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Numerical solution for the Schrodinger equation with potential in graphene structures
L. A. Nhat1'2
1Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya str., 117198, Moscow, Russia 2 Tan Trao University, 22227, Tuyen Quang, Vietnam [email protected]
DOI 10.17586/2220-8054-2019-10-2-124-130
This paper presents a different numerical solution to compute eigenvalues of the Schrodinger equation with the potentials in graphene structures [1]. The research subjects include the Schrodinger equation and the exchange-correlation energy of the graphene structures in Grachev's article. Specifically, we used the pseudospectral method basing on the Chebyshev-Gauss-Lobatto grid to determine the approximate numerical results of the problem. The results are the discrete energy spectra and the corresponding eigenfunctions of the nonlinear spin waves in the graphene structure. Additionally, these results can be applied to create the nonlinear spin waves in the graphene structures. Keywords: graphene, kinks, breathers, spin, pseudospectral method, Schrodinger equation, Chebyshev, eigenvalue problems, nonlinear models. Received: 12 April 2019 Revised: 17 April 2019
1. Introduction
In 2010, D. D. Grachev and L. A. Sevastyanov invented the method for generating spin waves [2,3]. This method formed the quantum collective excitations of spin density and magnetization density in graphene films. It may be used in quantum nanoelectronics, spintronics, for creating spin-processors, memory cells, physical field sensors, other devices and systems for processing and storing information of terahertz (and higher) range that have nanometric dimensions and work in a broad temperature range with minimum energy consumption.
Since the publication of this invention, the authors studied, built and perfected the theory of the method for generating spin waves. In particular, Grachev constructed the quantum field model to adequately describe ferromagnetic properties in graphene structures to match the results of physical and numerical experiments. This model described properties of monoatomic graphene layers, which connected with the presence of a nontrivial function of the distribution of the spin density, formed as a result of the spontaneous breakdown of the spin symmetry of valence electrons in atoms of carbon [4]; in [5], the authors provided the nonlinear field model to describe of the spin density distribution of the valence electrons in graphene films. This model describes experimentally observable ferromagnetic properties of such films; prior work [6] claimed that the offered variant of the nonlinear field model, in which carriers of spin density are not fermions (electrons), but bosons (spinons), is quite adequate for describing the magnetic properties of graphene structures. The authors proposed in two articles [1, 7] the desirability of a nonlinear model that describes a possible mechanism of ferromagnetism in graphene structures, resulting from electron-electron interaction and spontaneous breaking of spin symmetry of valence electrons. We investigated such spatially localized nonlinear spin of the valence electron density on the graphene surface such as kinks, and their interactions, as well as quasibound metastable states of the interacting kinks and antikinks, that are breathers [8].
In addition, many scientists have also studied how to spin waves in graphene, and they have gained certain achievements such as the excitations with spin reversal such as spin-flip and spin-wave excitations were studied in [9]. They showed that these excitations were correctly accounted for in the time-dependent Hartree-Fock and strong magnetic field approximations; F. J. Culchac investigated spin excitations and electronic properties of graphene nanoribbon devices with zigzag edges [10,11]. Those studies showed that a central point of a finite zigzag nanoribbon, when spin excitations are damped at all finite energies, their energy dispersion at small wave vector is dominated by antiferromagnetic correlations between the ribbons edges, in accordance with previous calculations. A. Matthew studied the collective excitations of doped graphene in the presence of in-plane magnetic fields and calculated the dispersions of charge and spin plasmons using time-dependent density-functional methods within a standard tight-binding approach [12].
Evidently, in the articles [1,4-8], it is shown that the density of spin symmetry was broken by the spontaneous breaking, which obeyed a nonlinear equation, and the offered nonlinear models with the limits will exist exact and
approximate solutions for the distribution of the spin density and magnetization on the surface of the graphene structures, as the method of the scattering matrix [5]; the finite element method (FEM) and the Ritz method [1,7]; the Ritz method using Hermitian functions as coordinate functions [8].
Based on the density functional theory and the perturbation methods, in [1] proposed a physically reasonable nonlinear model of interacting massless Dirac fermions in graphene structures for practical applications. This model described the spin density (s) distribution of the valence electrons of the carbon atoms in graphene structures. This corresponds to the equation of the well-known nonlinear model (A^4):
s" = A(s2 - s0)s, (1)
where so - the zero-order local spin density, A - the self-interaction constant.
To illustrate this, consider the simplest nonlinear model A^4 when the envelopes depend on one spatial coordinate. It is well known that the Hamiltonian density of the model has the form:
H[s] = (dvsdvs)/2 + A(s2 - s0)/4, v = 0,1, 2.
In this case, equation (1) was known to have two stable vacuum solutions: s± = ±s0 and the kink-antikink solutions:
s± = ±sot^yA20^ . (2)
Due to the non-linearity of equation (1), for qualitative estimates, the author determined approximate solutions of equation (1) by choosing the field function of interacting kinkantikink pair in such simple form:
$(x, a) = s+(x + a) + s_(x — a) — so (3)
where a is the parameter, and the function (3) has the following asymptotic behavior:
$(x, = +so,
a) = a) = —s0
a) = a) = 0
$(x, —to) = —3s0;
(4)
or
+to) = +so, $(+to, a) = $(—to, a) = —s0
(5)
a) = a) = 0
$(x, 0) = —so;
It can easily be seen that there are two models. We call the conditions (4) - model A, and the conditions (5) - model B.
We can write a system of the Hamiltonian with the field function in the form (3) that satisfies the equation of the type (1) [1,7]:
H {$,a} = dxj [$X(x,a)j2 + ^[$(x,a)2 — s2]2} . (6)
Similarly, we can write the energy density of the kink [7]:
E[s] = |~ dxj^ + A[s2 — s0]2} . (7) Combining (7) and (2) gives the result:
E [s] = (8)
Then, the sum of the mass-energy equivalent of free kink and antikink equals the potential energy of the breather depending on a with U{$, a} computed from (6) and the function H{$, a} possesses a minimum. We have the mass of the breather:
m[A, so] = —32As3. (9)
The Schrodinger equation for the quantum-mechanical wave function ^(a) of a stationary breather state with corresponding eigenenergy E has the form:
K2 d2
^(a) + U($, a)^(a) = E^(a), (10)
2m {A, so} da2
here mm {A, s0} = m{A, s0}/c2 is the effective mass of the breather, c is the speed of light, U($, a) is the potential energy, ^(a) is the wave function and E is the energy of the system.
Grachev's articles used the finite element method (FEM) and the conventional Ritz method to determine the eigenvalues and eigenfunctions of the Schrodinger equation (10) for the quantum-mechanical wave.
In this paper, we study the numerical solution to compute eigenvalues of the Schrodinger equation (10) with the energies of the system in graphene structures. That is the pseudospectral method basing on the Chebyshev polynomials of the first kind and using the Chebyshev-Gauss-Lobatto grid in the integration interval [-1,1].
2. Chebyshev differentiation matrix (CDM)
A grid function v(x) is defined on the Chebyshev-Gauss-Lobatto points x = {x0, x^ ..., xn} such that xk = cos(kn/n), k = 0, n. They are the extrema of the n-th order in the Chebyshev polynomial Tn(x) = cos(n cos-1 x). The function v(x) is interpolated by constructing the n-th order interpolation polynomial gj (x) such that gj(xk) = ,
n
p(x) = 13 Pj gj(x) (11)
j=0
where p(x) is the unique polynomial of degree n and pj = v(xj), j = 0, n. The following can be shown:
( ) (-lj+1(1 - x2)Tn(x) . 0- (12)
gj(x) =-^ 2( _—7;-, j = 0,n, (12)
C j (x* x• j )
where
2, j = 0 or n, 1 , otherwise.
(13)
As we know the values of p(x) at n+1 points, we would like to find approximately the values of the derivative
of p(x) at those points p'(x) = — p(x). We can write the same in the matrix form:
dx
p' = Dp (14)
where D = {dj is an (n +1) x (n +1) differentiation matrix. Evidently, the derivative of p(xj) becomes:
n
p'(xj ) = Dj,fcp(xfc), j = 0, n. (15)
k=0
We have the entries d(j = gi(xj) which are [13,14]
d(i) = -d(D = 2n2 + 1
d0,0 = dn,n = g , 7(1) xi • "i-r
d- = - 2(T-~xf), i = 1,n - 1, (16)
d(j = ^ , i = j, i,j = im—r,
K^jiAs i j
where ck is determined by the formula (13).
Similarly, p'(x) is a polynomial of degree n — 1; it exists the second differentiation matrix D2,
p'' = D2p, (17)
and
n
j2
k=0
p''(xj) = £ D2,fcp(xfc), j = 0, n. (18)
3. Pseudospectral method using the CDM for the Schrodinger equation
Suppose the Schrodinger equation has simple form as the following:
d2
—t~o u(x) + q(x)u(x) + Au(x) = 0, u( —1) = u(1) = 0 dx2
and collocation points xj such that 1 = xo >xi > ... > x„ = —1. We know that:
d2
dx2 U™(Xi) — ^ ] Di,fcun(xfc).
fc=Q
Thus, equation (19) takes the following form:
n
— y^ D2,kMn(xfc) + q(xj)u(xj) = —Au(xj), i = 1, n — 1
k=Q
with u„(x„) = 0 and u„(xo) = 0.
Alternately, we partition the matrix D into matrices [13,15]:
( d(1)
j^d)
(1)
1,1 1,2 d2 , 1 d2,2
d(1) N d(1)
d2,n— 1
d
(1)
n-1,1
d(1) dn-1,2
d
(1)
'n-1,n-1 /
d(1)
(1)
1,Q (1)
2,Q
(1)
d(1)
2(1) , en
\ dn-1,Q )
(1)
(1)
\ dn-1,n )
(19)
(20)
(21)
(22)
We can rewrite in the matrix form: = {d^}, = {d(1))}, e,1 = {d(1„}, here i, j = 1, n — 1. Similarly, we partition matrix D2 into matrices ei)2'1, So, (21) can then be written in the matrix
form:
- ^w„(xo)eQ12) + J5(2)w + w„(xn)ê42)j + Qu — -am.
Since un(x0) = 0 and un(xn) = 0, we have derived the following:
(-£(2) + q m — -am,
(23)
(24)
where u denotes the vectors with elements {«„(xj)}, while Q denotes the diagonal matrix with elements
{g(xj)} , i = 1,n — 1.
4. Numerical solutions and results
In our research, the program has used the Mathematica 10.4 language [16]. Our numerical results are computed by the pseudospectral method using the CDM (CPSM), as shown in the CPSM columns. These numerical results as per Grachev's article have been included in the columns FEM for the sake of completeness.
The potential energy U($, a) in nonlinear spin waves in graphene structures has been calculated; it had the analytic form [1,7]:
U j$,a} —
where
4V2Â
t —
6 36%/2Äsoa — 24 48\/2ÄsQa \ 1 - t + t2 + t3 J ,
s2V2ASoa _ i for model A,
(25)
32V2as°m _ i for model B.
1
Q
3
s
Q
3
From section 4 of the article [1], we can deduce the following: if the vicinity of a = 0 and |a| < 1/(5s0\/2A), this potential has the following asymptotic form of a power series:
U (S, a) = 8s5AV2Âa2(2 - ^ + s0Aa2(4 + 7g0V2Âa) 5 15 105
16s4A2a4 32s5A^V2Aa5 416s6A3a6 32s7A3V2Aa7)
525
1575
+
3465
+
6237
(26)
if a ^ +to then Umax = U($, +to) = 4\/2As3)/3; for model A, at a ^ —to then potential U ($, —to) = ^V2As3 (6 |a| — 17) /3. With s0 = 2 and A = 1, we have graphs of potential energy U($, a) in the cases model A (denoted by UA) and model B (denoted by UB) as depicted in Fig. 1.
Fig. 1. Two cases model A and B of the potential energy U (S, a) in nonlinear spin waves in graphene structures with s0 = 2 and A = 1
Now, we consider the Schrodinger equation (10) in the case h = c = 1:
d2 - -
^(a) + 2U(S, aWa) = 2E^(a),
da2
(27)
here
and
2U(S, a) = 2m{A, s0}U(S, a) =
8V2As
-U(S, a)
2E = 2m{A,s0}E =
8T2Âs 3"~
-E.
We apply the section 3 for the equation (27), we can thus rewrite in the matrix form:
(-E(2) + G ^ = 2E^.
(28)
where G is the diagonal matrix of order n — 1, with the elements:
Î8V2Âs
3
-U (S,ai)
1,n- 1.
Thus, to find the energy 2E of the system (27) and find eigenvalues 2E in the equation (28) are equivalent, we
~ 3E have to deduce the eigenvalues of matrix + G and the total energy E =-— as is shown in the Tab.1
4s0v 2A
for the two case models A (denoted by EA) and B (denoted by EB), with A = 1and and s0 = 2. Therefore, we have the graphics illustrating the first four eigenfunctions of the breather states of models A (denoted by ^A(a)) -Fig. 2 and B (denoted by (a)) - Fig. 3.
Remarks: From the numerical results in Table 1, we see that: the numerical results of CPSM and FEM are equivalent. In addition, it provides many arbitrary numerical results based on practical applications. It is clear that this numerical solution is reliable and very accurate when k < 2n/n [17]. Hence, it may become our calculation
3
3
3
3
Table 1. First ten eigenvalues of the total energy EA and EB in the cases model A and B with A =1 and so = 2.
CPSM FEM
k ea EB ea EB
1 2.118495 1.870306 2.11 1.87
2 6.063236 5.232607 6.06 5.23
3 9.495058 7.990805 9.49 7.99
4 12.300544 10.277019 12.29 10.27
5 14.375397 12.113396 14.28 12.11
6 16.168130 13.535662 13.51
7 18.343813 14.647713 14.48
8 20.916405 15.694926 15.00
9 23.816396 16.893204
10 27.002109 18.303307
tool for future studies. Furthermore, we contend that the offered nonlinear model existence of metastable kink-antikink bound states for the function of spin density on a two-dimensional graphene surface is possible. Finally, the numerical calculations show that the interval change between the next levels generally decreases with the energy growth, and since some value of energy, the spectrum becomes continuous.
v
0.6
rA N. <l4 \ s
f". f { \ 0.2 • \ \
-0.5 \ -0 Î -0.4 \ 0.5 .______________
Fig. 2. First four eigenfunctions of model A with s0 = 2 and A = 1
5. Conclusion
We proposed a reasonable numerical model which offered approximate solutions for the spin density's distribution of the stationary pseudo-spin waves on the surface of the graphene monoatomic film. We have obtained the discrete energy spectra and the corresponding eigenfunctions of nonlinear spin waves in the graphene structure.
Fig. 3. First four eigenfunctions of model B with so = 2 and A = 1
More complete numerical results may be obtained by the Chebyshev pseudospectral method. These results may be applied to create the nonlinear spin waves in the graphene structures.
Acknowledgements
The author is greatly indebted to Prof. D. S. Kulaybov and to K. P. Lovetskiy for suggesting the problem and for many stimulating conversations.
The publication was prepared with the support of the RUDN University Program 5-100.
References
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