CC BY
23
UDC 538.958
COULOMB EFFECTS ON THE SPIN POLARIZATION
OF QUANTUM RINGS
A. Manolescu1, C. Daday1'2, V. Gudmundsson2 1 School of Science and Engineering, Reykjavik University, Reykjavik, Iceland 2Science Institute, University of Iceland, Reykjavik, Iceland
PACS 71.70.Ej, 73.21.Hb,
We consider electrons in a circular nanoring of zero width, in magnetic field, and with Rashba spin-orbit interaction. We include the Coulomb interaction in the the "exact diagonalization" manner. The Coulomb interaction has strong effects on the spin polarization which may be totally different than for noninteracting electrons. Our current results include up to four electrons, but this number can easily be increased. Keywords: quantum ring, spin-orbit interaction, Coulomb interaction.
1. Introduction
The spin-orbit interaction (SOI) of electrons in semiconductor nanostructures is in the center of the research in spintronics. This interaction mediates the influence of electric fields on the spin orientation of electrons, and thus the conversion of electric currents into spin polarization. Two well-known models of SOI are usually considered, one produced by the two-dimensional confinement of the electrons or Rashba [1] and another one due to the inversion asymmetry of the bulk crystal structure or Dresselhaus [2]. The strength of the Rashba SOI, a, may be modified with external electric fields created by external electrodes or gates, which is the main idea of the Datta-Das transistor [3], whereas the strength of the Dresselhaus SOI, f3, is fixed by the crystal structure and by the thickness of the quasi two-dimensional electron system [4,5]. In addition, in the presence of a magnetic field, an important effect on the electron spin is played by the Zeeman interaction which is determined by the effective g—factor, g*, which depends on the material energy-band structure, being very small in GaAs but more that 100 times larger in InSb. Initial suggestions for spin-operated electronic devices were based on the Zeeman effect, but later this was deemed impractical because in general it is more desirable to achieve spin manipulation via an electric field than a magnetic field [6].
These three parameters, a, f3, and g*, determine together the spin polarization of electrons in quite a complicated way. Spin states and spin currents have been intensively studied on ring models [7-12]. Most works, however, do not include Coulomb interaction, or if they do, they are restricted to the simplest case of exactly two electrons like Refs. [11,12]. Our model can treat a larger number of electrons, in principle limited only by the computational restrictions. In this paper we include up to four electrons.
If neither the Rashba nor the Dresselhaus coupling strength are negligible, then there will be a charge inhomogeneity on the ring. The density will have two maxima at angles ^ and [10]. However, in our paper we consider only the Rashba SOI in combination with the Zeeman effect and the Coulomb repulsion. We use a discrete model of a 1D quantum ring. We describe the Coulomb effects completely using the method of exact diagonalization, also known as configuration interaction.
2. Ring model
We consider a one-dimensional circular quantum ring of radius R in a magnetic field B perpendicular to the plane of the ring described by the vector potential A = B/2(—y, x, 0). The general Hamiltonian of one electron with spin-orbit interaction is,
H = H0 + Hz + Hso , (1)
where H0 = p2/2m* is the orbital Hamiltonian, with p = —ihV + eA and m* the effective mass of the electron in the semiconductor material, Hz = (1/2)g*^BBaz is the Zeeman Hamiltonian, g* being the (material specific) effective ^-factor. HSo = (®/h)(axpy — aypx) is the Rashba spin-orbit Hamiltonian, with a the SOI coupling parameter and ax,y,z being the Pauli matrices.
We will use a discrete representation of the Hamiltonian, by defining Nv sites around the ring. All sites have the same radial coordinate R. The angular coordinates are pj = (j — 1)5p, where j = 1, 2, and 5p = 2n/Nv is the angle between consecutive sites with the same
radius. After a standard (symmetrized) discretization of the first and second angular and radial derivatives on this grid1, we obtain the Hamiltonian in the position representation where the Hilbert space is spanned by the ket-vectors \ja), where the integer j corresponds to the angular coordinate and a = ±1 denotes the spin projection in the z direction. Such a discrete Hamiltonian has been often used in the recent literature [8,9,14].
In the basis {\ju)} the matrix elements of the orbital Hamiltonian are obtained as follows:
{ja\H0\j 'a') = T5a
(tv + Sjjt - (tv + 8jjl+1
+ h.c. . (2)
We denoted T = h2/(2m*R2) which we will use as energy unit. We will also consider R as length unit. We obtain the angular hopping energy, tv = l/Sp2 (in units of T), and tB = heB/(m*T) the magnetic cyclotron energy. As usual h.c. denotes the Hermitian conjugation of the previous terms.
The matrix elements of the Zeeman Hamiltonian are simply diagonal in the spatial coordinates,
{jv\Hz\j'a') = ^TtsnfaSj? , (3)
where 7 = g*m*/(2me) is the ratio between the Zeeman gap and the cyclotron energy, me being the free electron mass.
For the spin-orbit Hamiltonian we obtain:
(j(i\HSo\fa') = iTta\ y ¿j? + <2 Wr(<Pj) + <rr{<Pj+i)U/ Sjj'+i \ + h. c.. (4)
where ta = a/(RT) is the spin-orbit relative energy, while ar (p) = ax cos p + ay sin p and (tp) = —ax sin p + ay cos p are the radial and angular Pauli matrices, respectively.
The single particle states corresponding to the Hamiltonian (1), = ta^a, are calculated as eigenvalues and eigenvectors of the matrices (2)-(4). The eigenvectors are in fact spinors, meaning that the wave function of a quantum state a at any lattice site has the form
1The first and second derivatives of any generic function f (x), are approximated as f(x) &
[f (x + h) — f (x — h)} /2h and f "(x) & [f (x + h) + f (x — h) — 2f (x)j /h2, respectively, where h is considered sufficiently small.
^a(pj) = Y1 a (j)|^), where ^(j) are c—numbers. Using now the single-particle states as a basis we write the many-body Hamiltonian in the second quantization as
n = Yl £qC«C« + \ Vabcdclclcdcc , (5)
a abed
where c^ and ca are the creation and annihilation operators on the single-particle state a. The matrix elements of the Coulomb potential V(r — r') = e2/(n\r — r'|), k being the dielectric constant of the material, are in general given by
Vabcd = <^0(r)^b(r')|V(r — r')^c(r)^d(r')) . (6)
In the present discrete model the double scalar product is in fact a double summation over all lattice sites and spin labels
D
Vabed = Ttc Y, r r ,;/;.. (7)
j'a'
The new energy parameter introduced by the Coulomb repulsion is tc = e2/(kRT). In the above summation over the sites the contact terms (j = j') are avoided, their contribution vanishing in the continuous limit.
The many-body states are found by solving the eigenvalue problem for the Hamiltonian (5), = . We do that by using the Hamiltonian matrix in the noninteracting many-body basis (Slater determinants), ||a) = ,i%,...,iK)}, where i" = 0,1 is the occupation number of the single-particle state and K is the number of single-particle states considered. The occupation numbers are listed in the increasing energy order, so tK is the highest energy of the single-particle states included the many-body basis. For any |a) we have a ^a = N, which is the number of electrons in the ring. It is straightforward to derive the matrix elements Vaa' using the action of the creation and annihilation operators on the many-body basis. This procedure is known as "exact diagonalization" or "configuration interaction". It does not rely on any mean field description of the Coulomb effects (like Hartree, Hartree-Fock, etc.), and it can be made convergent with K for a sufficiently small number of electrons, and a sufficiently small ratio of Coulomb to confinement energies, tc.
To be able to carry the numerical calculations in a reasonable time we consider a small ring of radius R = 50 nm containing N < 4 electrons. The discretization is done in 100 angular points (sites). Common semiconductor materials used in the experimental spintronics are: InAs with m* = 0.023me, g* = —14.9, k = 14.6, and estimated (or possible) values for the Rashba SOI a & 20 meVnm; InSb with m* = 0.014me,g* = —51.6, ^ = 17.9, and a & 50 meVnm [4,5]. The relative energies which we defined are: for InAs ta = 0.60, tc = 2.9,7 = —0.17; for InSb ta = 0.92, tc = 1.5,7 = —0.36. In our calculations we have considered material parameters somewhere in between these to sets, 7 = —0.2 and tc = 2. Since in general the SOI coupling constant may be tunable we have considered it variable, such that 0 < ta < 2,
3. Results
3.1. Single-particle states
The single-electron eigenstates of the one-dimensional ring are analytically known. The Hamiltonian commutes with the total angular momentum Jz = Lz + Sz and in the continuous limit (Np ^ to) the eigenvectors are (see for example [10])
—aci-a el
(a) N=1 (b) N=1 (c) N=1
Fig. 1. (a) The lowest 10 energies of the single particle states vs. the SOI strength ta. The magnetic energy is small, tB = 0.1, such that the spin spitting is small. The spin spitting of the ground state is only barely visible. (b) The expected value of the spin projection along the z direction Sz, in units of h, in the ground state, vs. ta, for tB = 0.1,0.5,1.0,1.5, 2.0. The thicker line corresponds to the energy spectrum shown in (a). (c) The spin orientation around the ring in the ground state, for ta = 0.5 and tb = 0.1 The angle with the z axis is about 27°.
where I = 0, ±1, ±2,... is the orbital quantum number, and we have used the notations ci,+1 =
cos(di/2), ci—1 = sin(di/2). The angle di is defined as
9i = tan 1 (t.a—-' ^ B- | ,
V 4l + 2 + tB-2jtBJ
(9)
which is the angle between the direction of the spin and the z axis. We see that for zero magnetic field (tß = 0), or for zero g-factor (7 = 0), this angle is independent on the state. Obviously, in the absence of the SOI (ta = 0), 61 = 0 or n, i. e. the spin is parallel or antiparallel with the z axis. The energy spectrum corresponding to the eigenfunctions (8) is
1 + \ + \tB - v^T) 2 - I (tan^)2 + ■ <10)
2 4 2 cos 0i / 4 2 cos Ui
/11 a V
In Fig. 1 we show the energies of the single particle states obtained with the discrete model with 100 sites, in the interval of SOI strength 0 < ta < 2 (units of T). The magnetic energy is ts = 0.1 (units of T), which corresponds to a magnetic field strength of 0.013 Tesla. In this case the spin splitting is small, as seen in Fig. 1(a). The states are (doubly) spin-degenerate in zero magnetic field (Kramers degeneracy) for any ta = 0, but for ta = 0 only the ground state is doubly degenerate, the other states being degenerate four times. Not all intersections of the energy lines shown in Fig. 1(a) are real intersections. States with the same total angular momentum in reality repel each other and lead to avoiding crossings, but for the parameters used in this work this effect is not visible. (See for example Ref. [12].) Due to the SOI the energy of the states with Sz parallel to the effective magnetic field produced by the orbital motion decreases with increasing ta, i. e. the states with a = +1 and I > 0, or a = —1 and I < 0. For the other states, with a = —1 and I > 0, or a = +1 and I < 0, the energy increases. In Fig. 1(b) we show the total spin in the ground state for several strengths of the SOI and magnetic field where a transition from spin-up to spin-down can be seen at strong SOI. The thicker line is calculated for the energies shown in Fig. 1(a), with tB = 0.1 and a spin transition at ta & 1.7. The interval 0 < tB < 2 corresponds to 0 < B < 0.26 Tesla. In Fig. 1(c) one can see the spin orientation in
a
the ground state for tB = 0.1 and ta = 0.5. Since the magnetic field is small this angle is in this case close to tan-1 ta = 27°, according to Eq. (9). So we see that the results obtained with the discretized ring coincide with those corresponding to the continuous ring, given for example in Ref. [10].
3.2. Many-particle states
We will discuss now the results obtained for more electrons. In Fig. 2 we compare the energy spectra for the first 10 states vs. the magnetic energy for N = 2, 3, and 4 electrons, without and with Coulomb interaction. Since the Coulomb interaction is invariant at spatial rotations and independent on spin the total angular momentum is still a good quantum number. In other words the Coulomb term mixes many-body states with the same total Jz, and thus possibly with different spins. The spectra shown look different from the single-particle case, more or less like "spaghetti", but they are not very different with or without interaction. Most of the low-energy states decrease with increasing ta because they are built on single-particle states with Sz along the effective magnetic field. There are two main effects of the interaction: One is the shift all energies to higher values, due to the net Coulomb charging. The other effect is that different states have different energy dispersion with ta and so the level intersections or level repulsion may occur totally differently than in the absence of the Coulomb interaction. Consequently the spin states and the spin transitions may be totally different.
In Fig. 3 we show the total spin projected along the z direction, in the ground state, for several values of the magnetic energy. Because of the spin degeneracy, the results at zero magnetic field are not shown, but for the states shown they are very similar to those at tB = 0.01, As before the thicker lines are calculated with the same magnetic energy used for the energy spectra, here tB = 0.5. At this magnetic energy, the ground state for two electrons is close to a singlet type, or a spin compensated state, both with and without interaction, with 0 < Sz < 0.08, as long as ta < 1.55, Fig. 3(a) and (b). Around this SOI strength, an intersection of the ground state with another state occurs in both cases. For the noninteracting case the new ground state is still of a singlet type, whereas for the interacting case it is mixed with a triplet state such that Sz = 0.55. At a lower magnetic energy, like tB = 0.01, the ground state in the interacting case becomes again singlet at tB = 1.86. At higher magnetic energies the ground state becomes triplet at zero or low SOI first in the interacting case (for example at tB = 1), and then also for the noninteracting case (for example at tB = 1.5).
More transitions can be seen for N = 3, for the interacting vs. the noninteracting case. For example at tB = 0.5 in both cases the total spin is 0.5 at zero or low SOI strength, but at ta & 0.3 Sz becomes almost 1.5 with interaction, but almost —0.5 without interaction, as seen in Fig. 3(c) and (d). For N = 4 and tB = 0.5 again a spin compensated state is preferred by the noninteracting system in the ground state for any SOI strength shown, whereas the interacting system has a configuration with Sz & 1 at zero and low SOI strength (three spins up and one down), as shown in Fig. 3(e) and (f). All these results are consequences of the mixing of different spin states with the same total angular momentum due to the Coulomb interaction. Examples with two electrons are also published in Refs. [11,12].
Finally in Fig. 4 we compare the total spin for N = 3 and N = 4 electrons with and without Coulomb interaction, for equal SOI and magnetic energies, ta = tB = 0.5. For N = 3, as described above, the projection of the total spin in the z direction is close to —0.5 (actually Sz = —0.42), but for the interacting state is close to 1.5 (actually Sz = 1.39). Obviously the quantized half-integer values are mentioned here only as reference values which have a meaning only in the absence of the SOI. In the next example, for N = 4 we compare an almost spin compensated state in the absence of the Coulomb interaction, Sz = 0.01, with a a spin polarized
Fig. 2. Energy spectra of the lowest 10 states for N = 2, 3, and 4 electrons without Coulomb interaction, tc = 0 in panels (a),(c),(e), and with Coulomb interaction, tc = 2, in panels (b),(d),(f). The magnetic energy is tB = 0.5. The variation of the energy with the SOI strength increases with the number of electrons and so the scale on the energy axes has been enlarged with N.
Fig. 3. Total spin of the many body states with N = 2, 3,4 electrons, without interaction, i. e. tc = 0 in panels (a),(c),(e), and with interaction, with tc = 2, in panels (b),(d),(f), in the ground state, for magnetic energies tB = 0.01,0.5,1.0,1.5, 2.0. The thicker lines are for the same magnetic field as the energy spectra shown in Fig. 2.
(a) N=3 tC=0
(c) N=3 tC=0
y/// / M \ \ \\v
^/lliww^
(b) N=3 tC=2
m
(d) N=4 tc=2
1j\\\\
\\
Fig. 4. The total spin orientation for N = 3 and N = 4 electrons, without Coulomb interaction, tc = 0, and with Coulomb interaction, tc = 2, in the ground state. The SOI energy is ta = 0.5 and magnetic energy is tB = 0.5, like in Fig. 2.
state in the presence of the Coulomb interaction with Sz = 0.91. The excited states (not shown) may have totally different spin magnitude and orientations. In principle the direction of the total spin, which for N =1 is given by the angle Bi, is also determined by a mixing of states in the presence of the interaction. At low magnetic fields this angle is only slightly dependent on the state for the single particle states, but this variation may become much larger for the many-body states.
4. Conclusions
We calculated the many body states of a one-dimensional ring with N < 4, electrons with Rashba spin-orbit and with Coulomb interactions, in the presence of a magnetic field perpendicular to the surface of the ring. The Coulomb effects are fully included in the calculation via the "exact diagonalization" method. We used material parameters comparable to InAs and InSb materials. We observed strong (even dramatic) effects of the Coulomb interaction on the spin polarization both in the ground state and in the excited states which are illustrated with few selected examples.
Acknowledgments
This material was presented at the workshop Mathematical Challenges of Quantum Transport in Nano-Optoelectronic Systems, WIAS, Berlin, February 4-5, 2011 (OptoTrans2011). We wish to thank again to the organizers for this opportunity. We are also thankful to Catalina
Marinescu, Marian Nita, and Sigurdur Erlingsson for valuable discussions. This research was supported by the Icelandic Research Fund.
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