ivchenko_bookreg

2.2 Electron States in Quantum Wires and Nanotubes

41

It turns out that, for ν = 0 when na + nb is a multiple of 3, a carbon nanotube is gapless (metallic, or semimetallic) since the valence and conduction subbands with n = 0 still touch each other: Ec,v (0, kz ; K) = ±γ|kz |. For ν = ±1, a nanotube is semiconducting with the band gap ∆ = (4πγ/3L). In the following we focus on the nanotubes characterized by a finite band gap and assume ν = 0.

According to (2.77), for small values of kz satisfying the condition |kz | |k | in the K valley and similar condition in the K valley, the electron spectrum has a parabolic form with terms linear in kz

Ec,v (n, kz ; K) = ±

∆

2k2

+ βnkz

,

(2.82)

2

+ 2mn

n

z

Ec,v (n, kz ; K ) = ±

2

+

2mn

+ βn

kz

,

∆

2k 2

n

z

where

= 2γ k

, ∆

= 2γ k

(2.83)

∆

n

|

,

|

n

|

|

m

n

=

2|k |

, m

=

2|k |

,

(2.84)

γ

n

γ

√

√

β

=

−

3

γa k

|

sin 3θ , β

=

3

γa k

sin 3θ ,

(2.85)

4

4

n

|

n

| |

and k , k are defined in (2.78). Note that the identity Ec,v (n, kz ; K ) = Ec,v (−n, −kz ; K) follows directly from the time inversion symmetry.

In the presence of an external magnetic field B, the electron energy is

modified just by changing k , k from (2.78) into							+ Φ0
k = L		n − 3	+ Φ0		; k = L n +	3			,	(2.86)
	2π	ν		Φ	2π	ν		Φ

where Φ is the magnetic flux passing through the cross section of a carbon nanotube, Φ = Bz L2/(4π), and Φ0 is the magnetic flux quantum, ch/e. Now the consequence of time inversion symmetry takes the form

Ec,v (n, kz ; Φ; K ) = Ec,v (−n, −kz ; −Φ; K) .

Chirality (or spirality) of a nanotube manifests itself in a particular coupling between the angular momentum as described by n and the directed translational motion as described by kz : due to the linear-in-kz terms in (2.82) or, in general, due to odd-in-kz terms in (2.77) the energy has a contribution which depends both on the sign of kz and the sign of n (propeller e ect).

It is interesting to analyze how this particular coupling disappears for zigzag and armchair tubes which are achiral from the symmetry point of view. In zigzag tubes, the angle θ between the circumferential vector L and

42 2 Quantum Confinement in Low-Dimensional Systems

the vector a is an integer multiple of 60◦, sin 3θ is zero and odd-in-kz terms in (2.77, 2.82) vanish. In armchair tubes, the angle θ equals 30◦ + N ·60◦ with N integer leading to one of the following three relations between na and nb: na = 2nb or nb = 2na or nb = −na. This means that the sum na + nb is an integer multiple of 3 and the parameter ν is zero. As a result values of |k | become independent of the sign of n, and the coupling between signs of n and kz in the odd-in-kz terms vanishes. This follows also from the symmetry considerations. In Sect. 8.6 we will show that the odd-in-kz terms in the electron energy spectrum govern the chirality e ects in carbon nanotubes.

It is worth to mention that, in actual nanotubes, σ and π bands of graphene are slightly mixed due to the presence of a finite cylinder curvature. This will lead to a small band-gap for the metallic nanotubes and a possible shift in the minimum position of the conduction band and the maximum position of the valence band in the 1D Brillouin zone. Moreover, local twists and bends may have an additional e ect on electronic structure [2.55]. Reich et al. [2.56] compared the nearest-neighbor tight-binding approximation for the electron dispersion in graphene and carbon nanotubes with first-principles calculations and the tight-binding dispersion including up to third-nearest neighbors. They concluded that the nearest-neighbor tight-binding dispersion predicts the electronic energies correctly only for a very limited range of wave vectors. In order to establish the theoretical similarities between Si and C, Fagan et al. [2.57] performed first-principles calculations for a hypothetical material, silicon nanotubes. The band-structure calculations show that, similar to carbon structures, they may present metallic (armchair) or semiconductor (zigzag and mixed) behaviors.

2.3 Size Quantization in Quantum Dots

Di erent schemes are used for calculation of electronic states in QDs. They depend on the dot shape and size which, in their turn, are determined by the growth conditions, see the book [2.58] and references therein. Theoretically, the shape of a dot was modelled by spheres [2.59], cones [2.60], pyramids [2.61, 2.62], disks (or cylinders) [2.63, 2.64] and lenses [2.65]. The dots made from a QW heterostructure through a series of masking and etching steps are modelled by a continual lateral confining potential, as a rule a parabolic lateral potential (parabolic QDs) [2.66].

2.3.1 Rectangular and Spherical Quantum Dots

The simplest models are rectangular and spherical QDs with infinite barriers treated in the isotropic-e ective-mass approximation. Similarly to (2.60), the envelope function of an electron confined in a ax × ay × az rectangular QD, or a quantum box, is a product

2.3 Size Quantization in Quantum Dots

43

ψ(r) = ϕνx (x, ax) ϕνy (y, ay ) ϕνz (z, az )

(2.87)

of three envelopes (2.61) describing the size-quantization in one particular direction, respectively x, y or z. The electron energy levels are

Eeνx νy νz = 2mA ax		2	+ ay		2	+ az		.	(2.88)
	π2 2 νx			νy			νz

The electronic states in a spherical QD dot are characterized by the electron orbital angular momentum l. The lowest-energy electron state corresponds to l = 0. For an infinitely high barrier, the ground-state electron

wave function has the form
1			sin (πr/R)				(2.89)
ψs(r) = f (r) |s , f (r) =	√					,
				r
		2πR

where the spin index s assumes the values ±1/2 and the Bloch function |s = csS. The confinement energy is

			E =	π2 2			(2.90)
					.
				2mAR2
	Refining the procedure by finite barriers we obtain for the ground state
	f (r) =	C		sin kr		for r ≤ R ,	(2.91)
		r	e−æ(r−R) sin kR
						for r ≥ R ,

where C is a normalizing factor and k, æ are defined by (2.58) taken at kz = 0. The eigenenergies satisfy the equation

1 − kR cot kR =	mA	(1 + æR) .	(2.92)
	mB

In the Kane model, the influence of nonparabolicity is reduced to a replacement of mA,B by mA,B(E), the scalar envelope us(r) ≡ f (r)cs is given by (2.91), and the vector envelope is determined from

P v

(r) =

2

1 r

i

g(E) − g0

σ

r

df

c

.

2

m(E) r

−

2m0

× r

s

dr

s

The product V R2 should exceed some critical value to allow at least one confined state. For mA = mB ≡ m , this critical value equals

(V R2)cr = π2 2 .

8m

It is worth to mention that, for the ground electron state in a rectangular box (the point symmetry D2h), all four functions in (2.64) are nonzero and allow the representation

44 2 Quantum Confinement in Low-Dimensional Systems

f (r) = f (x2, y2, z2), hx(r) = yz Mx(x2, y2, z2) , hy (r) = zx My (x2, y2, z2), hz (r) = xy Mz (x2, y2, z2) ,

where f and Mα are arbitrary functions of x2, y2 and z2.

A confined-hole state formed from the four-fold spin-degenerate band of Γ8 symmetry (the hole spin J = 3/2, its projection m = ±3/2, ±1/2) cannot be characterized by any definite value of the hole orbital angular momentum L. In the spherical approximation for the Luttinger Hamiltonian (γ2 = γ3 ≡ γ), this is the total hole angular momentum F = J + L which serves as a good quantum number. The hole state is, therefore, (2F + 1)-fold degenerate due to the projection Fz of the angular momentum F along an arbitrary axis, z.

For the ground state we have F = 3/2, Fz = ±3/2, ±1/2. Orbital states with L = 0, 2 are involved in the formation of this four-degenerate state. Following (2.32) we present the hole wave functions at the ground level as

(h)
ψFz (r) = Rm,Fz (r) |3/2, m .		(2.93)
	m

As a function of r and Ji, the matrix ˆ(r) with the components m,F must

R R z

be a spherical invariant. There are only two linearly independent invariants, r2I and (J · r)2, which can be constructed from products of the bilinear functions rirj and the 4×4 basic matrices presented in Table 2.1. Therefore,

one has for the matrix ˆ(r) inside the dot [2.67–2.69]

R

Rˆ(r) = √4πR3/2	f0	R	− f2	R	J · r			− 4
1		r		r	r		2	5

The radial functions fl(x) are defined as

fl(x) = C	jl (φx) − (−1)l/2 j0 j0√βφ jl					βφx
		(φ)

.

,

(2.94)

(2.95)

where jl(x) are the spherical Bessel functions, C is determined by the normalization condition

1

f02(x) + f22(x) x2 dx = 1 ,

0

β is the light-to-heavy hole mass ratio (in the spherical approximation), φ is the dimensionless parameter which determines, according to

	2		φ2
	Eh =			,	(2.96)
		2mhhR2

2.3 Size Quantization in Quantum Dots

45

the hole quantum-confinement energy at the ground state. For the infinite barrier potential the envelope function should vanish at r = R which leads to a simple equation [2.59]

j0(φ) j2
		βφ + j2(φ) j0		βφ			= 0 .	(2.97)

Note that the spherical Bessel functions allow the representation

jl(x) =

2x

Jl+1/2(x)

π

and the first three of them are

sin x

sin x

cos x

j0 =

, j1

=

−

,

x

x2

x

j2 =

x3

− x

sin x − 3

x2 .

3

1

cos x

For β → 1, the lightand heavy-hole subbands in a bulk semiconductor merge to form one four-fold degenerate band. In this hypothetic limit the envelope function is spin-independent and, for the ground state, coincides with the electron envelope (2.89).

2.3.2 Parabolic Quantum Dots

Vertical QDs are artificial semiconductor ‘atoms’ produced by a combination of MBE and nanolithography. They are MBE-grown QW structures with a lateral confining potential due to electrostatic gates or some etching processing. The confinement along the growth direction z is much stronger than in the x-y plane. This allows to separate the vertical and in-plane motions and approximate the electron envelope by

ψ(r) = Φ(x, y)f (z) ,

(2.98)

where f (z) is one of the subband functions (2.8, 2.13) in a 2D QW, Φ(x, y) describes the in-plane confinement. In many vertical QDs the lateral potential, to a good approximation, has a cylindrical symmetry with a parabolic profile. For these dots, the function Φ satisfies the Schr¨odinger equation for a 2D harmonic oscillator

−

2

1

∂

ρ

∂

+

1 ∂2

+

1

m ω02ρ2

Φ(ρ, ϕ) = EΦ(ρ, ϕ) . (2.99)

2m

ρ

∂ρ

∂ρ

ρ2 ∂ϕ2

2

Here m is the in-plane electron e ective mass, the last term in the square brackets is the 2D confining potential expressed via the oscillator characteristic frequency ω0 and we use the cylindrical coordinates. The function Φ(ρ, ϕ) is separable into azimuthal and radial parts

46	2 Quantum Confinement in Low-Dimensional Systems
		eimϕ
	Φ(ρ, ϕ) =	√2π	R(ρ) ,	(2.100)

where m is the angular momentum component along z. The radial function R satisfies (2.99) where the operator ∂2/∂ϕ2 is changed by −m2. The solutions have the form [2.66]

(nρ + |m|)!

l0

|m|

−2l02

nρ +|m| l02

2n

ρ

ρ2

L|m|

ρ2

Rnρ ,m(ρ) =

ρ

exp

, (2.101)

where l0 is the natural harmonic oscillator length scale,

/(m ω0), nρ =

±

1,

±

2... is the quantum num-

0, 1, 2... is the radial quantum number, m = 0,

m

ber for angular momentum component and Ln+m(ξ) is the associated La-

guerre polynomial

Lm

(ξ) =

(n + m)!

eξ ξ−m

dn

(e−ξ ξn+m) .

(2.102)

n!

n+m

dξn

The energy levels of the 2D oscillator are

Enρ m = (2nρ + |m| + 1) ω0 .

(2.103)

It follows then that the 2D parabolic potential forms shells, or degenerate orbitals including ±m and spin degeneracy. The shells N = 1, 2, 3... are equally spaced in energy with the intershell spacing ω0 and the degeneracy 2N .

Let the parabolic potential be anisotropic,

V (x, y) = 12 m (ωx2x2 + ωy2y2) ,

where the resonant frequencies ωx and ωy are di erent. The solutions are, as before, separable

Φ(x, y) = Φnx (x; ωx) Φny (y; ωy ) ,

where nx and ny are non-negative integers enumerating the eigenstates of a

1D oscillator,
Φn(x; ω0) =	1	Hn		x	exp −	x2
							,
	π1/4(2nn!l0)1/2			l0		2l02

l0 = /(m ω0) and Hn(ξ) is the Hermite polynomial

Hn(ξ) = (−1)neξ2 dn e−ξ2 . dξn

The energy in the state (nx, ny ) is given by [2.70]

	Enx ,ny = ωx	nx +	1	+ ωy	ny +		1	.
			2			2

2.3 Size Quantization in Quantum Dots

47

At ωx = ωy ≡ ω0, this set of levels reduces to the equidistant set ω0(nx + ny + 1) equivalent to (2.103).

The above results are simply extended to the problem of an electron in a parabolic QD subject to a magnetic field B z. In this case the eigenenergies

are given by [2.71, 2.72]
Enρ m(B) = (2nρ + |m| + 1)			+		2 ωc ,	(2.104)
	ω02 + 4	ωc2
1				m

where ωc = |e|B/m c is the cyclotron frequency. This equation can be understood taking into account that, in the presence of magnetic field, two additional terms

18 m ωc2ρ2 − i 2ωc ∂ϕ∂ = 18 m ωc2ρ2 + m2 ωc

appear in the electron e ective Hamiltonian in (2.99). Therefore, as compared to (2.103) one should add the term (m/2) ωc to the energy Enρ m and change the oscillator resonance frequency ω02 by [ω02 + (ωc2/4)]1/2 which gives immediately (2.104).

For calculation of hole energy levels in a parabolic potential one needs to take into account the valence-band mixing. Pederson and Chang [2.66] proposed to expand the in-plane envelopes in terms of heavy-hole oscillator states. In this basis the light-hole block of the Luttinger Hamiltonian is not diagonal. However, the advantage of such a formulation is that all matrix elements can be calculated analytically. Then the energy spectrum is found by using a standard numerical diagonalization technique.

2.3.3 Cone-, Lensand Pyramid-Shaped Quantum Dots

The first step of the calculation of InAs/GaAs or InGaAs/GaAs QDs formed on a wetting layer is to approximate pyramids with a square base by cones or lenses having the same height, h, and base surface. For a cone on the wetting layer of thickness d, the confining potential is modelled by the function

Vcone(ρ, z) = V [1 − θ(z) θ(d − z) − θ(z − d) θ[h(1 − ρ/ρ0) + d − z]] ,

where ρ0 is the base radius. The products of Heaviside functions in square brackets define the wetting-layer and cone regions. A lens is modelled as a part of a sphere with fixed height h and radius at the base ρ0. The radius of the sphere R is related to h, ρ0 by R2 = (R − h)2 + ρ20, or R = (h2 + ρ20)/2h. The resulting confining potential is given by

Vlens(ρ, z) = V 1 − θ(z) θ(d − z) − θ(z − d) θ[R2 − (z − d − h + R)2 − ρ2] .

It is clear that Hamiltonians with the above confining potentials are not separable in z and ρ and full 3D numerical diagonalization is needed to solve the

48 2 Quantum Confinement in Low-Dimensional Systems

problem. However, complete numerical approaches justify the use of separable approximate treatments [2.60, 2.65], the latter being by far more physically intuitive. For illustration, we consider the adiabatic approximation applied by Wojs et al. [2.65] to calculate the electronic structure of a lens-shaped QD. They take advantage of the fact that the electron wave function is strongly confined to the lowest subband of the wetting layer and write the function in the separable form

eimϕ
ψ(ρ, ϕ, z) = √2π Rm(ρ)fρ(z) ,	(2.105)

where fρ(z) is a slowly varying function of ρ and m is the angular momentum component along z. The functions fρ(z) and Rm(ρ) satisfy the separate equations

2

∂2

V (ρ, z) fρ(z) = E0(ρ)fρ(z) , (2.106)

−

+

2m

∂z2

2

−

1 ∂

∂

M 2

+ E0(ρ) Rm(ρ) = ERm(ρ) .

ρ

+

2m

ρ

∂ρ

∂ρ

ρ2

For simplicity, the e ective masses in the structure are assumed to be material independent, mA = mB ≡ m . The calculation starts with finding energy E0(ρ) corresponding to the electron motion in the z direction for a given thickness of the A layer,

a(ρ) = R2 − ρ2 + d + h − R .

Next, the radial motion in the potential E0(ρ) is solved for each angular momentum component m. The electron energy spectrum of a lens-shaped

˚	˚	˚
QD In0.5Ga0.5As/GaAs with h = 44 A, ρ0	= 180 A, d = 16 A consists of five

shells, almost degenerate and almost equally spaced with intershell spacing of ω0 = 30 meV [2.66]. Thus, the energy spectrum is well approximated by the cuto spectrum of the parabolic lateral potential.

Strain caused by the di erences in the lattice constants a0 of dot and matrix materials is decisive for Stranski-Krastanow self-organized growth. On the other hand, the strain distribution in and around dots modifies the electron band structure, due to deformation potential and piezoelectric fields. To illustrate, we remind the strain distribution in pseudomorphic slab and sphere. The term ‘pseudomorphic’ is used for heterostructures when they are lattice mismatched to their substrate, and the strain is accommodated entirely elastically without relaxation via dislocations and other defects. If the slab extends along the x and y directions the outer material B imposes

	the strain					a0(B) − a0(A)
	u	xx	= u	yy	=		≡	u		(2.107)
						a0(A)

2.3 Size Quantization in Quantum Dots

49

onto the inner material A, a0(A) and a0(B) being the lattice constants of bulk materials A and B. Since the slab can freely extend in the growth direction the stress along z is absent,

σzz = C11uzz + C12(uxx + uyy ) = 0 ,

and one finds

2C12

uzz = − C11 u ,

while the shear strain is zero and the outer material remains completely unstrained. Here C11 = Cxxxx, C12 = Cxxyy are the elastic sti ness constants. In general, the sti ness tensor C relates the stress and strain components by

σkl = Cklmnumn ,

(2.108)

For cubic crystal classes, it has three linearly independent components: Cxxxx, Cxxyy and Cxyxy ≡ C44. Furthermore, in isotropic materials these three co- e cients are related by 2C44 = C11 − C12 and then the stress-strain relation is written in terms of the Poisson ratio ν = C12/(C11 + C12) and the Young modulus E = (C11 − C12)(1 + ν).

For a pseudomorphic sphere of the radius R, under the simplifying assumption of isotropic materials and material-independent elastic constants,

the strains are [2.61]

uin

= uin

= uin

=

2

u

1 − 2ν

,

3

1 − ν

rr

θθ

ϕϕ

2

1

+ ν

R

3

urrout = −2uθθout = −2uϕϕout =

u

,

3

1

− ν

r

where the superscripts in, out refer to the inner (A) and outer (B) materials and we present the strain tensor components in the spherical coordinates,

urr =

∂ur

, uθθ =

1 ∂uθ

+

ur

,

∂r

r ∂θ

r

uϕϕ =

1

∂uϕ

+

uθ

cot θ +

ur

.

r sin θ ∂ϕ

r

r

One can check that components of uin and uout at the dot surface, r = R,

satisfy the conditions

(uinr − uoutr )|r=R = u R ,

a0(A)(1 + uinθθ) = a0(A)(1 + uinϕϕ) = a0(B)(1 + uoutθθ ) = a0(B)(1 + uoutϕϕ ) .

The strain distribution of a 3D dot of arbitrary shape and elastic properties cannot be presented in a closed analytical form but has to be solved numerically [2.61].

50 2 Quantum Confinement in Low-Dimensional Systems

The strain gives rise to deformation-potential and piezoelectric contributions to the electron e ective Hamiltonian. The former contribution is taken into account by including the Bir-Pikus Hamiltonian (2.30) for the Γ8 valence band or the similar 6×6 matrix Hu for the valence bands Γ8 + Γ7 considered together. For the conduction band Γ6, this contribution is a scalar depending on the hydrostatic components of the strain,

δEc = C1 (uxx + uyy + uzz ) ,

(2.109)

where C1 is the conduction-band deformation-potential constant. In crystals with a zinc-blende lattice, the shear strain induces a piezoelectric polarization P with

Pi = eijkujk ,

eijk being the piezoelectric tensor and having only one linearly independent component exyz ≡ e14. The strain-induced polarization creates fixed charges

ρ(r) = −divP (r)

(2.110)

and gives rise to the piezoelectric potential acting both on electrons and holes. Equation (2.110) reduces the symmetry of the electron Hamiltonian to C2v even when the tensor uij possesses the C4v point symmetry.

Fig. 2.6. 3D view of the isosurface (the probability of finding the electron inside is 70%), and cross section with isolines (30%, 70%, and 96% probabilities of finding the electron inside) of the squared electron ground-state wave function for an InAs QD. The cross section is a (010) plane through the dot center. From [2.61].

In Fig. 2.6 the calculated electron ground-state wave function in a pyramid QD is shown in a perspective view and in a (010) cross section [2.61]. The isosurface containing 70% of the wave-function probability lays almost completely within the dot. In [2.61] the calculation is performed in the simple