1.

N

m mi i ii

u c x+=

= Φ∑ (29)

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Determining i+Φ from equations (27) and substituting it into formula (29), for the quantity 1mu we

derive

2 2 21 1 2 2

21 22 2

1 21

c c c 1 1 1

m m mN N

N

N N NNm

x x xc c c

c c cu = −

. . .

21 22 2

1 2

1,2, , .

N

N N NN

m N

c c c

c c c

= …

. . .

(30)

Let us take the m-th equation from the system of equations (10), multiply in by small d and, changing the index I from unity to N, construct a new system of equations for determination of 1v m ( )1,2, ,m N= … . In order to obtain this system of equations, one must tend d to zero and K to infinity so that the expression dK is changed by P and, hence, ikdV by v .ik By solving the obtained system of equations with respect to 1v m we can check that 1 1vm mu = for 1,2, , .m N= … Thus, formulas (28) can be rewritten in the following form:

( ) ( )

1

1 1

211 111 1

1 1

1 1

v v 1 , ,2 2

v v, , 2,3, , .2 2

N

m N m N

ik x

i k k x i k k xm mm m

m m

i iD D ek k

i iD e D e m Nk k

−

− − − +

= + =

= = = … (31)

Substituting expressions (31) into formulas (7), we can determine the scattering amplitudes for the problem with potential (24).

4. DISCUSSION OF RESULTS Using expressions (22) for the functions mD and ,mD from formulas (7) we can derive the scattering

amplitudes and coefficients of transmission and reflection for the multichannel scattering both for potentials (1) and (24). The obtained expressions of scattering amplitudes for potential (1) are very cumbersome; therefore here we present their expressions only for potential (24). By substituting the expressions for mD and mD from expression (31) into formula (7) we get the following expressions for the scattering amplitudes:

( ) ( )

1

1 1

211 11 11

1 1 11 1 112

221 22111

22 121 1

1 1

2211

2 11

v v v1 12 2 2

, ,v vv14 4 4

v v2 2,

v v4 4

N

m N m N

ik x

NNm

mm m

m m

i k k x i k k xm m

m mm mN

mm

m m

i ii ek k k

t r

k k k k k

i e i ek kt r

k k k

==

− +

=

⎛ ⎞ ⎛ ⎞− − −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠= =⎡ ⎤ ⎡ ⎤⎛ ⎞Γ +⎢ ⎥ + Γ +⎜ ⎟ ⎢ ⎥⎣ ⎦ ⎝ ⎠ ⎣ ⎦

= − = −⎡ ⎤Γ +⎢ ⎥ Γ +⎣ ⎦

∑ ∑

∑12

2

, 2,3, , ,N

m m

m N

k=

=⎡ ⎤⎢ ⎥⎣ ⎦

∑…

(32)

where ( )2 211 11 v 4 .kΓ = +

Using formula (32), one can easily check the equality

( )2 2 2 21 1

2 1

1,N

mm m

m

kt r t rk=

+ + + =∑ (33)

i.e., the condition of continuity of particle flow.

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As seen from solutions (32), the scattering amplitudes depend only on the potentials 11v , 1v m and are independent of vmm . The cause of such a behavior of the scattering amplitudes is in the fact that the scattering potential (24) is localized at the point 0x x= , 0y y= . The incident flow has a momentum 1k and, hence, the scattering along the first channel should be described by the potential 11v . On the other hand, the scattering along the channel m is a transition of the particle at the point 0x , 0y into the state with the longitudinal momentum mk and the scattering along two opposite directions of x. Therefore the scattering amplitudes should be proportional to only 1v .m If the potential has a finite width d as in case (1), then the potentials vmm will enter the expressions for scattering amplitudes.

Note also one more important property of N-channel scattering. It follows from formulas (7), (22), and (31) that the transmission amplitude 1t does not depend on the coordinate of the potential center 0 ,x whereas the reflection amplitude 1r has a phase factor ( )1 0exp 2 .ik x As to the scattering along the channel m, then the transmission and reflection amplitudes mt and mr have phase factors ( )( )1 0exp mi k k x− and

( )( )1 0exp ,mi k k x+ respectively. Hence it follows that the phase factor of the product m mr t is equal to ( )1 0exp 2 ,ik x i.e., it does not depend on the scattering channel. This property of N-channel scattering will

be used in solution of the problem of electron localization on a quasi-periodic system of potentials.

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Nanoelectronics, Tsakhcadzor, Armenia, 2009, p. 11. 6. Sedrakian, L.R., Doklady NAN Armenii, 2009, vol. 109, p. 214. 7. Sedrakian, D.M., Kazaryan, E.M., and Sedrakian, L.R., J. Contemp. Phys. (Armenian Ac. Sci.), 2010, vol. 45,

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