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NANOSYSTEMS: Blinova I.V., et al. Nanosystems:
PHYSICS, CHEMISTRY, MATHEMATICS Phys. Chem. Math., 2023,14 (4), 413-417.
http://nanojournal.ifmo.ru
Original article DOI 10.17586/2220-8054-2023-14-4-413-417
On spin flip for electron scattering by several delta-potentials for 1D Hamiltonian with spin-orbit interaction
IrinaV. Blinova1", EvgenyN. Grishanov2'6, Anton I. Popov1c, IgorY. Popov1d, MariaO. Smolkina1'6
1ITMO University, St. Petersburg, Russia 2Ogarev Mordovia State University, Saransk, Russia
[email protected], b [email protected], [email protected], [email protected], [email protected] Corresponding author: I. Y. Popov, [email protected]
PACS 73.22.-f,71.15.Dx,71.70.Di,81.05.Uw
Abstract One-dimensional Rashba and Dresselhaus Hamiltonians with spin-orbit interaction are studied. It is assumed that there are point-like potentials on the line. The scattering problem is solved and the possibility of spin-flip is discussed.
Keywords spin-orbit interaction; point-like potential; spin filtering; Schrodinger equation Acknowledgements The work was supported by Russian Science Foundation (grant number 23-21-00096 (https://rscf.ru/en/project/23-21-00096/)).
For citation Blinova I.V., Grishanov E.N., Popov A.I., Popov I.Y., Smolkina M.O. On spin flip for electron scattering by several delta-potentials for 1D Hamiltonian with spin-orbit interaction. Nanosystems: Phys. Chem. Math., 2023, 14 (4), 413-417.
1. Introduction
Conventional electronics is based on the charge transport. However, during last decades, a new branch of electronics, spintronics, was developing rapidly. In contrast to electronics which ignores the electron spin, spintronics deals with spinbased electron transport. It is not the electron charge but the electron spin that carries information [1]. The new wave of interest was inspired by the development of quantum computing and quantum computations (see, e.g., [2,3]). Correspondingly, one needs electronic devices which can distinguish the spin orientation and can control it. The most popular theoretical idea for creating the background for such device is taking into account the spin-orbit interaction. There are two types of spin-orbit coupling: the Rashba Hamiltonian [4] and the Dresselhaus Hamiltonian [5]. The Rashba effect is a direct result of inversion symmetry breaking in the direction perpendicular to the two-dimensional plane. The Dresselhaus Hamiltonian describes the spin-orbit coupling in a two-dimensional semiconductor thin film grown with appropriate geometry. A number of works were devoted to spin-orbit coupling in different dimensions (see, e.g., [6-10]). Point-like potentials are rather useful in this situation [11-14]. As for one-dimensional models, quantum graph is very effective in this situation (see, e.g., [15-17]). Particularly, to ensure the spin filtering, spin flip, control of the spin transport one uses, usually, systems of coupled quantum rings (see, e.g., [14,18-27]). In the present paper, we show an example of the spin-flip in 1D system without rings for the Hamiltonian with spin-orbit interaction.
2. Model construction
We start with the Schrodinger equation for the Rashba Hamiltonian in one-dimensional case. In the case of spin-orbit
p1
interaction, we deal with 2-vector functions ^
^2
dp1 h2 d2 p1 dp2
dt 2m dx2 R dx ' (i)
dp2 _ h2 d2 p2 ^ dp1 (1)
dt 2m dx2 R dx '
where h is the Plank constant, aR is the parameter of the Rashba spin-orbit interaction, |p|2 _ |p^2 + |p2|2. Let us separate variables x, t. It means that p(x,t) _ eiWtp(x). Correspondingly, the system (1) transforms to the following form:
T h2 d2p 1 dp
-hwipi _ ----—2 + aR——,
2m ox2 dx (2)
t T h2 d2'P>2 dP1 ( )
-hwp2 _ --KT - aR^—,
2m ox2 ox
© Blinova I.V., Grishanov E.N., Popov A.I., Popov I.Y., Smolkina M.O., 2023
414
I. V. Blinova, E. N. Grishanov, A. I. Popov, I. Y. Popov, M. O. Smolkina
In the case of the Dresselhaus spin-orbit interaction, the corresponding system has the following form:
, r h2 d Vi .
-ftw^i = ----- —,
2m dx2 dx
—ftw/ = —
fi2 d2/2 2m dx2
iac
d//i dx
(3)
where is the parameter of the Dresselhaus spin-orbit interaction.
As for the point-like potential at point x0, it is determined by the coupling conditions at this point. There are many variants of such conditions (see, e.g., [14]). We choose the condition which corresponds to 1D delta-potential at the point:
/(xo + 0) = /(xo — 0), /'(xo +0) — /(xo — 0) = — ß/(xo + 0).
(4)
Let us choose the atomic system of units with h = 1, m = 1/2. Consider the case of finite number of point-like potentials at the line. Then, at each half-axis and at each segment between the potentials, one has the following system of differential equations:
^i' + k2^i + a.R ^2 = 0, ^2' + k2^2 - an = 0,
where k is the wavenumber. Taking = Cj eAx, j = 1,2, one obtains the following characteristic equation
(A2 + k2)2 + aRA2 = 0
with the following four roots and the corresponding vectors
Ci C2
(5)
(6)
A = iki,
A = —ik,
A = ik,
A = —iki
where
aR + 4k2 — aR ), ki = -(J aR + 4k2 + afl).
Correspondingly, at each segment and half-axis, one has a solution in the form of linear combination of standard solutions. Coupling conditions (4) gives one a system for determination the coefficients of the linear combination.
3. Results and discussion
Consider the scattering problem for the case of one point-like potential at point x0 = 0. System (5) has solution of the form
/1 /2
Akx
1
+ Bie
-ikx
i
+ Die
-ik i x
/i
i
+ C2ei
i
i
i
x > 0.
x < 0,
Coupling conditions (4) at point x0 = 0 gives one the following system for coefficients of (7), (8):
1 + Bi + Di = A + C2, 1 - Bi + Di = A2 - C2, k - kBi - kiDi = kA2 + kiC2 - -—k - kBi + ki D i = -kA + kiC2 + i^A.2 -We are interested in coefficients of the outgoing solution, which are as follows:
k + ki
i.e. the outgoing solution has the form:
A =
/i /2
k + ki — iß
k + ki k + ki — iß
C2 =0,
ikx
(7)
(8)
(9)
(10)
i
i
i
i
—i
i
i
—i
The second scattering problem with the solution of the following form
Pi p2
= eikix
1
+ B' e
—i kx
1
+ D' e
— i ki
x < 0,
P 1 p2
: A'eikx
iki X
+ C2 ei k 1
1
—i
x > 0,
is solved analogously and gives one the following outgoing solution:
P 1 p2
k + & 1
i ki x
k + ki — ifi
Using the linearity of the problem, one can incorporate (10), (13) and obtain the solution for the general case:
Pi p2
ei k x j ei k i x
i(ei kx — ei kix)
+ (Bi + Bi )e
-ikx I 1 J + (Di + d')e—ikix I 1
x < 0,
Pi \ k + k' I + e"
k + ki — iM i(ei kx — ei kix)
x > 0.
Pk2
One can see that
ei x j ei i x i(ei kx — ei kix )
1 + eiaRx i(1 — eiaR x)
(11)
(12)
(13)
(14)
(15)
(16)
Here we took into account that T1 - T _ aR.
To obtain the spin flip, one should choose the proper input and output points. Namely, let the input point xin be such
1
that
,i«Rxin _
1. Then, keeping in mind (16), one obtains that the input wave function in (14) has the form
0
can choose the output point xout in such a way that
0iaRxout _
—1.
One
(17)
We remember that does not depend on the electron energy (i.e. on k). It means that the relation (17) is valid for all
energies. Correspondingly, according to (15), the output state at this point has the form
Thus, one obtains the
spin flip.
One observes the analogous situation for the case of several delta-potentials. It can be solved analytically, but the expressions are too large. It is more convenient to solve the equations numerically. Fig. 1 shows |p11 and |p21 at x _ xout
as functions of k for fixed xin _ -5 and xout _ 5 (dimensionless units) for cases of 1, 3, 5, 9 point-like potentials posed
|p2|
at integer points symmetrically in respect to x _ 0. One can see that in all cases the ratio —-r does not depend on k
|p1|
as has been obtained analytically (see (15) and (16)) for the case of one point-like potential at point x _ 0. We mention that we deal with the diapason of k outside resonances induced by 1D resonators formed by several delta-potentials (see, e.g., [14]).
The case of the Dresselhaus spin-orbit interaction (3) can be considered analogously. Particularly, one obtains the following characteristic equation which is analogous to (6):
(A2 + k2)2 + aD a2 =0
(18)
with the following four roots and the corresponding vectors
Ci C2
A = iki,
A = — ik,
A = ik,
1 —1
A = —iki
where
k = aD + 4k2 — «D ), ki = ô(\/ «D + 4k2 + «D ).
1
i
i
1
1
i
i x
e
1
416
I. V. Blinova, E. N. Grishanov, A. I. Popov, I. Y. Popov, M. O. Smolkina
c d
Fig. 1. r = | (curve 1) and r = |^21 (curve 2) as functions of k: a - for one center, b- for 3 centers, c - for 5 centers, d - for 9 centers
Using the same procedure as in the Rashba case, one comes to the solution of the scattering problem for the Dresselhaus Hamiltonian:
„ifcx
ki (k + ki - ¿в)
(k + ki)(k + ki - ¿в - (k - ¿в)eiaDx) + 2kkieiaDx -(k + ki)(k + ki - ¿в + (k - ¿в)eiaDx) + 2kkieiaDx
x > 0.
One can see that there is no energy independent condition for spin flip as in the case of the Rashba spin-orbit interaction. Moreover, calculations show that the values of transmission coefficients is significantly smaller than in the Rashba case. Correspondingly, the Dresselhaus case is not good for spin-flip applications.
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Submitted 23 May 2023; accepted 27 July 2023
Information about the authors:
IrinaV. Blinova - Center of Mathematics, ITMO University, Kroverkskiy, 49, St. Petersburg, 197101, Russia; ORCID 0000-0003-2115-2479; [email protected]
EvgenyN. Grishanov - Department of Mathematics and IT, Ogarev Mordovia State University Bolshevistskaya Str. 68, Saransk, 430005, Russia; [email protected]
AntonI. Popov - Center of Mathematics, ITMO University, Kroverkskiy, 49, St. Petersburg, 197101, Russia; ORCID 0000-0001-7137-4067; [email protected]
IgorY. Popov - Center of Mathematics, ITMO University, Kroverkskiy, 49, St. Petersburg, 197101, Russia; ORCID 0000-0002-5251-5327; [email protected]
Maria O. Smolkina - Center of Mathematics, ITMO University, Kroverkskiy, 49, St. Petersburg, 197101, Russia; [email protected]
Conflict of interest: the authors declare no conflict of interest.
