Journal of Computational Electronics 2: 487–490, 2003 c
2003 Kluwer Academic Publishers. Manufactured in The Netherlands.
Effect of Shape and Size on Electron Transition Energies for Nanoscale
InAs/GaAs Quantum Rings
YIMING LI∗
Department of Nano Device Technology, National Nano Device Laboratories, Hsinchu 300, Taiwan; Microelectronics and Information Systems Research Center, National Chiao Tung University,
Hsinchu 300, Taiwan
HSIAO-MEI LU
Department of Bioengineering, University of Illinois at Chicago, Chicago, Illinois 60612, USA
Abstract. In this paper, we study the impact of the sizes and the shapes of nanoscale semiconductor quantum rings on the electron and hole energy states. A three-dimensional effective one band Schr¨odinger equation is solved numerically for semiconductor quantum rings with disk, cut-bottom-elliptical, and conical shapes. For small InAs/GaAs quantum rings we have found a sufficient difference in the ground state and excited state (l = −1) electron energies for rings with the same volume but different shapes. Volume dependence of the electron and hole energies can vary over a wide range and depends significantly on the ring shapes. It is found that a non-periodical oscillation of the energy band gap between the lowest electron and hole states as a function of external magnetic fields.
Keywords: nanoscale quantum ring, InAs/GaAs, energy spectra, geometry and magnetic field effects, computer simulation
1. Introduction
Advances in the fabrication of semiconductor nanos-tructures have generated a huge quantity of experi-mental and theoretical data in this topic [1–13]. The three-dimensional (3D) confinement of charge carriers in those structures allows very rich optical and mag-netic characteristics which potentially may have very important device applications [1–6]. The study of semi-conductor nanoscale quantum rings significantly con-nects the gap between quantum dots and meso-scopic quantum ring structures. The spectral variation in semi-conductor quantum rings caused by the non-uniformity in the size and shape is important for magneto-optical properties and practical device applications. Various experimental results suggest that InAs/GaAs quantum
∗To whom correspondence should be addressed.
rings can have disk (DI), cut-bottom-elliptical (EL), or conical (CO) shapes with a circular top view cross sec-tion and a large area-to-height aspect ratio [7–10]. To the best of our knowledge, analysis of the influence of the ring size and shape on the electron energy states has not been done yet.
In this study, we calculate and compare the electron energy spectra for 3D nanoscale InAs/GaAs quantum rings of three different shapes (see Fig. 1): DI, EL, and CO shapes. Our model considers: (1) the position de-pendent effective mass Hamiltonian in non-parabolic approximation for electrons; (2) the position dependent effective mass Hamiltonian in parabolic approximation for holes; (3) the finite hard wall confinement poten-tial; and (4) the Ben Daniel-Duke boundary conditions. To solve this 3D nonlinear problem, the nonlinear it-erative method [11–13] is improved to calculate “self-consistent” solutions more efficiently. It is found that
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Figure 1. Quantum rings with different shapes: (a) DI, (b) EL and (c) CO.
the volume dependence of electron and hole energies can vary over a wide range and depends on the ring shapes. The variations of the ring size and shape pro-duce an energy change up to an order of 0.15 eV in the strong confinement region. The energy band gap versus the magnetic field depends on the ring volume and transits non-periodically.
This paper is organized as follows. Section 2 is the modeling and simulation. Section 3 describes results of calculations. Section 4 draws the conclusion.
2. Theoretical Model and Computational Method
We consider the DI-, EL-, and CO-shaped quantum rings with the hard-wall confinement potential [12,13].
The effective mass Hamiltonian for electrons (c = e) and holes (c= h) is ˆ Hc= r 1 2mc(E, r) r+ Vc(r)+ 1 2gc(E, r)µBBσ, (1) wherer = −i∇r+ eA(r) represents the electron
momentum vector. ∇r is the spatial gradient, A(r) is
the vector potential, and B = curlA is the magnetic field. For electrons, the electron effective mass follows
1 me(E, r) = 2P2 32 2 E+ Eg(r)− Ve(r) + 1 E+ Eg(r)− Ve(r)+ (r) , (2)
and Land´e factor is
ge(E, r) = 2 1− m0 me(E, r) (r) 3(E+ Eg(r))+ 2(r) . (3)
Ve(r) is the confinement potential, Eg(r) and(r) are
the position-dependent band gap and spin-orbit split-ting in the valence band, P is the momentum matrix element, σ is the vector of the Pauli matrixes, and
m0 and e are the free-electron elementary mass and
charge, respectively. For holes, mh(E, r) and gh(E, r)
are assumed to be only position dependent. We con-sider the hard-wall confinement potential Vc(r) = 0
for r inside the ring (I ) and Vc(r) = Vc0 for r
out-side the ring (II), where Vc0 is the band offset. The
Ben Daniel-Duke boundary conditions for the electron and hole wave functions(r) are c I(rs)= c I I(rs)
and (2/2m
c(E, r))∇r|ni(rs) = constant, where rs
is the position of the system interface. All rings are cylindrically symmetric with respect to the base radius and height in the coordinates (R,φ, z), so the wave function can be written asc(r)= c(R, z) exp(ilφ),
where l = 0, ±1, ±2,. . . is the orbital quantum num-ber, and the problem is in the coordinate (R, z). The Schr¨odinger equation for electrons and holes is
− 2 2mci(E) ∂2 ∂ R2 + ∂ R∂ R + ∂2 ∂z2 − l2 R2 ci(R, z) + mci(E)2ci(E)R2 8 + s µB 2 gci(E)B +ci(E) 2 l+ Vc0δi I I ci(R, z) = Eci(R, z), (4)
Effect of Shape and Size on Electron Transition Energies 489
where i = I , II, ci(E) = eB/mci(E), and s = ±1
is the orientation of the electron spin along the z-axis. The boundary conditions arec I(R, z) = c I I(R, z),
and 1 mc I(E) ∂c I(R, z) ∂ R + d f (R) d R ∂c I(R, z) ∂z z= f (R) = 1 mc I I(E) ∂ c I I(R, z) ∂ R +d f (R) d R ∂c I I(R, z) ∂z z= f (R) , (5)
where z = f (R) on the {R, z} plane is a contour generator for all ring structures.
The energy band gap Eg(B)= Ege(B)+ Egh(B)+ Eg R, where Ege and Egh are the ground state
ener-gies for electrons and holes, and Eg R is the energy
gap in the quantum ring. The energy dependence of the electron effective mass and Land´e factor compli-cates the analytical solution considerably. Computer simulation of energy spectra for quantum rings is sug-gested. For each applied magnetic field, the effective mass and Land´e factor are calculated with an arbitrary initial energy E = E0. The finite volume [14]
dis-cretized Schr¨odinger equation is solved to calculate all bounded energy levels. To solve the correspond-ing matrix eigenvalue problem more efficiently, a hy-brid computational scheme is suggested. This scheme combines the robust balanced and shifted QR method [15] with the fast implicitly restarted Arnoldi [16]. If
E converges, we calculate Eg(B); else we update the
newer E and perform the next iteration. The general-ized method for solving different shape quantum rings converges monotonically and is cost effective
3. Results and Discussion
In Fig. 2 we present the calculated electron energy lev-els for InAs/GaAs quantum rings as functions of the ring volume. For InAs, E1gis 0.42 eV,1is 0.38 eV,
and m1e(0) = 0.024m0. For GaAs, E2g is 1.52 eV, 2 = 0.34 eV, m2e(0)= 0.067m0, and V0= 0.77 eV.
From experimental data [6–9], the base radius of the rings R0 = 20 nm and the inner radius Rin = 10 nm
for all shapes. Our model predicts electron energy de-pendences on the volume for rings of different shapes. When the ring volume increases, the energy states of different shapes converge. The most sensitive to the
Figure 2. The electron ground state (the left figure) and the l= −1 excited state (the right one) energies versus the ring volume at B= 0 T.
ring volume variation is the DI shape and, the least one is the CO shape rings. This is no surprise since the electron wave function is the best confined for the disk geometry when the volume and the radius are fixed. The wave function shape confirms weaker confinement for CO-shaped rings. The excited state (l = −1), how-ever, has demonstrated a weaker sensitivity to the ring shape and volume (see the right one of Fig. 2). This is because that the electron wave functions of the exited states are less confined and, therefore, are less sensitive to the ring shape and size.
Using the same calculation method, we obtained hole energy states for rings of the same shapes. The hole effective mass is taken as m1h = 0.4m0 and m2h= 0.5m0, and band offset V0= 0.33 eV. Shown in
Fig. 3 is the hole energy states for the ground (the left figure) and l= −1 excited (the right one) states. With this same parameter setting Fig. 4 shows the electron and hole energy versus B for the DI-shaped quantum ring, where the ring volume 3000 nm3is fixed. The
en-ergy states are numerated by a set of quantum numbers
{n, l, s}, where n = 0, 1, 2,. . . is the main quantum
Figure 3. The hole ground state (the left figure) and the l = −1 excited state (the right one) energies versus the ring volume at B= 0 T.
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Figure 4. Plots of electron (the left figure) and hole (the right one) energies versus B for the DI-shaped quantum ring with volume= 3000 nm3.
Figure 5. The energy band gap versus B for the DI-shaped quantum ring with volume= 3000 nm3(the left figure) and 6000 nm3(the
right one).
number. The left figure of Fig. 4 is the electron energy
E0,l,+1for l = 0, −1, and −2. The calculated results of the energy band gap between the lowest electron and hole states for the InAs/GaAs quantum ring with volumes 3000 and 6000 nm3 are shown in Fig. 5. We
find a non-periodical oscillation of Eg(B) between the
lowest electron and hole states as a function of B. The transition of Eg(B) depends on the ring volume. Due
to the wave function penetration into the torus region, it is found Eg(B) does not follow the 1D periodical
rule: π(Rin + R)2B/0 = n, where n is an integer
number and0is the quantum of magnetic flux [13]. Eg(B) depends on ring volume, and its non-periodical
oscillation directs to our 3D modeling and simula-tion. The results should be examined in the magnetic-photo-luminescence spectra for nanoscale quantum rings.
4. Conclusions
We have presented a computational approach that al-lows us to study the electron and hole energy states for nanoscale semiconductor quantum rings of three different shapes. This method is useful to analyze the dependence of the quantum ring spectra on the ring volume and shape distributions. We found a large difference for the electron ground state energy in InAs/GaAs rings of the same volume but different shapes. The energy states of holes and transition en-ergies under external magnetic fields were estimated. Our results advised that the non-periodical oscilla-tion of energy band gap depends on the ring volume.
Acknowledgment
This work is supported in part by the National Science Council of TAIWAN under under con-tract numbers 2112-M-429-001 and NSC-92-2815-C-492-001-E. It is also supported in part by the grant of the Ministry of Economic Affairs, Taiwan un-der contract No. 91-EC-17-A-07-S1-0011.
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