Finger-gate manipulated quantum transport in a semiconductor narrow constriction with spin-orbit
interactions and Zeeman effect
Chi-Shung Tang,1,*Shu-Yu Chang,2and Shun-Jen Cheng2,†
1Department of Mechanical Engineering, National United University, Miaoli 36003, Taiwan, Republic of China 2Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan, Republic of China
(Received 19 July 2012; published 24 September 2012)
The authors investigate quantum transport in a narrow constriction fabricated by narrow-band-gap semicon-ductor materials with spin-orbit couplings. We consider the Rashba-Dresselhaus (RD) spin-orbit interactions (SOIs) and the Zeeman effect induced by an in-plane magnetic field along the transport direction. The interplay of the RD SOI and the Zeeman effect may induce a SOI–Zeeman gap and influence the transport properties. We demonstrate that an attractive scattering potential may induce an electronlike quasi-bound-state feature and manifest the RD–SOI–Zeeman induced Fano line shape in conductance. Furthermore, a repulsive scattering potential may induce a holelike quasi-bound-state feature on the subband top of the lower spin branch. DOI:10.1103/PhysRevB.86.125321 PACS number(s): 73.23.−b, 72.25.Dc, 72.30.+q
I. INTRODUCTION
Quantum transport involving the interference nature of a charged particle can be realized by using the split-gates-induced narrow constriction connecting the source and drain Ohmic contacts. The conductance through the narrow con-striction is known to be quantized when the Fermi level of the system is tuned energetically by applying a voltage to a nearby gate.1,2 The quantization features can be explained within the framework of simple noninteracting models,3–5 and the conductance depends only on the transmission coefficient. The related quantum devices can be utilized in various applications including the prototypes of quantum information processing.6 Spin-orbit interaction (SOI) is a relativistic effect, in which a charged particle moving with direction perpendicular to an electric field experiences an effective magnetic field that couples to the spin degree of freedom of the moving particle. Various spin-orbit (SO) effects present in semiconductor structures provide a promising way to spin manipulation in two-dimensional (2D) electron gases.7,8Band-structure behav-iors and transport properties involving SOI in semiconductor quantum structures have received much interest due to their important application in the emerging field of spintronic devices.9–11 Manipulating the spin degree of freedom offers the possibility of devices with high-speed and very-low-power dissipation that is one of the essential requirements for the applications in quantum computing and memory storage.12,13
The SOI can be induced when the transporting electron experiences a strong electric field due to the asymmetry in the confinement potential, namely, the structure inversion asymmetry (SIA) induced Rashba SOI.14 Especially, the Rashba SOI may be significantly induced in two-dimensional electron gases (2DEGs) confined by asymmetric potential in semiconductor materials. Experimentally, the Rashba interac-tion has been shown to achieve electron spin manipulainterac-tion by using bias-controlled gate contacts.15
In addition to the Rashba effect, there is also a Dresselhaus SOI caused by the microscopic electric field arising from the lack of inversion symmetry in the Bravais lattice, namely, the bulk inversion asymmetry (BIA).16 The combined effect of the Rashba and Dresselhaus SOI affects significantly the spin-related properties and should be considered when
analyzing the performance of spin-resolved electronic devices. Recently, several approaches were proposed to engineer the spin-resolved subband structure utilizing magnetic fields17–20 or ferromagnetic materials.21,22The SOI and in-plane magnetic field induced Zeeman effect may modify the subband structure leading to a SOI–Zeeman subband gap feature.23,24However, how the scattering potentials influence the spin-resolved quantum transport and its interplay with the SOI–Zeeman interactions has not yet been explored.
In this work, we consider a split-gate-induced narrow constriction that is fabricated in a 2D quantum well with narrow-band-gap semiconductor material. Both the Rashba and Dresselhaus SOIs as well as an applied external in-plane magnetic field are taken into account to investigate the influ-ences of the subband structures. Moreover, we apply a narrow finger gate to affect the ballistic transport properties. In the following, we shall demonstrate analytically and numerically that tuning the strength of the applied in-plane magnetic field as well as the Rashba and the Dresselhaus SO coupling constants to manipulate the subband structures, leading to fruitful quantum transport properties.
This article is organized as follows. In Sec. II, we shall describe our theoretical model including the Rashba and Dres-selhaus SOIs as well as an external in-plane magnetic field. SectionIII investigates the spin-resolved quantum transport properties. Concluding remarks will be presented in Sec. IV.
II. MODEL AND SUBBAND STRUCTURES
The system under investigation is assumed to be a narrow-band-gap InAs-In1−xGaxAs semiconductor heterostructure
grown in [0,0,1] crystallographic direction. We consider the conduction band of a 2D quantum well within the effective mass approximation. We select the length unit l∗= 1/kF as
the inverse of the Fermi wave number kF, and the energy unit
E∗= EFis the Fermi energy EF = ¯h2k2F/2m∗with m∗and ¯h
being, respectively, the effective mass of an electron and the reduced Planck constant. Correspondingly, the magnetic field is in units of B∗= E∗/μBwith μBbeing the Bohr magneton,
and the Rashba and Dresselhaus SO coupling constants are in units of α∗= β∗= E∗l∗. By using the above units, all physical quantities presented in the following are dimensionless.25
FIG. 1. (Color online) Schematic illustration of the quantum channel defined by a pair of split gates that is fabricated by a narrow-band-gap InAs-In1−xGaxAs semiconductor heterostructure forming the two-dimensional electron gas (2DEG). An external in-plane magnetic field B= B ˆx and a top finger gate are applied to influence the spin-resolved quantum transport properties.
A pair of split gates is applied in the transverse direction forming a quantum channel described by the unperturbed Hamiltonian
H0= k2+ Vc(y) (1)
that consists of a 2D kinetic energy term k2= k2
x+ k2y and a
confining potential energy term Vc(y)=
0, |y| < W/2
∞, otherwise (2)
where W indicates the width of the quantum channel. The transported electron is supposed to be affected by the effects of SO interaction and the external in-plane magnetic field, and hence can be described by the effective unperturbed Hamiltonian
H0= H0+ HSO+ HZ. (3)
Here, we have assumed that the magnetic field is applied in the transport direction ˆx such that the Zeeman interaction is simply HZ= gBσx, in which the factor g= gs/2 with gs being the
effective gyromagnetic factor (gs = −15 for InAs). In order
to manipulate the spin-resolved quantum transport properties, we apply a finger gate on top of split gate with an insulator in-between, as illustrated in Fig.1. We assume that the finger gate is sufficiently narrow and then can be described by a delta scattering potential form Vsc(x)= V0δ(x). The whole
quantum channel system under investigation is thus described by the total Hamiltonian H = H0+ Vsc(x).
The SOI term in Eq. (3) consists of the Rashba and Dresselhaus SOI effects HSO= HR+ HD. For the transport
direction ˆx [1,0,0], the Rashba SO Hamiltonian is given by the k-linear form
HR= α(σxky− σykx), (4)
where σi(i= {x,y,z}) are the Pauli matrices and k = (kx,ky)
is the 2D electron wave vector. The Rashba coupling strength α is proportional to the electric field along the ˆz direction perpendicular to the 2D electron gas. In general, the Dressel-haus interaction has a cubic dependence on the momentum of the carriers. For a narrow semiconductor quantum well
grown along the [0,0,1] direction, it reduces to a 2D linear momentum-dependent form
HD= β(σxkx− σyky), (5)
where the Dresselhaus coupling strength β is determined by the semiconductor material and the geometry of the sample. The spin-orbit coupling contributions can be simplified as HSO =
(−ασy+ βσx)kx in a narrow quantum channel.
The eigenfunction of Eq. (3) can be expressed as the multiplication of the spatial wave functions and the spinor state χn:
(x,y)=
n
φn(y)eikxxχn, (6)
where the transverse wave function in the subband n is of the form φn(y)= π W sin nπ Wy (7) with unperturbed subband energy εn= (nπ/W)2 due to the
bare confining potential. The corresponding eigenenergies can be obtained.
A. Rashba-Zeeman effects
In the absence of the Dresselhaus SOI, the Dresselhaus coupling strength β is identically zero. In this section, we focus on the the Rashba-Zeeman (RZ) effect, in which the spin-resolved subband energies can be obtained analytically26:
Enσ = εn+ kx2+ σ
(gB)2+ (2αkx)2 (8)
and the spinor states χnσ = √1 2 1 σ eiθ(kx) , (9)
where σ = ± indicates the upper (+) and lower (−) spin branches and θ (kx)= tan−1(2αkx/|gB|) describes the
momentum-dependent spin orientation of an electron. Defin-ing the group velocity of an electron in the σ spin branch
vgσ =dE σ n dkx = 2kx+ σ 4α2k x g2B2+ 4α2k2 x (10) allows us to determine the local minimum (subband bottom) and local maximum (subband top) in the subband structures by setting the group velocity to be identically zero.
The calculations presented below are carried out under the assumption that the electron effective mass m∗= 0.023m0,
which is appropriate to the InAs-In1−xGaxAs semiconductor
interface with the typical electron density ne∼ 1012 cm−2.15
Accordingly, the length unit is l∗= 1/kF= 5.0 nm, the energy
unit is E∗ = EF = 66 meV, and the spin-orbit coupling
parameters are in units of α∗ = β∗= 3.3 × 10−10eV m. In the following, we select the width of the narrow constriction W= πl∗= 15.7 nm so that the unperturbed subband energy is simply εn= n2. Moreover, the range of the variation of
energy E is smaller than the second unperturbed subband energy, namely, E < ε2EF = 4EF. Furthermore, sufficient
low temperature is required to avoid thermal broadening effect, that is, kBT <0.1 ε (or T < 23 K). We note in passing that
FIG. 2. (Color online) Energy spectrum versus wave number with magnetic field strength gB= 0.02 with different values of Rashba coupling constant: (a) α= 0.05 (weak Rashba effect, 2α2< gB); (b)
α= 0.1 (critical Rashba effect, 2α2= gB); and (c) α = 0.2 (strong Rashba effect, 2α2> gB). The Fermi energy E
F= 66 meV and the Fermi wave number kF= 2 × 106cm−1. The magnetic field strength is approximately 3 T when gB = 0.02 (gs = −15 for InAs). The black and red curves indicate the plus (σ = +) and minus (σ = −) spin branches, respectively.
the width of the scattering potential Vsc(x) should be less than
the Fermi wave length λF = 31.4 nm to be described as a delta
potential. We assume high-mobility semiconductor materials and, hence, the impurities and defects can be neglected.
The energy spectrum for the case of RZ effect with different coupling strength regimes is illustrated in Fig. 2. Before we illustrate the subband gap features, it should be reminded that the Zeeman effect is to induce an energy gap EZ= 2gB between the opposite spin branches. For the case
of weak Rashba SO coupling, namely, 2α2< gB, both the spin
branches have only subband bottoms at kx = 0 with energies
at Eσ
n = εn+ σgB. The subband energy spacing between the
upper (+) and the lower (−) branches is the Zeeman splitting ERZ = EZ. Hence, the RZ–SO gap is dominated by the
Zeeman effect in the weak SO coupling regime.
For the case of strong Rashba coupling 2α2> gB, the
subband bottom of the upper spin branch is still at En+=
εn+ gB. However, the subband bottom at kx = 0 of the
lower spin branch becomes a subband top with the same
FIG. 3. (Color online) Energy (in units of Fermi energy EF) as a function of complex wave number kx = kR+ ikI(in units of Fermi wave number kF). The physical parameters are α= 0.2, β = 0, gB = 0.02. The black curves represent the propagating modes and the red curves denote the evanescent modes.
energy En+= εn− gB. Therefore, the subband energy spacing
between the + and − branches is still EZ= 2gB, but
forming a subband gap. In addition to the subband top in the lower subband branch, there are two subband bottoms at kx = ±[α2− (gB/2α)2]1/2 with the same energy E−n =
εn− [α2+ (gB/2α)2].
We note in passing that if only the Rashba effect is consid-ered, the subband structure is simply Eσ
n = εn+ k2x+ σ2αkx.
Also, the subband structure manifests only lateral splitting in momentum kx = 2α, where the subband bottoms of σ spin
branches are at the wave numbers kxσ = −σ α with the same
energy En= εn− α2.
In order to investigate the transport properties, one has to determine the propagating and evanescent modes for a given energy. To this end, it is convenient to rewrite the energy dispersion relation in the form
k2x = (E + 2α2− εn)
∓
(E+ 2α2− εn)2+ (gB)2− (E − ε
n)2. (11)
In general, this equation determines four complex kx values
corresponding to either propagating or evanescent modes. In Fig.3, we show the energy dispersion obtained from Eq. (11) in the complex wave-number space for the case of 2α2> gB so that subband gaps can be generated. It is clearly shown that there are four evanescent modes when the electron energy is less than the lower subband bottom. When the electron energy is greater than the lower subband bottom and below the subband gap, there are four propagating modes. It is interesting to notice that when electron energy is within the subband energy gap regime, there are two propagating modes and two evanescent modes (the red bubble in Fig.3). Although the conductance calculated later only sums over the propagating modes, a sufficient number of evanescent modes should be taken into account to achieve numerical accuracy when we calculate the intermediate scattering processes.
B. Rashba-Dresselhaus-Zeeman effects
In the presence of Rashba and Dresselhaus SOI with an in-plane magnetic field along the transport direction, the
FIG. 4. (Color online) Energy spectrum versus wave number in the presence of in-plane magnetic field (gB = 0.02) with different Rashba and Dresselhaus SO coupling constants: (a) α= β = 0.02 (weak coupling); (b) α= β = 0.1 (mediate coupling); (c) α = β = 0.2 (strong coupling). The Fermi energy EF = 66 meV and the Fermi wave vector kF = 2 × 106cm−1. The magnetic field strength is approximately 3 T when gB = 0.02 (gs = −15 for InAs). The black and red curves indicate the plus (σ = +) and minus (σ = −) spin branches, respectively.
electronic system can be described by εn+
k2x− 2αkxσy+ 2βkxσx+ gBσx
= E. (12) For electrons incident from the subband n, the spinor states χn
satisfy the 2× 2 matrix equation k2 x gB+ 2βkx+ i2αkx gB+ 2βkx− i2αkx kx2 χn = (E − εn)χn. (13)
The energy spectrum can be easily obtained of the form E= εn+ kx2+ σ
(2βkx+ gB)2+ (2αkx)2. (14)
This equation is convenient to obtain energy spectrum as a function of real wave vector for propagating modes.
In Fig.4, we show the Dresselhaus effect correction to the Rashba SOI by fixing the strength of Zeeman effect (gB = 0.02) and tuning the strength of Rashba and Dresselhaus
SOI. To analyze the the subband structures, it is convenient to introduce the Rashba-Dresselhaus SO coupling constant γ, namely, γ2= α2+ β2, for defining different coupling
regimes. Figure 4(a) demonstrates the weak SO coupling regime γ2< gB. It is clearly shown that the spin-split subband
structure is slightly asymmetric due to the Dresselhaus effect. The subband bottoms of both subband branches are no longer at zero wave number. Instead, the energy bottoms are located at (kx,E1+)= (−0.02,1.02) and (kx,E1−)= (0.02,0.980). The
two spin branches of a subband n manifest a Zeeman split-ting EZ= En+− En−= 2gB = 0.04. Hence, in the weak
coupling regime, the Zeeman effect dominates the subband structure and the RD coupling slightly lets the subband structure form an asymmetric lateral shift in the opposite direction for the spin branches.
Figure 4(b) illustrates the case of intermediate SO cou-pling regime γ2= gB: it is shown that the lower spin
branch becomes a shoulder subband structure at (kx,E−1)=
(−0.1002,0.99) and a clear subband bottom at (kx,E−1)=
(0.136,0.964). On the other hand, the subband bottom of the upper branch is at (kx,E1+)= (−0.0361,1.016), and hence the
spin branches form a shoulder gap feature Esg = 0.026 <
EZ.
In Fig. 4(c), we show the case of strong SO coupling regime (γ2> gB). In this regime, it is interesting that the
lower spin branch manifests three extreme values in energy. First, the left subband bottom of the upper spin branch is at (kx,E1−)= (−0.28,0.934). Second, the right subband bottom
is at (kx,E1−)= (0.28,0.905). Third, the subband top of the
lower subband branch is at (kx,E1−)= (−0.02,0.986). On
the other hand, the subband bottom of the upper branch is at (kx−,E−1)= (−0.02,1.015). Therefore, the subband gap in
the strong coupling regime is around Eg = 0.029 > Esg.
This implies that reduction of the subband gap due to the Dresselhaus effect is a nontrivial effect.
III. QUANTUM TRANSPORT PROPERTIES In this section, we shall investigate the quantum transport properties subject to spin-orbit interactions and Zeeman effect in a narrow constriction. We assume that the quantum channel is sufficiently narrow and focus on the first two conductance steps associated with the two spin branches of an electron. In the following, we shall explore how the spin-mixing effect due to the SOI–Zeeman coupling influences the transport properties.
A. Rahsba-Zeeman effects
To consider an electron incident along the transport direc-tion x, it is convenient to denote the wave number of rightgoing (leftgoing) modes as kσ (qσ), in which the subscript σ could
be by “1” or “2” indicating the “outer” or the “inner” modes, as illustrated in Fig.5. The scattering wave function for an electron incident from the source electrode can be written in the form ψ(x)= eikσxχ(k σ)+ σ rσeiqσxχ(qσ) if x < 0, (15) ψ(x)= σ tσeikσxχ(kσ) if x > 0. (16)
FIG. 5. (Color online) Schematic illustration of the energy spectrum with labeling modes. Here, k1 and q1 indicate the outer rightgoing and outer leftgoing modes, respectively. Similarly, k2 and q2 indicate the inner rightgoing and inner leftgoing modes, respectively. In the gap energy regime, both the inner modes are evanescent modes.
Here, we have omitted the subband index for simplicity. By taking into account the spin branches as well as the spin-flip scattering mechanisms, the scattering wave functions can be generally expressed as ψ(x)= eikσx aσ bσ + rσeiqσx cσ dσ + rσ¯eiqσ¯x cσ¯ dσ¯ if x >0, (17) ψ(x)= tσeikσx aσ bσ + tσ¯eikσx¯ aσ¯ bσ¯ if x >0, (18) where aσ = 1 √
2 and bσ = −eiθ(kσ)/
√
2 with θ (kσ) =
tan−1[2αkσ/(gB)] are the spinor elements of the incident
wave. In addition, cσ = 1/
√
2 and dσ = −eiθ(qσ)/
√ 2 with θ(qσ) = tan−1[2αqσ/(gB)] are the spin-state elements of
the reflection wave. Moreover, the spin-state elements of the spin-flip transmission wave are given by
aσ¯ = (gB+ 2αkσ¯)2 (gB+ 2αkσ¯)2+g2B2− 4α2kσ2¯ 1/2 (19) and bσ¯ = aσ¯ g2B2− 4α2k2 ¯ σ gB+ 2αkσ¯ . (20)
Similarly, we can obtain the spin-state elements of the spin-flip reflection wave, given by
cσ¯ = (gB+ 2αqσ¯)2 (gB+ 2αqσ¯)2+g2B2− 4α2qσ2¯ 1/2 (21) and dσ¯ = cσ¯ g2B2− 4α2q2 ¯ σ gB+ 2αqσ¯ . (22)
By matching the boundary conditions at around the scat-tering potential induced by the finger gate, it is easy to obtain the matrix equation of motion for the spin-resolved transport
involving the finger-gate induced spin-flip scattering ⎡ ⎢ ⎢ ⎢ ⎣ −cσ −cσ¯ aσ aσ¯ −dσ −dσ¯ bσ bσ¯ −qσcσ −qσ¯cσ¯ (kσ+ iV0)aσ (kσ¯ + iV0)aσ¯ −qσdσ −qσ¯dσ¯ (kσ+ iV0)bσ (kσ¯ + iV0)bσ¯ ⎤ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎣ rσ rσ¯ tσ tσ¯ ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎣ aσ bσ kσaσ kσbσ ⎤ ⎥ ⎥ ⎥ ⎦. (23)
To calculate conductance in the noninteracting electron model, we employ the framework of the Landauer-B¨uttiker formula.3,4 For a given energy, solving for the spin-nonflip and -flip reflection coefficients rσ and rσ¯, as well as the
spin-nonflip and -flip transmission coefficients tσ and tσ¯, we
can thus express the zero-temperature conductance as G= G0
σL,σR
vσR
vσL
|tσL,σR|2. (24)
Here, G0= e2/ his the conductance quantum of a single spin
branch, and σLand σRindicate, respectively, the spin branches
of the incident and transmission waves in the left and right leads. Therefore, vσR and vσL represent the group velocity at
the corresponding modes.
In Fig.6, we demonstrate the transport properties in the presence of an attractive scattering potential due to the finger gate by fixing the in-plane magnetic field (gB= 0.02) while tuning the strength of Rashba SOI. In general, the attractive scattering potential plays a role to suppress the conductance. We present the conductance as a function of electron energy with different strength of attractive potential: V0= 0.0 (dotted
line), V0= −0.1 (dashed line), V0= −0.2 (dashed-dotted
line), V0= −0.3 (solid line). Here, we fix the Zeeman effect to
be gB= 0.02, in other words the magnetic field B = 3 T if the factor gs = −15 for InAs-based material. Moreover, we tune
the Rashba parameter as follows: (a) α= 0.05 (2α2< gB, weak SO coupling); (b) α= 0.1 (2α2= gB, intermediate SO coupling); (c) α= 0.2 (2α2> gB, strong SO coupling).
Figure6(a)illustrates the transport properties in the weak SO coupling regime (2α2 < gB). When the attractive potential
is weak (V0= −0.1), the conductance manifests a clear dip
structure and forms an electronlike quasi-bound-state at the subband bottom of the upper spin branch. When the potential strength increases (V0= −0.3), the dip structure becomes a
broad valley structure and is shifted toward the low-energy regime, indicating the shorter lifetime. This broadening effect is suppressed in the mediate SO coupling regime (2α2= gB), as shown in Fig.6(b). It is interesting to notice in Fig.6(c)that the conductance manifests an abrupt drop to unity in the energy regime 0.98 < E < 1.02 due to the spin-gap feature as shown previously in Fig.2(c). It is interesting that the conductance dip structure is not broadened for larger scattering potential V0. This indicates that the lifetime of the quasi-bound-state
feature is enhanced in the strong Rashba SO coupling regime. In Fig.7, we demonstrate the transport properties in the presence of a repulsive scattering potential due to the finger gate by fixing the in-plane magnetic field (gB= 0.02) while tuning the strength of Rashba SOI. In general, the repulsive
FIG. 6. (Color online) Conductance as a function of electron energy with different strength of attractive potential: V0= 0.0 (dotted black line), V0= −0.1 (dashed red line), V0= −0.2 (dashed-dotted blue line), V0= −0.3 (solid green line). We fix the Zeeman effect
gB= 0.02 (or B = 3 T if gs = −15 for InAs-based material), while we tune the Rashba parameter: (a) α= 0.05 (2α2< gB, weak SO coupling); (b) α = 0.1 (2α2= gB, intermediate SO coupling); (c)
α= 0.2 (2α2> gB, strong SO coupling).
potential plays a role to strongly suppress the conductance in the low-kinetic-energy regime. We exhibit the conductance as a function of electron energy with different strength of repulsive scattering potential: V0= 0.0 (dotted line), V0= 0.1
(dashed line), V0= 0.2 (dashed-dotted line), V0= 0.3 (solid
line). Here, we fix the Zeeman effect to be gB= 0.02, namely, the magnetic field B= 3 T if the factor gs = −15 for
InAs-based material. Then, we tune the strength of the Rashba SOI: (a) α = 0.05 (2α2< gB, weak SO coupling); (b) α = 0.1
(2α2= gB, intermediate SO coupling); (c) α = 0.2 (2α2>
gB, strong SO coupling).
For the case of weak SO coupling regime shown in Fig. 7(a), the conductance is strongly suppressed in the
FIG. 7. (Color online) Conductance as a function of electron energy with different strength of repulsive scattering potential:
V0= 0.0 (dotted black line), V0= 0.1 (dashed red line), V0= 0.2 (dashed-dotted blue line), V0= 0.3 (solid green line). We fix the Zeeman effect gB= 0.02 and tune the parameter of Rashba SOI: (a)
α= 0.05 (2α2< gB, weak SO coupling); (b) α= 0.1 (2α2= gB, intermediate SO coupling); (c) α = 0.2 (2α2> gB, strong SO coupling).
low-kinetic-energy regime and behaves monotonically in-creasing. For the case of intermediate SO coupling regime shown in Fig.7(b), the conductance is more strongly sup-pressed in the low-kinetic-energy-regime than in the case of the weak SO coupling regime. It is interesting to notice when the repulsive potential is strong enough (V0 = 0.3),
the conductance is even suppressed to zero at energy E≈ 0.984E∗. This is a clue of a holelike quasi-bound-state feature with very short lifetime due to the shoulderlike structure of the lower subband branch shown in Fig.7(b). For the case of strong SO coupling regime, since the subband structure can form a subband gap, as is shown in Fig. 7(c), it allows us to form a significant holelike quasi-bound-state feature at the
subband top of the lower spin branch. The conductance thus manifests a dip structure energy E≈ 0.984E∗.
B. Rashba-Dresselhaus-Zeeman effects
In this section, we shall explore the transport properties of a narrow constriction by fixing the Zeeman effect while manipulating the strength of the RD–SOI and tuning the amplitude of scattering potential that can be either attractive or repulsive. All the physical parameters shown in the following are the same with the physical parameters in the previous section discussing the case of R–SOI. The transport calculation for the case of RD–SOI is similar to the case of R–SOI but has to be solved numerically, which is not shown here.
In Fig. 8, we investigate how an attractive scattering potential influences the transport properties by tuning Rashba and the Dresselhaus effects and fixing the in-plane magnetic field; the corresponding energy spectra are shown in Fig.4. The conductance is presented as a function of electron energy with different strengths of attractive scattering potential: V0= 0.0
(dotted line), V0= −0.1 (dashed line), V0= −0.2
(dashed-dotted line), V0= −0.3 (solid line). We fix the in-plane
magnetic field so that the Zeeman effect gB = 0.02. In addition, the Rashba and Dresselhaus SO coupling constants are selected to cover three coupling regimes: (a) α= β = 0.02 (γ2< gB, weak SO coupling regime); (b) α= β = 0.1
(γ2= gB, intermediate SO coupling regime); (c) α = β =
0.2 (γ2> gB, strong SO coupling regime).
For the case of the weak SO coupling regime shown in Fig.8(a), the attractive scattering potential may induce a Fano structure in conductance. This is because a true bound state can be induced by the attractive scattering potential at energy E= E1+− V02/4, in which the binding energy Eb= V02/4= 0.0025
(dashed line), 0.01 (dashed-dotted line), and 0.0225 (solid line). The Fano structure is at E≈ 0.99E∗for potential V0=
−0.3. It is interesting to notice that the bounded upper-spin-branch electron bounded energy interfere with the extended lower-spin-branch electron and form the RD–Zeeman induced Fano structure. For the case of the intermediate SO coupling regime shown in Fig.8(b), we can see a clear quasi-bound-state feature at the subband bottom of the upper spin branch. For the case of strong SO coupling regime shown in Fig. 8(c), the Fano structure is red-shifted to E≈ 0.91E∗for potential V0= −0.3. In the SOI–Zeeman induced subband gap region,
we see a more significant quasi-bound-state formed at around E≈ 1.02E∗.
In Fig.9, we investigate how a repulsive scattering potential influences the transport properties by tuning Rashba and the Dresselhaus effects and fixing the in-plane magnetic field; the corresponding energy spectra are shown in Fig. 4. The conductance is plotted as a function of electron energy with different strength of repulsive scattering potential: V0= 0.0
(dotted line), V0= 0.1 (dashed line), V0= 0.2 (dashed-dotted
line), V0= 0.3 (solid line). We fix the in-plane magnetic
field so that the Zeeman effect gB = 0.02. In addition, we tune the Rashba and Dresselhaus SO coupling constants as (a) α= β = 0.02 (γ2< gB, weak spin-orbit coupling regime);
(b) α= β = 0.1 (γ2= gB, intermediate spin-orbit coupling
regime); (c) α= β = 0.2 (γ2> gB, strong spin-orbit coupling
regime).
FIG. 8. (Color online) Conductance as a function of electron energy with different strength of attractive scattering potential:
V0= 0.0 (dotted black line), V0= −0.1 (dashed red line), V0= −0.2 (dashed-dotted blue line), V0= −0.3 (solid green line). We fix the in-plane magnetic field (gB = 0.02) and tune the Rashba and Dresselhaus SO-coupling constants: (a) α= β = 0.02 (γ2< gB, weak SO coupling regime); (b) α= β = 0.1 (γ2= gB, intermediate SO coupling regime); (c) α= β = 0.2 (γ2> gB, strong SO coupling regime).
For the case of the weak SO coupling regime shown in Fig.9(a), the repulsive scattering potential can not form bound states even for the case of strong potential amplitude V0 = 0.3,
in which the conductance behaves monotonically increasing and the conductance is suppressed to G≈ 0.7G0. For the case
of the intermediate SO coupling regime shown in Fig.9(b), it is interesting to note that the conductance manifests a holelike quasi-bound-state feature on the top of shoulder subband top (E= 0.99E∗), as is shown in Fig. 4(b). For the case of the strong SO coupling regime shown in Fig.9(c), the general feature in conductance is the strong drop from 2G0 to G0
in the subband gap of the two spin branches. Moreover, it is clearly shown that the holelike quasi-bound-state feature can
FIG. 9. (Color online) Conductance as a function of electron energy with different strength of repulsive scattering potential:
V0= 0.0 (dotted black line), V0= 0.1 (dashed red line), V0= 0.2 (dashed-dotted blue line), V0= 0.3 (solid green line). We fix the in-plane magnetic field (gB = 0.02) and tune the Rashba and Dresselhaus SO coupling constants: (a) α= β = 0.02 (γ2< gB, weak SO coupling regime); (b) α= β = 0.1 (γ2= gB, intermediate SO coupling regime); (c) α= β = 0.2 (γ2> gB, strong SO coupling regime).
be induced on the subband top of the lower spin branch and form a very clear dip structure in conductance.
IV. CONCLUDING REMARKS
We consider a narrow constriction with the Rashba and Dresselhaus spin-orbit interactions under an in-plane magnetic field applied in the transport direction. A top finger gate is used to generate an attractive or a repulsive scattering potential. This allows us to investigate the coherent quantum transport properties involving spin-flip scattering. The competition of the spin-orbit scattering and the Zeeman effect plays an important role to the subband structures and the transport properties. The Zeeman effect allows us to separate the R–SOI
and RD–SOI into three regimes: the weak, mediate, and strong SO coupling regimes.
In the weak SO coupling regime with Zeeman effect, the subband structure remains the quadratic form. It is symmet-ric if only the Rashba SOI dominates, while asymmetsymmet-ric if both the Rashba and Dresselhaus SOIs are significant. For the case of attractive potential with only the Rashba SOI, it allows electron occupying the upper spin branch to form a true-bound-state feature with binding energy V2
0/4, and the conductance manifests a valley structure. It
is important to note that the presence of both the Rashba and the Dresselhaus SOIs may enhance the interference between the localized upper-spin-branch state and the ex-tended lower-spin-branch state, and hence the conductance manifests a Fano structure. For the case of repulsive potential, the conductance behaves monotonically increasing for both R–SOI and RD–SOI.
In the intermediate SO coupling regime with Zeeman effect, the subband structure of the lower spin branch exhibits a quadratic structure for R–SOI and a shoulderlike structure for RD–SOI. For the case of attractive potential with R– SOI, the conductance manifests a quasi-bound-state feature below the upper branch. Moreover, for the case of attractive potential with RD–SOI, we have found a kink structure in conductance at the shoulder of the lower spin branch. For the case of repulsive potential with R–SOI, the conductance is strongly suppressed and monotonically increasing. How-ever, for the case of repulsive potential with RD–SOI, the conductance can manifest a clear holelike quasi-bound-state feature.
In the strong SO coupling regime with Zeeman effect, the subband structure of the lower spin branch exhibits a subband top structure for both the R–SOI and RD–SOI. In addition, the two subband bottoms of the lower spin branch with the same energy for R–SOI and with different energy for RD–SOI. For the case of attractive potential with R–SOI, the conductance manifests a quasi-bound-state feature below the upper branch. However, the conductance structure for the case of attractive potential with RD–SOI is more complicated. We have found a true-bound-state feature in conductance with Fano line shape depending on the strength of the scattering potential. This behavior is due to the different energy of two subband bottoms in the lower spin branch. In addition, an electronlike quasi-bound-state can be found at the subband bottom of the upper spin branch. For the case of the repulsive potential with R–SOI, we have found a clear holelike quasi-bound-state feature at the subband top of the lower spin branch. This holelike quasi-bound-state feature is more significant with longer lifetime for the case of repulsive potential with RD–SOI.
In conclusion, we have investigated the interplay of the Rashba-Dresselhaus sporbit interaction and the in-plane magnetic field induced Zeeman effect to influence the spin-resolved coherent transport. By tuning the fin-ger gate, we have demonstrated how the attractive and repulsive scattering potentials affect the conductance fea-tures. We have analyzed in detail the nontrivial subband and quantum transport properties concerning the SOI– Zeeman induced electronlike and holelike quasi-bound-state features.
ACKNOWLEDGMENTS
This work was supported by the National Science Coun-cil in Taiwan under Grants No.
NSC100-2112-M-239-001-MY3, No. 98-2112-M-009-011-MY2, and No. NSC-100-2112-M-009-013-MY2. We are thankful to the technical support from S.-J. Yu.
*[email protected] †[email protected]
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