Introduction to semiconductors

The term “semiconductor” represents a certain class of solid materials. It suggests that the electrical conductivity is intermediate in magnitude between a conductor and an insulator. Semiconductor materials are numerous and versatile. We can distinguish it into elementary semiconductors and compound semiconductors.

Elementary semiconductors are Silicon (Si) and germanium (Ge), phosphorous (P), sulfur (S), selenium (Se), and tellurium (Te). Compound semiconductors are categorized following by the group of their constituents in the periodic table of elements. Such as gallium arsenide (GaAs), aluminium arsenide (AlAs), indium arsenide (InAs), indium antimonide (InSb), gallium antimonide (GaSb), gallium phosphide (GaP), gallium nitride (GaN), aluminium antimonide (AlSb), and indium phosphide (InP) are all belong to the so-called III-V semiconductors. There are also II-VI semiconductors, such as zinc sulfide (ZnS), zinc selenide (ZnSe) and cadmium telluride (CdTe), III-VI compounds, such as gallium sulfide (GaS) and indium selenide (InSe), as well as IV-VI compounds, such as lead sulfide (PbS), lead telluride (PbTe), lead selenide (PbSe), germanium telluride (GeTe), tin selenide (SnSe), and tin telluride (SnTe).

For compound semiconductors, there are two chemical constituents are called binary compounds. Additionally, there are compound semiconductors with three constituents, such as AlxGa1−xAs (aluminium gallium arsenide), InxGa1−xAs (indium gallium arsenide), and also InxGa1−xP (indium gallium phosphide). In this situation, it is called about ternary semiconductors or semiconductor alloys.

( , , ) _{x y z}

H x y z =H +H +H (1.1.1)

1.2 Low dimensional semiconductor systems

1.2.1 Introduction to heterostructure semiconductors

For heterostructure, since the two different materials will have two different energy bandgaps, the energy band will have a discontinuity at the junction interface.

We may have an abrupt junction in which the semiconductor changes abruptly from a narrow bandgap material to a wide-band gap material. In Fig. 1.2.1.1 shows the energy-band diagram of a GaAs-AlGaAs heterojunction in thermal equilibrium. The

AlGaAs is moderately to heavily doped n type, while the GaAs is more lightly doped or even intrinsic. In order to achieve thermal equilibrium, electrons flow from the wide-bandgap AlGaAs into the GaAs, forming an accumulation layer of electrons in the potential well adjacent to the interface. The electrons contained in a potential well are quantized. The two-dimensional electron gas refers to the condition in which the electrons have quantized energy levels in one spatial direction (perpendicular to the interface), but are free to move in the other two spatial directions.

Fig. 1.2.1.1. The bandage profile of semiconductor heterostructures.

Since the GaAs is lightly doped or intrinsic, the two-dimensional electron gas is in a region of low impurity doping so that impurity scattering effects are minimized.

The electron mobility will be much larger than if the electrons were in the same region with the ionized donors. The movement of the electrons parallel to the interface will still be influenced by the coulomb attraction of the ionized impurities in the AlGaAs. The effect of these forces can be further reduced by using a graded AlGaAs-GaAs heterojunction. The graded layer is AlxGa1-xAs in which the fraction x varies with distance. In this situation, an intrinsic layer of graded AlGaAs can be sandwiched between the N-type AlGaAs and the intrinsic GaAs. Fig. 1.2.1.2 shows the conduction-band edge across a graded AlGaAs-GaAs heterojunction in thermal equilibrium. The electrons in the potential well are further separated from the ionized impurities so that the electron mobility is increased above that in an abrupt heterojunction.

Fig. 1.2.1.2. the conduction-band edge across a graded AlGaAs-GaAs heterojunction in thermal equilibrium.

The two-dimensional electron gas (2DEG) trapped at a doped heterostructure is the most important low-dimensional system for electronic transport. It forms the kernel of a field-effect transistor. The high electron mobility transistor has many acronyms including modulation-doped field-effect transistor (MODFET) and high electron mobility transistor (HEMT).

1.2.2 Modeling the low dimensional semiconductor systems

Fig. 1.2.2.1 is the GaAs/AlGaAs high electron mobility transistor. The cap layer in the transistor can prevent the n-type AlGaAs from oxidizing. Above the cap layer, we use two metal gates to define a quasi-one dimensional quantum channel. The Hamiltonian of a semiconductor with heterostructure can be written separately in the vertical and lateral parts of form

( , , ) _z

Vc(z) is the quantum well at the interface of the heterostructure. The electrons underneath the gate oxide are confined to the heterostructure interface, and thus occupy well defined energy levels. Nearly always, only the lowest level is occupied, and so the motion of the electrons perpendicular to the interface can be ignored. While, the electron can be free to move in the other two spatial directions. Hence, we can ignore the z-part Hamiltonian and emphasize the x, y dependant Hamiltonian (Eq.

1.2.2.2).

Fig. 1.2.2.1. The GaAs/AlGaAs high electron mobility transistor.

1.3 Quantum transport in quasi-one-dimensional quantum systems 1.3.1 Introduction to quantum transport

In macroscopic systems, the conductance obeys an ohmic scaling law:

G W L

=σ . (1.3.1.1)

As the dimensions become smaller, there are two corrections to this law. Firstly there is an interface resistance independent of the length L of the sample. Secondly the conductance does not decrease linearly with the width W anymore. Instead it depends on the number of transverse modes in the quantum channel. The Landauer-Buttiker formula incorporates both of these features [1, 2]:

2e2

G NT

= h . (1.3.1.2)

The factor T is the average probability that an electron incident from the source will transmit to the drain, the factor 2 is for the spin and N is the number of propagating modes with positive group velocity due to transverse confinement. The Landauer-Buttiker formalism only applies to coherent transport. In this paper, we assume that the phase-coherent length is larger than the sample of linear size L, in which l_φ > and the elastic mean free path is larger than the sample size L l_e> . L Namely, our system is in the coherent quantum transport regime.

1.3.2 Quasi-one-dimensional quantum systems

To form a quasi-one-dimensional quantum system (Fig. 1.2.2.1), we use two split top gates above the HEMT. We can rewrite the Eq. 1.2.2.2 in the following form:

( )

Since the two split top gates are quite near each another, electrons will be confined in the quantum channel and can only propagate along the x direction. Hence,the single particle Hamiltonian in the narrow channel can be described by.

2 2

* ( , ) 2

H k V x y

= m + . (1.3.2.2)

This Hamiltonian can be separated into two parts:

2 2

VC(y) indicates the confining potential in the transverse direction. The corresponding eigenvalue of Hy is the sub-band energy. In the narrow channel, the electron propagates along x direction whose kinetic energy will be the total energy of an incident electron subtracting the subband energy Ek = Etot - εn, εn depends on which subband the electron occupying. V(x) exhibits the x dependant potential which can be the spin orbit interaction or the scattering potential in longitudinal direction. In this chapter, we consider the system is only with the static scattering potential along x direction without spin orbit interaction. In the following chapters, we will discuss the spin-resolved transport properties including both the static scattering potential and spin orbit interaction.

Fig. 1.3.2.1. System figuration.

1.3.3 Analytical approach

The system figuration is shown in Fig. 1.3.2.1. A static finger gate is in the middle of the narrow channel. The system under investigation can be described by the Hamiltonian:

In order to simplify the calculation, the dimensionless Hamiltonian is introduced by choosing appropriate physical units: the length unit 1

* Following performing standard dimensionless the Hamiltonian becomes:

2 2 2

The wave function can factorize into functions of x and y, as follows:

( )

^{ψ( ) ( )x ϕ y}

Ψ r = . (1.3.3.3)

Since the confining potential in the transverse direction is a parabolic potential, the wavefunction and the subband energy will be

(2 1)

The electrons incident from the left source will be scattered by the static delta potential in the middle of the quantum channel. The electrons may be back scattered or forward scattered. Therefore, the x-part wave functions can be written in the form:

0, ( ) ^{ikx ikx}

x< ψ x =e +re⁻ (1.3.3.6)

and

0, ( ) ^ikx, _n

x> ψ x =te k= E−ε . (1.3.3.7)

r, t represent the reflected and transmitted coefficients. E is the total energy of the electron and εn is the subband energy. The wavefunctions should satisfy these boundary conditions:

(i) (ψ x=0 )⁻ =ψ(x=0 )⁺ (1.3.3.8) and

(ii)ψ′(x=0 )⁻ =ψ′(x=0 )⁺ −V₀ψ(x=0 )⁺ . (1.3.3.9) Substituting the x-part wave functions into these boundary conditions can obtain:

1

r t= − (1.3.3.10)

and

(1 ) 0

ik − =r ikt V t− . (1.3.3.11) Combining these two equations and using linear algebra, the transmitted coefficient can be expressed as:

Once obtaining the transmitted coefficient, we can substitute it into the Landauer-Büttiker equation and acquire the conductance.

2 2

1.3.4 Numerical approach

In this section, we show the numerical results and discussion of the variation of conductance with the potential strength V0. The numerical calculations presented below are carried out under the assumption that the electron effective mass m^* = 0.067m0, which is appropriate to the GaAs-based semiconductors. The typical electron density is n ~ 10¹¹ cm^-2. Accordingly, the energy unit E^* = 9 meV , the length unit L^* = 7.96 nm, and the frequency unit ω^*= E^* =13.6 THz[3].

In Fig. 1.3.4.1, we demonstrate the conductance at different scattering potential strength and the frequency remaining at ωy = 13.6 THz. In the absence of scattering potential, the conductance is ideally quantized. The conductance regularly increases 2e²/h as the energy raises 2EF, since the transverse modes will increase one mode whenever the energy raises 2EF and we need to take account of another subband (the subband energy level spacing is 2EF.). As the magnitude of the scattering potential increases, the electrons may be reflected by the scattering potential and successfully transmitted. Then, the conductance can not transmit completely anymore. When the scattering potential strength changes into stronger, the probability for electrons to transmit is more difficult therefore the conductance is significantly suppressed and the degree of suppression will increase with the stronger of magnitude of the scattering potential.

(a) (b)

Fig. 1.3.4.1. Conductance (in units of 2e²/h) versus kinetic energy in a quantum channel with tunable potential strength V0 (a) The potential is repulsive (b) The potential is attractive. The Fermi energy EF = 9 meV

Chapter 2 Spin-resolved quantum transport

2.1 Introduction to spintronics

In the recent years, there has been growing interest in the emerging field of spin electronics or “spintronics”. Spintronics, where the spin of electrons is used to carry information, is a rapidly growing area of research [4−6]. There are several techniques for generating pure spin currents [7–9]; Spintronics involves exploration of the extra degrees of freedom provided by the electron spin, in addition to those due to electron charge, with a new view to realize the new functionalities in future electronic devices.

Spin-orbit interaction (SOI) is considered as an efficient manipulation via gate voltages, which is a relativistic effect that couples the electron spin, momentum, and electric field (or momentum dependant effective magnetic field in the electron frame.) The SOI has been utilized to devise various spintronics devices such as spin transistors, spin logic, and spin filters [10−13].

In 1990, Datta and Das proposed to control the strength of Rashba spin-orbit interaction using gate voltage as a spin-field transistor based on spin rotation, which can be a significant strong effect in narrow gap semiconductor heterostructures [14].

The gate control of the spin current employing the Aronov-Casher effect was considered. The electric dipole spin resonance controlled by the time-dependant gate was also studied. Furthermore, spin-orbit interaction is likely to be important in Einstein-Podolsky-Rosen type spin-dependant entangled electronic states for quantum information processing [15, 16]. Considering semiconductor systems, there are two main types of spin-orbit interaction. The Dresselhaus spin-orbit interaction [17]

appears due to the asymmetry present in certain crystal lattices.

The Rashba spin-orbit interaction [18] arises due to the asymmetry associated with the confining potential of the heterostructure quantum well. The perpendicular electric fields inside heterostructure quantum wells are important for understanding spin-orbit coupling, which is sample-specific and adjustable. In narrow gap semiconducting quantum wells, a variation of about 50% of the spin-orbit coupling coefficient was observed experimentally by adjusting the voltage on adjacent gate electrodes, in which a quantum well is populated only by donor-layer electrons.

Consequently, much interest has been attracted to the realization of spin polarized transistors, spin filter devices, and other devices based on electrical gate control to the spin-dependant transport.

2.2 The spin-orbit interactions and the Zeeman effect

To realize a spin device, it is important to utilize the spin-orbit interaction since it provides a way of controlling the spin degree of freedom electrically in semiconductor-based systems. Moreover, for a quasi-one-dimensional ballistic it is found that the SOI could significantly modify the band structure, thus additional subband extrema and energy gaps are produced. Effects of SOI and Zeeman splitting on the physical properties of quantum wires, e.g., photovoltaic effect [19] and shot noise [20] have been investigated in detail. Li et al [21]. have presented that the SOI and the Zeeman effect could result in significant variations of the conductance and the thermopower which are spin-dependent.

We will consider the transport properties in the presence of the SOI and the in-plane magnetic field. The spin-orbit interaction can be caused by structural inversion asymmetry (SIA), which can be artificially controlled by the applied gate voltages or by the specific design of the heterostructure, or by bulk inversion asymmetry (BIA), which is determined by the semiconductor material and the geometry of the sample. Both HBIA and HSIA lead to spin splitting of the conduction band linear in k. The in-plane magnetic field will cause the energy splitting that is independent of k.

The structural inversion asymmetry results in the Rashba spin-orbit interaction.

The Rashba spin-orbit interaction depends on the gradient of the potential and is therefore more important the higher the nuclear charge of the element. In Ch. 1.2.1, we have mentioned that the electrons are confined at the heterostructure interface. For the purpose of confining electrons to nanostructure devices, potential well is necessary.

The potential well at the interface results in the non-negligible Rashba spin–orbit interaction (SOI), especially in systems with structural inversion asymmetry (SIA) like e.g. semiconductor heterostructures. Heavy elements in the periodic table show stronger effects. This is also valid in crystals. For instance, in silicon the spin-orbit interaction is much weaker than in Ge or GaAs. It is even more important in InAs and InSb. In a two-dimensional electron gas (2DEG) obtained by a strong confinement in the z-direction (Fig. 1.3.3.1), the Rashba SOI is described by the Rashba term

( )

R y x x y z

H =α pσ −pσ . (2.2.1)

The components of the electron momentum operator are denoted by pi, the Pauli matrices are represented by σi, and α proportional to Ez is the SOI coupling coefficient set by the confining electric field or by the applied gate voltage.

In III-V or II-VI the heterostructure semiconductors, such as, the difference between cations and anions breaks the degeneracy of the band structure with respect to the spin degree of freedom, and is present in both bulk materials and semiconductor nanostructures. The electric fields resulting from the lack of an inversion centre lead to bulk inversion asymmetry (BIA) and to the Dresselhaus term in the Hamiltonian. In the conduction band, the spin splitting Hamiltonian is given by

(

^{2 2}

) (

^{2 2}

) (

^{2 2}

)

,

bulk D c x x y z y y z x z z x y

H =γ σ^⎡⎣ k k −k +σ k k −k +σ k k −k ^⎤⎦ . (2.2.2) To obtain the effective Hamiltonian of the two-dimensional quantum channel, we take the average of the above bulk Hamiltonian with respect to the ground state wave function along the vertical z direction.

( )

D x x y y

H = β pσ −p σ . (2.2.3)

The components of the electron momentum operator are denoted by pi, the Pauli matrices are represented by σi, and β is the Dresselhaus spin-orbit interaction strength.

An external magnetic field lifts time inversion symmetry so that we can obtain a finite Zeeman energy splitting ΔEZ = g^∗μBB, where g^∗ is the effective g factor and μB

the Bohr magneton of the electron or hole states. It was first shown by Roth et al. [22]

that electrons can have an effective g factor g^∗ that differs substantially from the free-electron value g0 = 2. The effective g factor g^∗ ≠ 2 results from the spin–orbit interaction, which couples the orbital motion with the spin degree of freedom.

Because of without SOI, the motion of spin-up electrons would be completely decoupled from the motion of spin-down electrons, and there would be identical Hamiltonians for spin-up and spin-down electrons except for the trivial Zeeman term

±(g0/2)μBB, so that in this case Zeeman splitting would be controlled by the g factor, in which g0 = 2 of free electrons. Recently, calculations and experiments have shown that g^∗ can have different values for B applied in the direction normal to the plane of the 2D system and for B in the plane of the quantum wire [23−26].

In Ch3 and Ch4, we will analyze the transport properties in a quantum channel in the presence of the spin-orbit interactions and in-plane magnetic field.

Chapter 3 Quantum transport in the presence of the Rashba spin-orbit interaction with in-plane magnetic field

In this chapter, we will use the analytical approach to investigate how the Rashba spin-orbit interaction and an in-plane magnetic field affect the electron transport. We will introduce the system Hamiltonian and analyze the energy spectrum and the wavefunction in the first section. In the second section we will use the Landauer-Buttiker formula by the matching method to calculate the conductance. At last, we will demonstrate the numerical results under different strengths of the Rashba spin-orbit interaction, the magnetic field and the gate voltage.

3.1 Theory

In this section, we use the numerical approach to calculate the energy spectrum and the spinor states of the system considering both the Rashba and the Dresselhaus spin-orbit coupling and an in-plane magnetic field.

3.1.1 System and Formulation

In this paragraph, we use the analytical approach to derive the energy spectrum and the spinor states of the system considering the Rashba spin-orbit coupling and an in-plane magnetic field [27].

We use a transverse hard wall potential to simulate the confinement potential along y direction. The transverse potential is a narrow constriction therefore we can neglect the momentum py along y direction. Then, the Rashba term can be reduced from Eq.(2.2.1) to

R x y

H = −α σp . (3.1.1.1)

The Hamiltonian for the quantum channel in the presence of the Rashba spin-orbit interaction and the Zeeman effect which is due to an applied magnetic field along x direction has the form

where α is the Rashba strength, B is the magnetic field strength and Vc is the confining potential. In the middle of the quantum channel there is a scattering potential in forms of delta potential. Then the total single particle Hamiltonian is

0 _s( )

H =H +V x , V x_s( )=V0δ

( )

x . (3.1.1.3)

x

For convenience, we choose the following units: length unit ^* 1

F

≡ μ , the Rashba coefficient unit

2

g≡ g . In the following way, we can obtain the dimensionless unperturbed Hamiltonian:

2

0 2 _{x y x} ( )

H =k − α σk +gBσ +V y . (3.1.1.4)

Separating the unperturbed Hamiltonian into the x-dependant and y-dependant parts can get:

is a potential that confines the electron in the transverse direction and we suppose that the confining potential with only the lowest occupied subband.

The wavefunction of the unperturbed Hamiltonian can be expanded by the spatial wavefunction and spinor state,

( , )x y φ_n( )y e^{ik xx} χ

Ψ = . (3.1.1.9)

Since the transverse confinement potential is the hard wall potential, the transverse wavefunction will be

and the subband energy will be

2

Here, we only consider the lowest occupied subband. That is n is equal to 1. Then, substituting the transverse wavefunction and the subband energy into Eq. (3.1.1.4) and Eq. (3.1.1.9) obtain

( 2− α σk_{x y}+gBσ χ_x) =(E− −ε_n k_x2)χ. (3.1.1.12) Expanding the above equation by the Pauli matrices:

0 2 2

The spinor state and the eigen-energy can be obtained by solving the above eigenvalue problem. The spinor state is

( )

For an ideal wire without scattering potential, it is convenient to use Eq. (3.1.1.16) to obtain energy spectrum as a function of wave vector for propagating modes, as shown

in Secs. 3.1.2 and 3.1.3.

In general, there are four extreme values in the energy dispersion. For convenience, we define Pbσ = (kbσ, Ebσ) and Ptσ = (ktσ, Etσ) to denote the extreme values of the energy dispersion at the subband bottom (b) and subband top (t), respectively. We also define ΔE^Z to represent the pseudo-gap or the branch level

In general, there are four extreme values in the energy dispersion. For convenience, we define Pbσ = (kbσ, Ebσ) and Ptσ = (ktσ, Etσ) to denote the extreme values of the energy dispersion at the subband bottom (b) and subband top (t), respectively. We also define ΔE^Z to represent the pseudo-gap or the branch level

在文檔中 在共平面磁場中與Rashba 及Dresselhaus 自旋軌道交互作用之窄通道的量子傳輸 (頁 16-0)