Download Quantum Rings with Two Deeply Bound Electrons under a Magnetic

Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 733–736
c International Academic Publishers
Vol. 47, No. 4, April 15, 2007
Quantum Rings with Two Deeply Bound Electrons under a Magnetic Field∗
SITU Shu-Ping
Department of Physics, Sun Yat-Sen University, Guangzhou 510275, China
(Received June 2, 2006; Revised August 17, 2006)
Abstract A model is proposed to study the quantum rings with two deeply bound electrons under a variable
magnetic field. The emphasis is placed to clarify the effect of the size (diameter) and the width of the ring on the
fractional Aharonov–Bohm oscillation. It was found that the reduction of size will lead to a very strong oscillation in
the ground state energy and in the persistent current. The electronic correlation has also been demonstrated by showing
the nodal structures of wave functions.
PACS numbers: 73.23.Ra, 73.63.Hs
Key words: quantum ring, fractional Aharonov–Bohm oscillation, persistent current
Quantum rings are a kind of well-known mesoscopic
systems having a great potential application.[1−6] Since
their physical properties can be controlled, the physics involved is very rich, therefore they are also attractive in the
academic aspect. Now the quantum rings containing only
a few electrons can be fabricated in laboratories.[4,5] In
these systems (with a radius about 20 to 120 nm) both the
electronic correlation and quantum effect are believed to
be important. When a magnetic field B is applied, interesting physical phenomena, e.g., Aharonov–Bohm oscillation (ABO) of the ground state energy and the persistent
current, have been observed.[6,7] In particular, the fractional ABO (FABO) has been observed recently, where
a period of the oscillation Φ0 /4 for 4-electron rings was
found (Φ0 = hc/e is the flux quantum).[2,8] In the theoretical aspect, a number of calculations have been performed,
which can in general reproduce the experimental data, including the fractional periodicity.[2,8−10] In these calculations, mostly the system is considered to be 2-dimensional
and the electrons are confined by a parabolic-like potential m∗ ω 2 (r − ro )2 /2. Obviously, this kind of confinement
is not sharp. However, when the ring is fabricated surrounded by appropriate insulator, the confinement can be
very sharp. Therefore, as a complement to the existing
theory, in this paper, an extreme case, namely, the electrons are strictly bound in an annular region, is considered. Accordingly, the confinement potential U (r) is chosen as U = 0 if a < r < b, and U is infinite otherwise,
where a and b are the inner and outer radii of the ring, respectively. The emphasis of this paper is placed to study
the effect of the size (or average diameter) (a + b)/2 and
the width b − a on the ABO.
Let us consider a two-electron ring lying on the XY plane (2-dimensional) subjected to a uniform magnetic
field B perpendicular to the X-Y plane. The Hamiltonian
reads
X e2 1
X 1 e ~ 2
H=
p~i + A
+ U (ri ) +
i
∗
2m
c
ε rij
+ HZeeman ,
(1)
∗ The
where m∗ is the effective mass of electron, and ε is the dielectric coefficient of the material. The vector potential is
~ = (B/2)~ez × ~r. HZeeman = − g ∗ µB BSZ is the
given as A
Zeeman energy, where SZ is the Z-component
pof the total
spin. By using the characteristic length λ = h̄/m∗ ωc as
the unit of length, the Hamiltonian can be simplified as
hX 1
1
1
α i
H = h̄ωc
− ∇2i − ˆli + ri2 + Ū (ri ) +
2
2
8
r12
+ HZeeman
α
= T1 + T2 + h̄ωc
+ HZeeman ,
(2)
r12
where ωc = eB/m∗ c is the cyclotron frequency, ˆli =
−i∂/∂φ
qi is the angular momentum operator and α =
2
(e /ε) m∗ /h̄3 ωc , Ti is the kinetic energy of i-th electron.
In order to diagonalize the Hamiltonian, basis functions will be introduced. In order to guarantee that the
wave functions do not exist outside the ring, it is convenient to define θ = π[(r − (a + b))/2]/(b − a) to replace
r. When r runs from a to b, θ runs from −π/2 to π/2.
Using θ and the azimuthal angle φ as variables, the basis
functions for a single electron can be introduced as
1
(3)
ψjk = √ e ikφ fj (θ) ,
2π
where fj (θ) = cos(jθ) if j is odd, or fj (θ) = sin(jθ) if j
is even, k is a positive or negative integer which denotes
the orbital angular momentum of the electron. Evidently,
these basis functions satisfy the boundary condition that
they are zero at r = a or b.
For a 2-electron state with a total angular momentum
L and a total spin S, the basis functions can be chosen as
Φj1 ,k1 ,j2 ,k2 (1, 2) = ψj1 k1 (1)ψj2 k2 (2)
± ψj1 k1 (2)ψj2 k2 (1) ,
(4)
where the plus (negative) sign is for S = 0(1), and
k1 + k2 = L. It is noted that these basis functions are
not orthonormalized. They can be simply denoted as Φi
project supported by the National Natural Science Foundation of China under Grant Nos. 10574163 and 90306016
734
SITU Shu-Ping
where i labels all the quantum numbers. With these basis functions, the eigenstates of the Hamiltonian can be
expanded as
X
Ψ=
Ci Φi ,
(5)
where the coefficients Ci together with the eigenenergies
can be obtained after the diagonalization.
The crucial point in the procedure of diagonalization
is the calculation of related matrix elements. They appear
as
hΦi0 |Φi i = 2(δk1 ,k10 δk2 ,k20 Xj1 j10 Xj2 j20
± δk1 ,k20 δk2 ,k10 Xj1 j20 Xj2 j10 ) ,
2δk1 ,k10 δk2 ,k20 (Xj1 j10 Tjk22j 0
2
hΦi0 |H|Φi i =
±
+
(6)
+
Xj2 j20 Tjk11j 0 )
1
2δk1 ,k20 δk2 ,k10 (Xj1 j20 Tjk22j 0 +
1
2δk1 +k2 ,k10 + k20 (va ± vb ) ,
Xj2 j10 Tjk11j 0 )
2
(7)
where
b/λ
Z
Xjj 0 =
fj 0 fj r dr ,
(8)
a/λ
Z
b/λ
Yjj 0 =
fj 0
a/λ
(−2)
Xjj 0
(+2)
Xjj 0
Z
1
gj r dr ,
θ + θ0
b/λ
=
1
fj r dr ,
(θ + θ0 )2
(10)
fj 0 (θ + θ0 )2 fj r dr ,
(11)
fj 0
a/λ
Z
(9)
b/λ
=
Vol. 47
Figure 1 is for a ring with a narrow width b − a = 10 nm,
where the evolution of the two lowest levels and the persistent current J of the ground states against B are plotted
(instead of J, it is the hJ that is plotted in the figure,
where h is the Planck’s constant. The dimension of hJ
is energy) . One can see a clear ABO in which the average period of B is about 0.073 Tesla. Since the flux
F = eπr2 B/hc, when r = (a + b)/2 is assumed, the period of the flux is close to 1/2, a typical period of the
FABO. During the oscillation, the orbital angular momentum of the ground state Lo decreases each step by one as
marked in the figure, while the total spin of the ground
state So matches Lo , namely, So is 0 (1) when Lo is even
(odd). Thus, the FABO is accompanied by the singlettriplet transitions occurring repeatedly. Furthermore, the
oscillation of the persistent current J matches exactly that
of the ground state energies. Once a transition of Lo occurs, J reverses its sign suddenly, i.e., from its maximum
jumps suddenly to its minimum as shown in the figure.
In fact the oscillation in the FABO region is not regular,
the period associated with an odd Lo becomes longer and
longer when B increases. This arises because the lowest
odd Lo states are associated with S = 1, they will benefit
more from the Zeeman energy and therefore have a larger
chance to appear as a ground state. Figure 1 is typical
for the region of the FABO characterized by the irregular
oscillation.
a/λ
b/λ
h 11 ∂ ∂
k2 k 1 2 i
r
−
− + r fj r dr
fj 0 −
2 r ∂r ∂r
r
2 8
a/λ
π 2 (−1)j j
1 πj 2
− k Xjj 0
=−
Yjj 0 +
h
2
2
h
1 πk 2 (−2) 1 h 2 (+2)
+
(12)
Xjj 0 +
Xjj 0 .
2 h
8 π
The interaction matrix elements
1
va = hψj10 k10 (1)ψj20 k20 (2)|
|ψj k (1)ψj2 k2 (2)i ,
r12 1 1
1
vb = hψj10 k10 (1)ψj20 k20 (2)|
|ψj k (1)ψj1 k1 (2)i ,
(13)
r12 2 2
k
Tjj
0 =
Z
(−2)
(+2)
and Xjj 0 and Xjj 0 are calculated numerically.
After the diagonalization, the eigenenergies and eigenwave functions are obtained, the e-e correlation can be
understood by observing the density-functions as shown
later. Furthermore, the angular persistent current of the
first electron e1 at r1 reads
J1 (r1 )
Z b/λ
h i
∂
r1 2 ωc
r2 dr2 dφ2 Ψ∗ −i
+
Ψ + c.c. . (14)
=
(2r1 ) a/λ
∂φ1
2
From Eq. (14), integrating over r1 from the inner radius a to the outer radius b, together with the current of
e2 , we obtain the total angular current J which would be
given later.
The numerical calculations have been carried out with
the parameters ε = 12.4 and m∗ = 0.067me for the typical
material GaAs. The results are presented by the figures.
Fig. 1 The evolution of the two lowest levels (upper
panel) and the persistent current of the ground states
(lower panel) against B. The inner and outer radii of the
ring are a = 90 nm and b = 100 nm. The orbital angular momenta of the ground states Lo are now negative,
their absolute value are marked by the curves. Instead
of the persistent current J itself, the quantity hJ in meV
is plotted.
Figure 2 is similar to Fig. 1 but B is given larger. In
this region, due to the Zeeman effect, the odd-Lo states
are dominant and the even-Lo states disappear completely
from the ground states. It implies that the ground states
are fully polarized. The oscillation in this region becomes
regular, both the period and amplitude remain unchanged.
No. 4
Quantum Rings with Two Deeply Bound Electrons under a Magnetic Field
Thus the normal ABO recovers with the period of the flux
just equal to 1.
Figure 3 is for a small ring, but the width b − a is the
same as the ring of Fig. 1. Compared with Fig. 1, the levels
are now in general higher because the electrons are closer
to each other, and therefore the e-e Coulomb repulsion
is stronger. The period of B is now much longer, however the period of the flux remains the same because the
area embraced by the ring is now much smaller. On the
other hand, it was found that the amplitudes are greatly
affected by the size of the ring. When a − b is equal to
90 ∼ 100 nm and B is small, the amplitude of the ground
state energy is about 0.007 meV, while the amplitude of
hJ is about 0.1 meV. However, when a − b is equal to
30 ∼ 40 nm and B is small, the amplitude of the ground
state energy is about 0.07 meV, while the amplitude of hJ
is about 1 meV, i.e., they are larger by one order. Thus,
the reduction of size will lead to a very strong oscillation.
735
has been strictly limited inside a very narrow domain (this
is similar to the case of a square well potential with a perfectly rigid wall discussed frequently in the text books of
quantum mechanics). Once this limitation is released, the
levels become much lower as shown in the figure. When B
is small, the amplitude of the ground state energy is about
0.07 meV, while the amplitude of hJ is about 0.17 meV.
Thus the increase of the width leads to an increase of the
current. Furthermore, by comparing Fig. 1 with Fig. 4,
the ground state energies of a narrow ring go down in accord with the increase of B, however those of a broad ring
go up as shown in Fig. 4, this is a noticeable point.
Fig. 4 The same as Fig. 1 but a = 50 nm and
b = 120 nm.
Fig. 2 The same as Fig. 1 but B is larger.
Fig. 3 The same as Fig. 1 but a = 30 nm and b = 40 nm.
Fig. 5 The probability density |Ψ|2 plotted in the X-Y
plane as a function of r2 when e1 is given at the positive
X-axis marked by a white circle. The eigenstates plotted in (a) to (c) are the lowest, second lowest, and third
lowest states with S = 0 and L = −4. B = 1 T and
a = 30 nm, b = 60 nm are assumed. The length is in nm.
Figure 4 is for a very broad ring with a width 70 nm.
It is noted that the levels in Figs. 1 ∼ 3 are very high,
this is mainly not contributed by the angular motion but
a quantum effect caused by the fact that the radial motion
Now let us study the e-e correlation in the low-lying
states. In Fig. 5 the square of the norm of the eigenstates
|Ψ|2 was directly plotted in the X-Y plane as a function of
~r2 when e1 is given at the X-axis. Although only L = −4
736
SITU Shu-Ping
and S = 0 states are plotted, however this figure is typi-
Vol. 47
not affected by the size and width of the rings.
cal to all (S + L)-even states. They are characterized by
having a peak opposites to e1 . It is noted that when the
electrons stay at the two ends of a diameter, the Coulomb
repulsion can be minimized. Therefore, it is natural that
the lowest state would pursue the geometric configuration
as shown in Fig. 5a. In fact, this most favorable configuration is common to all the ground states. On the other
hand, during excitation, internal motion becomes more
energetic, more nodes will therefore appear in the wave
functions as shown in Figs. 5(b) and 5(c).
Figure 6 is typical to the lowest one of a series of states
with given S and L, and with (S +L)-odd. Instead of having a peak, this figure is characterized by having a node
Fig. 6 The same as Fig. 5 but for the lowest state with
S = 1 and L = −4.
opposites to e1 , this feature is common to all (S + L)-odd
states appear only in the higher part of the spectra. One
In summary, a model has been proposed for tightly
bound 2-electron rings. The effect of the size and the
width on the fractional AB oscillation together with the
e-e correlations have been studied. The results obtained
in this paper would enrich the understanding of quantum
rings.
can see from Fig. 6 that, even in the lowest odd state, the
Acknowledgments
internal oscillation has already been excited. Incidentally,
The author wishes to thank Prof. C.G. Bao and Mr.
G.M. Huang for their helpful discussions.
states, the node implies an excitation of an inherent mode.
Owing to this additional inherent node arising from symmetry constraint,[11,12] the (S + L)-odd states would oneto-one higher than the even states. In fact, the (S+L)-odd
the qualitative features of both Figs. 5 and 6 are basically
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