Fig.4 shows the electron energy levels in quantum dot without spin-orbit term under a magnetic field. We calculate thatEnlσ = Ω= (2n + +l 1)−=ωBl + σEB, where effective diameter d=20 nm and d=40nm，spin orbit coupling is not considered，at least 12-levels 0,0,↑ , 0,0,↓ , 0,1,↑ , 0,1,↓ , 0, 1,− ↑ , 0, 1,− ↓ ,……. , are included. The quantum dot energy level for d=40 and d=20 ratio is 0.8 with B < 1T.
We calculated H ψe l =ε ψ , the eigenenergy l l of the total electron system . The calculated spectrum of GaAs quantum dot as a function of the magnetic field B with ,
ε
lHe
n= 1± l=0, 1,± andσ=±1.As shown in Figs.5-7 the simulations are presented for the cylindrical quantum dots, for radius of 10nm, 20nm, 40nm, and 60nm, respectively. The GaAs quantum dot is used, thus the effective mass is taken as 0.067m0.
∆E
Fig.8 shows the corresponding energy splitting versus the magnetic field for quantum dot with radius 60nm. We gradually increase the number of basis function when the energy is converged to 0.15% precision. In order to converge the lowest 2
levels, the quantum dot d=60 has to use 12 basis functions.
In Fig.9 The crossing levels occur at 0.45T, 0.67T, 1.1T, 1.8T, and 4.02T for quantum dot diameters d=60nm, d=50nm,d=40nm,d=30nm,d=20nm.
The eigenvalue problem of Eqs (2.2.10), (2.2.11), and (2.2.17) can be solved. A major feature of the confined modes is the quantization of the phonon wave in x and y directions. Figs.10-11 show the dispersion relation for the seven lowest different thickness modes and width modes. The thickness mode ( ) is calculated from the following three equations
h=h =h1 2 numerical approach to solve the three equations in (2.2.19) and (2.2.20). The dispersion relation for thickness phonon, for a rectangular wire with 130nmx260nm 、100nm x 200nm,respectively. We found the phonon mode with thickness with 1 mode and width with 2 mode as E∆ and phonon coupling numbers.
We can calculate the strength of the electric-phonon scattering matrix M(E E . f', i) In Fig.12 We present the
γ
c which is the spin-orbit coupling strengthH ( Dresselhaus coupling )and is the key to understand the spin flip. That is due to so
the fact that the spin-orbit coupling Hso is proportional ( 12 a ).
In Fig13, we investigate the magnetic field dependence of the SRT for different radius of the quantum dot at two different temperatures. For larger temperature, the spin relaxation time becomes much larger due to the stronger electron-phonon scattering and the wider range of energy space the electron occupies. The magnetic field dependence of the spin relaxation time for different quantum dot diameter is shown in Fig.13. It is seen that the SRT decreases rapidly with the magnetic field at each dot size. This is because the magnetic field helps to increase the spin-flip. From the Figure, it can be understood that for largest dot, more energy levels are involved in the spin-flip scattering and hence sharply reduce the SRT.
Fig.14 shows the spin relaxation rate gets larger with the increase of the temperature.
Moreover, the increase of the temperature comes the phonon number nqλ to become larger. This enhances the electron-phonon scattering and leads to a larger transition probability.
In Fig.15 we find the spin relaxation time is decreased with the magnetic field. This can be understood from the fact that the energy splitting E∆ increases with the applied magnetic field B and energy splitting required to couple with the phonon mode is increased.
In Figs. 16(a)(b) The spin relaxation time (SRT) of electrons due to the spin-orbit
coupling induces spin-flip electron-phonon scattering at low temperature, where the dominant electron-phonon scattering arises from the deformation potential. Smaller wire width corresponds to larger spin-orbit coupling, therefore, yields as smaller SRT.
For larger wire width, more subbands are involved and hence increases an opposite tendency of spin flip i.e a shorter SRT with the increase of the wire width and thickness.
Chapter 4 Conclusions
In this work, we obtained the solutions for the spin relaxation time from exact diagonalization of Hamiltonian to explore its dependence on different magnetic field, temperature, quantum dot, and different width and thickness of the quantum wires.
We found that SRT decreases with the applied magnetic field. This can be understood from the fact that the energy splitting ∆ increases with the applied magnetic field B E and more the phonon modes are required to be coupled. Therefore, the electron-phonon scattering probability is larger. The SRT decreases rapidly with the magnetic field at each dot size and temperature. We found that the SRT becomes smaller with the increase of the temperature. The features can be understood by noting that the increase of the temperature will make the phonon numbern to be larger. q This enhances the electron-phonon scattering and leads to the larger transition probability.
As the quantum dot is confined in quantum wire, it is necessary to study the width dependence of quantum wire of the SRT. Smaller wire width and thickness correspond to larger spin-orbit coupling and therefore smaller SRT. For larger wire width, more
subbands are involved and hence increase an opposite tendency for a shorter SRT with the increase of the wire width and thickness.
We found that the SRT decreases with the magnetic field. SRT decreases with the diameter of the quantum dot, but increases with the width (thickness) of the quantum wire. With high temperature, SRT becomes longer due to the stronger electron-phonon scattering and the wide-range of energy level the electron occupies.
Appendix A
The Hamiltonian of an electron in external magnetic field derived from a vector potential can be written as:
2 * 2
with the vector potential
A
, which is a function of the coordinates. Therefore, the Hamiltonian can be expressed as2 2 of the momentum operator with any function of the coordinates, we get
P A ⋅ − ⋅ = − ∇ ⋅ A P i A
(A.1.4)toP A⋅ , then
From the theory of classical electromagnetism, the vector potential corresponding to a uniform magnetic field may be written as
1 (
∂ with the Hamiltonian Eq(A.1.1) is a generalization of equation to the case where a magnetic field is applied. On the other hands, since
1( For a particle having a spin with intrinsic magnetic moment
µ
, the quantummechanical operator is proportional to the spin operator , and can be written as ˆs
ˆ sˆ
µ µ
= (A.1.11) The intrinsic magnetic moment of the particle interacts directly with the magnetic field. The correct expression for the Hamiltonian is obtained by including an extra term ˆµ
⋅ corresponding to the energy of the magnetic moment Bµ
in the field .For an electron in a quantum dot with finite confined potential and an external
magnetic field along z axis, the time-independent equation of the electron is: In cylindrical coordinate, the operators Pand
2
The equation can be solved by setting
ψ = nlσ =R( )ρ ⋅ Φ( )ϕ ξ⋅ ( )z ⋅χ (A.1.17) σ
, where ( )
φ ρ
represents the radial part of the electron wave function inside the quantum dot.We obtain eigenenergy for the quantum dot
Then the equation above can be written as
2
(A.1.22) at the origin and infinity in order to know if they are well behaved. We require the wave function to be finite everywhere. Dividing Eq. (A.2.22) by x, we get
2 2
By setting
And substituting Eq. (A.1.25) into Eq. (A.1.24), we get
2
The coefficients of polynomials should equal to zero to satisfy Eq. (A.1.25), that is, the case as x approaches zero.
(II) As x approaches infinity, Eq.(A.1.23) can be written as will diverge, so we choose
/ 2
Substituting Eq. (A.1.24) into Eq. (A.1.25) and simplifying the resulted equation, we
get
We compare the equation with the confluent hypergeometric function:
X(x) is satisfied by the confluent hypergeometric function, we have
1 , and may be determined by the boundary condition.
Therefore we obtain
where A is the normalized constant. Eq.( A.1.22) can be rewritten as
2/ 4 2 / 2 2
polynomial. Specifically, one usually writes
1 1
( 1)
( ) / ( 1) ( , 1, )
n !
L z n F n z
n
α = Γ
α
+ + Γα
+ ⋅ −α
+ (A.1.40) We obtain
2
2
n! ( )
2 2= = [ ]( ) exp( ) (
( )! 2
il l l
nl n
nl R e r r L r
n l
θ σ
α α
σ χ α α
π −
+ )
(A.1.41)
Appendix B
Definition of stress and strain:
The intensity of the force, the force per unit area, is defined to be the stress. Let the components of ∆F along x, y, z axes be ∆ , Fx ∆ ,Fy ∆ . Stress components are Fz
Normal stress is the intensity of a force perpendicular to a cut curve while the shear stress is parallel to the plane of the element.
There are three normal stressesσ ,x σy,σ where the axis along which is the normal to z
the cut plane. There are also six shear stresses τxy,τyx,τyz,τzy,τ ,zx τ , where the first xz
subscript denotes the axis perpendicular to the plane on which the stress acts and the second provides the direction of the stress. For example, the shear stress τyxacts on a plane normal to the y axis and in a direction parallel to the x axis.
In matrix form, the stress components appear as The matrix of stress is called a stress tensor.
Definition of strain:
The strain can be defined in terms of normal and shear strains. Normal strain is defined as the change in length per unit length of a line segment in the direction under consideration. Shear strain is defined as the tangent of the change in angle of a right angle in a member undergoing deformation.
Ifu, , are three displacement components at a point in a body for the x, y,z directions of coordinate axes in Figure 2, small strains are related to the displacements through the geometric relationships.
Let be the elastic displacement at x along the axis of the one-dimensional structure, and describes the uniform longitudinal displacement of the element dx . In the elastic model the dynamics of phonon cavity , dx, is described in terms of Newton’ laws. It follows from Hooke’s law that
( ) u x
( ) u x
T = ⋅Y S , (B.1.11) where Y is a Young’s modulus. The force equation describing the dynamics of the element dx of density
ρ
( )x is given by Newton’s law;stress and strains independent of variable.
1
[ ] [ ] [ ]
For the zincblende crystals, the stress-strain relation is the most general form, the matrix is of the form (the zincblende crystal have only three independent elastic constants, , ,C ).
⇒
y director : elastic-wave e
z director : where
ρ
is the density of a semiconductor and Tαβis the stress tensor.Then the stress tensor is
= 2
T
αβλ S
αα αβδ + µ S
αβ (B.1.20), where
λ
andµ
are elastic moduli, or Lame’ constant, and δαβ is Kronecker deltaFor an isotropic medium, Eq.(B.1.19) cab be rewritten in vector form as
2
2 2 2 2
t t L
2
= s +(s - s ) grad div( ) t
α α α
µ µ µ
∂ ∇
∂
(B.1.21), 2t 2L 2
where s
µ
and sλ µ
ρ ρ
= = +
are the velocities of the LA and the TA waves in bulk semiconductors
Appendix C.1
The Hamiltonian describing the harmonic oscillator associated with a phonon mode of wavevectorq is displacement associated with it, and p is kinetic momentum. q
Introducing the operators, aq and aq+,and
Appendix C.2
It will be convenient to express the normal-mode phonon displacement
µ
qin terms of the phonon creation and annihilation operators.(
Moreover, each incoming or outgoing phonon will be associated with a unit polarization vector, these unit polarization vectors will be denoted by for incoming waves and by for outgoing waves.
, where is summed over all wavevectors in the Brillouin zone and N is the number of q unit cells in the sample.
The interaction between the quantum dot and the acoustic phonon can be expressed as
(
†)
iq rep q q q
q
H
= ∑
M a+
a e i (C.2.3)For long wavelengths we can treat the one-dimensional chain and the strain becomes a derivative.
The longitudinal strain is
( ) 2 sin( q ) The deformation potential energy can be calculated as through the strain.
( ) 2 sin( ) ( ) This is the form of the perturbing potential caused by the phonon to be used in Fermi’s golden rule.
The piezoelectric interaction occurs in all polar crystal lacking an inversion symmetry. In rectangular coordinates, the polarization created by the piezoelectric interaction in cubic crystals, including zincblende crystals, may be written as
4 4 4
, where is the piezoelectric coupling constant and the factors multiplying are the components of the strain tensor that contribute to the piezoelectric polarization in a zincblende crystal.
4
ex ex4
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B=B z
ρ
zρ
ρ
Fig.1 The cylindrical-shaped quantum dot in an applied magnetic field.
Fig.2 For a cubic medium , volume of cube is ∆ ∆ ∆ x y z
∆ x
∆ y
∆ z
T
xyT
xyT
xxT
xxT
xyFig.3 A rectangular rod of infinite length in z-direction with a height 2a in x-direction and a width 2d in the y-direction.
-a a
b -b d
y z x
0.0 0.5 1.0 1.5 2.0
Fig.4 Energy levels for quantum dot with an effective diameter d=20 nm and d=40nm ，spin orbit coupling is not included，at least 12-levels
0,0,↑ , 0,0,↓ , 0,1,↑ , 0,1,↓ ,………, 1,1, , 1, 1, , 1, 1,↓ − ↑ − ↓
Fig.5 Energy levels for quantum dot with an effective diameter d=10 nm、 d=20nm, respectively with including Dresshauls effective interaction.
Fig.6 Energy levels for quantum dot with an effective diameter d=30 nm 、 d=40nm, respectively with including Dresshauls effective interaction.