2.1. Two-dimensional Electron Gas (2DEG)
By dynamically two-dimensional we mean that the electrons or holes have quan-tized energy levels for one spatial dimension, but are free to move in two spatial dimensions. Thus the wave vector is a good quantum number for two dimensions, but not for the third. These systems are not two-dimensional in a strict sense, both because wave functions have a …nite spatial extent in the third dimension and because electromagnetic …elds are not con…ned to a plane but spill out into the third dimension. Theoretical predictions for idealized two-dimensional systems must therefore be modi…ed before they can be compared with experiment.
Here we shall generally con…ne our discussion to systems for which parameters can be varied in a given sample, usually by application of an electrical stress. Sys-tems of this sort generally occur in what may broadly be called heterostructures.
The best known example are carriers con…ned to the vicinity of junctions between insulators and semiconductors, between layers of di¤erent semiconductors, and between vacuum and liquid helium. For most of these systems the carrier concen-tration can be varied, so that a wealth of information can be obtained from one
21
22
sample. They all have at least one well-de…ned interface which is usually sharp to a nanometer or less.
The e¤ects of changes in surface conditions on the conductance of a semicon-ductor sample have been studied for many years. Such measurements are usually called …eld-e¤ect measurements because a major physical variables is the electric
…eld normal to the semiconductor surface. One important way to change the sur-face condition, and therefore the sursur-face electric …eld, is through the control of gaseous ambients, see for example: Brattain, W. H.-Bardeen, J. cycle experiment in 1953, and Mary, A. experiment in 1974. A disadvantage of the early mea-surements was that the conductance of the entire sample was measured, and the surface e¤ects were extracted by taking di¤erences or derivatives as the ambient was changed. In conjuction with …eld-e¤ect measurements, theories for the depen-dence of the mobility of carriers near the surface on the surface conditions were developed and re…ned. Most of the early work was based on the phenomenological notion of di¤use and specular re‡ection at the surface, as …rst used by Fuchs, K.
in 1938 in studing transport in metal …lms. .
Investigation of space-charge layers on narrow-gap III-V semiconductors also started in the mid-1960s. However, the di¢ culty of obtaining samples with good quality has long prevented progress in this system. After many people e¤ort and many years later, the development on heterostructure growth techniques made it possible to fabricate high-quality-double heterostructures with ultrathin layers.
Two main methods of growth with very precise control of thickness, planarity,
Figure 2.1. Conduction and valence band line-up at a junction be-tween an n-type AlGaAs and intrinsic GaAs (a) before and (b) after charge transfer has taken place. Note that this is a cross-section view. Patterning is done on the surface (x-y plane) using litho-graphic techniques [1].
compositions etc. were developed in the 1970s. A modern molecular-beam epitaxy method became practically important for III-V heterostructure technology due …rst of all to the pioneering work of Cho, A. in 1971. Metal-organic chemical-vaper deposition originated from the early work of Manasevit, H. in 1968 and found broad application in III-V heterostructure research after Dupuis, R. and Dapkus, P. in 1977 reported the room-temperature injection of AlGaAs DH lasers which had been grown by the metal-organic chemical-vaper deposition method. Here we do not discuss the progress of the techniques development more forward, we turn our attention to the formation aspects of 2DEG in GaAs-AlGaAs heterojunctions.
Recent work on mesoscopic conductors has largely been based on GaAs-AlGaAs heterojunctions where a thin two-dimensional conducting layer is formed at the interface between GaAs and AlGaAs. To understand why this layer is formed
24
consider the conduction and valence band line-up in the z-direction when …rst bring the layers in contact Fig. 2.1 (a). The Fermi energy EF in the widegap AlGaAs layer is higher than that in the narrowgap GaAs layer. Consequently electrons spill over from the n-AlGaAs leaving behind positively charged donors.
This space charge gives rise to an electrostatic potential that causes the bands to bend as shown. At equilibrium the Fermi energy is constant everywhere. The electron density is sharply peaked near the GaAs-AlGaAs interface (where the Fermi energy is inside the conduction band) forming a thin conducting layer which is usually refered to as the two-dimensional electron gas Fig. 2.1 (b). The carrier concentration in a 2DEG typically ranges from 2:0 1011=cm2to 2:0 1012=cm2and can be depleted by applying a negative voltage to a metallic gate deposited on the surface. The practical importance of this structure lies in its use as a …eld e¤ect transistor which goes under a variety of names such as MODFET (Modulation Doped Field E¤ect Transistor) or HEMT (High Electron Mobility Transistor).
Note that this structure is similar to standard silicon MOSFETs, where the 2DEG is formed in silicon instead of GaAs. The role of the wide-gap AlGaAs is played by a thermally grown oxide layer (SiOx). Indeed much of the pioneering work on the properties of two-dimensional conductors was performed using silicon MOSFETs.
Except for the space-charge layers investigation in Si and narrow-gap III-V semiconductors, there are many other systems to be investigated, like two-dimensional electron crystal, InSb, InAs, InP, Hg1 xCdxTe systems and Ge, Te,
PbTe, ZnO systems. And the investigated systems also include heterojunctions, quantum wells, superlattices, thin …lm, and layer compounds, for example, GaSe and related materials and TaSe2 and related materials, and graphite and inter-calated graphite, and even for electron-hole system. And the electrons on liquid helium also constitute a special kind of quasi-two-dimensional space-charge layer.
And the magnetic-…eld-induced surface states in metals also be consider in point of view of inversion two-dimensional layer [1].
2.2. Proposed Sample Preparation
In this thesis, we explore mainly the spin relaxation in closed quantum sys-tems and the spin transport in open quantum syssys-tems. Here we just mention the minimal description to give a contour to understand our proposed sample for exploring.
Open quantum system
Figure 2.2 shows the layer structure of the inverted In0:53Ga0:47As/In0:52Al0:48As modulation doped structure. The heterostructure proposed to be used in this thesis was grown by molecular beam epitaxy (MBE) on a Fe-doped semi-insulating (100) InP substrate. All InGaAs and InAlAs layers were lattice matched to InP.
The doping density of the 7-nm-thick In0:52Al0:48As carrier supply layer which is underneath a 2DEG channel was 4:0 1018cm 3. The 2DEG channel was formed in an undoped In0:53Ga0:47As channel layer of 20-nm thickness. The channel layer is separated by an undoped In0:52Al0:48As spacer layer of 6-nm thickness to reduce
26
Figure 2.2. Schematic layer structure of an inverted In0.53Ga0.47/In0.52Al0.48As heterostructure and pro…le of the Datta-Das-like SFET.
ionized donor scattering. Then we can apply MBE technique to grow an about 300-nm-thick Ga1 xMnxAs layer with x 0:045. GaAs and (In, Ga) As layers were grown at T 540 580oC, while the (Ga, Mn) As layer was grown at T 250oC.
The 4:5% Mn concentration is determined from the lattice constant measured by X-ray di¤raction, and is expected to yield Curie temperature Tc in the range of 40 90K with a hole concentration p 1020cm 3. The easy axis of the (Ga, Mn) As magnetization is in the plane of the sample, veri…ed by a superconducting quantum interference device (SQUID) magnetometer. Then we may use electron beam (EB) lithography, Lift o¤ technique and Ar sputter etching to shape the desired 2DEG structure, for example, various sized mesoscopic rings, dots, and so on [27], and reveal the electrical spin injection source and spin detection drain electrodes. Next we grow about a 100-nm-thick SiO2 insulating layer which covers the shaped 2DEG structure, …nally the gate electrode was made on the top of the
Figure 2.3. Schematic layer structure of an inverted In0.53Ga0.47/In0.52Al0.48As heterostructure and pro…le of the quantum dot for simulation.
source and drain and the SiO2 insulating layer [28]. Here we see a Datta-Das-like spin …eld-e¤ect transistor (SFET) in Fig. 1.1 as described in Section Spintronics.
Closed quantum system
For the closed Quantum system, the procedure of the prepararion of sample is almost the same to the description of open quantum system, the only di¤erences are we need not fabricate the source and drain electrodes, and we just imagine that polarized electrons have existed before the simulation. Figure 2.3 shows the layer structure and pro…le of one of the simulation sample.
CHAPTER 3
Spin
3.1. Lead-in
Spin is a fundamental property of all elementary particle [29]. In classical me-chanics, a rigid object admits two kinds of angular momentum: orbital (L = r p), associated with the motion of the center of mass, and spin (S =I!), associated with motion about the center of mass. We have an example, the earth has orbital angular momentum attributable to its annual revolution around the sun, and spin angular momentum coming from its daily rotation about the north-south axis. We
…nd that in the classical context this distinction is largely a matter of convenience, for when you come right down to it, S is nothing but the sum total of the "or-bital" angular momenta of all the rocks and dirt clods that go to make up the earth, as they circle around the axis. But an analogous thing happens in quantum mechanics, and here the distinction is absolutely fundamental. In addition to or-bital angular momentum, associated (in the case of hydrogen) with the motion of the electron around the nucleus (and described by the spherical harmonics), the electron also carries another form of angular momentum, which has nothing to do with motion in space (and which is not, therefore, described by any function of the position variable r; ; ) but which is somewhat analogous to classical spin (and
28
for which, therefore, we use the same word). It doesn’t pay to press this analogy too far: The electron (as far as we know) is a structureless point particle, and its spin angular mementum cannot be decomposed into orbital angular momenta of constituent parts. Su¢ ce it to say that elementary particles carry intrinsic angular momentum (S) in addition to their "extrinsic" angular momentum (L) [30].
We also could …nd one of contrary interpretations about spin described by Ohanian, H. C. [31]. The point of view of Ohanian’s paper is stated below. The lack of a concrete picture of the spin leaves a grierous gap in our understanging of quantum mechanics. The prevailing acquiescence to this unsatisfactory situation becomes all the more puzzling when one realizes that the means for …lling the gap have been at hand since 1939, when Belinfante established that the spin could be regarded as due to a circulating ‡ow of energy, or a momentum density, in the electron wave …eld. He established that this picture of the spin is valid not only for electrons, but also for photons, vector mesons, and gravitons –in all cases the spin angular momentum is due to a circulating energy ‡ow in the …elds. Thus contrary to the common prejudice, the spin of the electron has a close classical analogy: It is an angular momentum of exactly the same kind as carried by the
…elds of a circularly polarized electromagnetic wave. Furthermore, according to a result established by Gordon in 1928, the magnetic moment of the electron is due to the circulating ‡ow of charge in the electron wave …eld. This means that neither the spin nor the magnetic moment are internal properties of the electron – they have nothing to do with the internal structure of the electron, but only with the
30
structure of its wave …eld. Here one thing must be emphasized that, in contrast to some other attempts at explaining the spin [32], the present explanation is completely consistent with the standard interpretation of quantum mechanics.
The algebraic theory of spin is a carbon copy of the theory of orbital angular mementum, beginning with the fundamental commutation relations [30]:
(3.1) [Sx; Sy] = i~Sz; [Sy; Sz] = i~Sx; [Sz; Sx] = i~Sy:
It follows that the eigenvectors of S2 and Sz satisfy
(3.2) S2 j smi = ~2s (s + 1)j smi; Sz j smi = ~m j smi;
and
(3.3) S j smi = ~p
s(s + 1) m(m 1)j s(m 1i;
where S Sx iSy. But here the eigenvectors are not spherical harmonics (they’re not functions of and at all), and there is no a priori reason to exclude the half-integer values of s and m:
(3.4) s = 0;1 2; 1;3
2; :::; m = s; s + 1; :::; s 1; s:
It so happens that every elementary particle has a speci…c and immutable value of s, which we call the spin of that particular species: pi mesons have spin 0;
electrons have spin 1=2; photons have spin 1; deltas have spin 3=2; gravitons have spin 2; and so on. By contrast, the orbital angular momentum quantum number l (for an electron in a hydrogen atom, say) can take on any (integer) value we please, and will change from one to another when the system is perturbed. But s is …xed, for any given particle, and this makes the theory of spin comparatively simple.
3.2. Spin 1/2
Here the most important case is s = 1=2, for this is the spin of the particles that make up ordinary matter (protons, neutron, and electrons), as well as all quarks and all leptons. Moreover, once we understand spin 1=2, it is a simple matter to work out the formalism for any higher spin. There are just two eigenstates: j 12
1 2i (or denoted as j +i ), which we call spin up (informally, "), and j 12( 12)i (or denoted as j i ), which we call spin down (informally, #). Using these as basis vectors, the general state of a spin-1=2 particle can be expressed as a two-element column matrix (or spinor ):
32
representing spin up and for spin down.
Meanwhile, the spin operators become 2 2 matrices, which we can work out by noting their e¤ect on + and ;Equation 3.2 says
(3.7) S2 += 3
(3.9) Sx = 1
34
Figure 3.1. n is a unit vector lies in 3D real space.
These are the famous Pauli spin matrices. Notice that Sx, Sy, Sz and S2 are all Hermitian (as they should be, since they represent observables). On the other hand, S+ and S are not hermitian –evidently they are not observable.
If we try to construct j S n; +i where it is one of the eigenstates which measured in the direction parallel to n , such that
(3.14) S nj S n; +i = ~
2 j S n; +i
where n is the unit vector and characterized by the angle shown in the Figure 3.1.
By applying the eigenvalue and eigenstate idea and skill, we can get
(3.15) j S n; +i = cos
2 j +i + sin
2 ei j i
Well! This equation can be used as the initial spinor input for calculation and simulation in this thesis.
3.3. Electron in A Magnetic Field
A spinning charged particle constitutes a magnetic dipole. Its magnetic dipole moment is proportional to its spin angular momentum S:
(3.16) = S;
the proportionality constant is called the gyromagnetic ratio. (Classically, the gyromagnetic ratio of a rigid objects is q=2m, where q is its charge and m is its mass. For reasons that are fully explained only in relativistic quantum theory, the gyromagnetic ratio of the electron is almost exactly twice the classical value [33].) When a magnetic dipole is placed in a magnetic …eld B, it experiences a torque, B, which tends to line it up parallel to the …eld (just like a compass needle).
The energy associated with this torque is [33]
(3.17) H = B;
so the Hamiltonian of a spinning changed particle, at least in a magnetic …eld B (If the particle is allowed to move, there will also be kinetic energy to consider;
36
Figure 3.2. Magnetic …eld sweeps around on a cone, at angular ve-locity !, Eq. 3.19.
moreover, it will be subject to the Lorentz force (qv B), which is not derivable from a potential energy function and hence does not …t the Schrödinger equation as we have formulated it so far. Anyhow, for the moment let’s just assume that the particle is free to rotate, but otherwise stationary.), becomes
(3.18) H = B S;
where S is the appropriate spin matrix (Eq. 3.12, in the case of spin 1=2).
Now let us see two cases which relevant to our thesis.
Cses 1
Imagine an electron (charge e, mass m) at rest at the origin in the presence of a magnetic …eld whose magnitude (B0) is constant but whose direction sweeps out a cone, of opening angle , at constant angular velocity !, Fig. 3.2.
The magnetic …eld is
(3.19) B(t) = B0[sin cos(!t)i + sin sin(!t)j + cos k] : The normalized eigenspinors of H(t) are
(3.22) +(t) =
38
they represent spin up and spin down, respectively, along the instantaneous direction of B(t). The corresponding eigenvalues are
(3.24) E = ~!1
2 :
Suppose the electron starts out with spin up, along B(0):
(3.25) (0) =
The exact solution to the time-dependent Schrödinger equation is
(3.26) (t) =
or, writing it as a linear combination of + and ;
Now if we assume an spin evolution operator Ss which with the relationship
(3.29) (t) = Ss (0); of someone extreme situation (i.e. the trajectory of someone particle (electron) exhibited as a smooth curve).
Case 2
40
An electron is at rest at the origin, in the presence of a magnetic …eld whose magnitude (B0) is constant but whose direction rides around at constant angular velocity ! on the lip of a case of opening angle as Case 1, Fig. 3.2:
(3.31) B(t) = B0[sin cos(!t)i + sin sin(!t)j + cos k] ;
where it is the same to Eq. 3.19.
Then assuming the particle starts out with spin up (says in z direction), we can …nd its exact solution to the time-dependent Schrödinger equation is
(3.32) (t) =
are also the same to Equations 3.21 and 3.27 [30].
As the Case 1, if we assume an spin evolution operator Ss which with the relationship (t) = Ss (0); we could get the similar equation as Eq. 3.30.
3.4. Time Evolution and Spin Rotation
Suppose we have a physical system whose state ket at t0 is represented by j ; t0i. At later times, we do not, in general, expect the system to remain in the same state j ; t0i. Let us denote the ket corresponding to the state at some later time by
(3.35) j ; t0; ti, t > t0:
The two kets are related by an operator which we call the time-evolution op-erator U (t; t0);
(3.36) j ; t0; ti = U(t; t0)j ; t0i:
Due to the unitary requirement and the composition property and borrow from classical mechanics idea that the Hamiltonian is the generator of time evolution [34], we can …nd out the in…nitesimal time-evolution operator is written as
42
(3.37) U (t0+ dt; t0) = 1 iHdt
~ ;
where H, the Hamiltonian operator, is assumed to be Hermitian, and then we exploit the composition property of the time-revolution operator, we could get
(3.38) i~@
@tU ( t; t0) = HU ( t; t0):
This is the Schrödinger equation for the time-revolution operator. If the Hamil-tonian operator is independent of time. By this we mean that even when the pa-rameter t is changed; the H operator remains unchanged. The Hamiltonian for a spin-magnetic moment interacting with a time-independent magnetic …eld is an example of this.
In the Schrödinger picture the operators corresponding to observables like x, py, and Sz are …xed in time, while state kets vary with time. By solving Eq. 3.38, we get
(3.39) U (t; t0) = exp iH (t t0)
~ :
In contrast, in the Heisenberg picture the operators corresponding to observ-ables vary with time; the state kets are …xed, frozen so to speak, at what they
were at to. It is convenient to set in U (t; t0) to zero for simplicity and work with U (t), which is de…ned by [35]
(3.40) U (t; t0 = 0) U (t) = exp iH (t)
~
Now let us talk about the spin rotation. Following the exploration, we think that because rotations a¤ect physical systems, the state ket corresponding to a rotated is expected to look di¤erent from the state ket corresponding to the original unrotated system. Given a rotation operator R, characterized by a 3 3 orthogonal matrix R, we associate an operator D (R) in the appropriate ket space such that
(3.41) j iR= D (R)j i
where j iR and j i stand for the kets of the rotated and original system, respectively. Note that the 3 3 orthogonal matrix R acts on a column matrix made up of the three components of a classical vector, while the operator D (R)
where j iR and j i stand for the kets of the rotated and original system, respectively. Note that the 3 3 orthogonal matrix R acts on a column matrix made up of the three components of a classical vector, while the operator D (R)