Optical properties of cylindrical nanowires

Optical properties of cylindrical nanowires L.A. Haverkate; L.F. Feiner 15th December 2006

Abstract A theoretical analysis is presented of the optical absorption of III-V semiconductor cylindrical nanowires. The optical properties are described by means of the dielectric function, calculated for band-to-band transitions close to the band gap. We have treated the electronic structure using effective mass theory, taking the degeneracy of the valence band into account. A strong polarization anisotropy is found which is due to quantum confinement, in agreement with atomistic methods. We show that the effective mass approach provides a fast and flexible tool to analyze the diameter dependent properties of nanowires for a wide range of semiconductor materials. In addition we discuss the effect of classical Mie scattering and show that it is negligible in the quantum regime.

Contents Introduction

6

I

8

Classical theory of light scattering by a cylinder

1 General solution 1.1 General theory . . . . . . . . . . . . . . . . . . . . 1.1.1 Maxwell equations . . . . . . . . . . . . . . 1.1.2 Boundary conditions . . . . . . . . . . . . . 1.2 Mie’s formal solution for circular cylinders . . . . . 1.3 Scattering problem . . . . . . . . . . . . . . . . . . 1.3.1 Scattering coefficients, general solution . . . 1.4 Far field theory . . . . . . . . . . . . . . . . . . . . 1.4.1 Far field approximation . . . . . . . . . . . 1.4.2 Poynting vector and electromagnetic energy 1.4.3 Cross sections and efficiencies . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . rates . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

9 9 9 10 11 12 15 17 17 18 20

2 Small dielectric cylinders 23 2.1 Coefficients in Rayleigh approximation . . . . . . . . . . . . . 23 2.2 Fields inside the wire . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Efficiency, polarization anisotropy and - contrast in Rayleigh approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Polarization anisotropy, polarization contrast . . . . . 28 2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.4.1 Efficiencies and polarization anisotropy at oblique incidence . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4.2 Efficiencies and polarization anisotropy as a function of wavelength . . . . . . . . . . . . . . . . . . . . . . . 36

II

Absorption

40

3 Electronic properties 41 3.1 The k · p method . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.1 Top valence bands in III-V semiconductors . . . . . . 44 3

Contents

3.2

. . . . . . . . . . . . .

46 47 48 49 51 53 54 54 56 59 59 61 64

. . . . . . . . . . . . . . . .

67 67 67 68 70 70 71 74 74 75 75 76 77 80 83 83 87

5 Dielectric function nanowire 5.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Atomic polarizability approach . . . . . . . . . . . . . 5.1.2 Transition rate method . . . . . . . . . . . . . . . . . 5.1.3 Dielectric function expressed in reduced effective mass 5.2 Dielectric function for finite group transitions . . . . . . . . . 5.3 Polarization anisotropy nanowire . . . . . . . . . . . . . . . . 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Estimation kz dependence of |Tcv |2 . . . . . . . . . . 5.4.2 Polarization anisotropy and R dependence . . . . . . . 5.4.3 Material dependence . . . . . . . . . . . . . . . . . . . 5.4.4 Effect of the dielectric background . . . . . . . . . . .

91 91 91 94 96 97 98 100 100 102 104 105

3.3

3.4

3.5

Effective mass approximation . . . . . . . . . . . . . . . 3.2.1 Crystal Hamiltonian in envelope representation . 3.2.2 Top valence bands in III-V semiconductors . . . Envelope description for infinite cylinders . . . . . . . . 3.3.1 Hole in III-V semiconductor nanowires . . . . . . 3.3.2 Electron in III-V semiconductor nanowires . . . Hole dispersion around kz = 0 . . . . . . . . . . . . . . . 3.4.1 Solutions at the wire zone center . . . . . . . . . 3.4.2 Hole dispersion around kz = 0 for |fz | = 12 , (−) . Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Hole energy bands of III-V material nanowires . 3.5.2 Hole wave functions of III-V material nanowires 3.5.3 Band gap in III-V material nanowires . . . . . .

4 EM transition matrix 4.1 General theory . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Radiation matter interaction . . . . . . . . . . 4.1.2 EM transition matrix . . . . . . . . . . . . . . 4.2 Bloch representation . . . . . . . . . . . . . . . . . . . 4.2.1 Total wavefunction in Bloch functions . . . . . 4.2.2 EM transition matrix in Bloch functions . . . . 4.3 Reformulation of transition matrix element . . . . . . 4.3.1 EM field in dipole approximation . . . . . . . . 4.3.2 EM field including Mie scattering . . . . . . . . 4.3.3 Polarization anisotropy of the transition matrix 4.4 Selection rules . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Polarization selection rules . . . . . . . . . . . 4.4.2 Selection rules on the envelope wavefunctions . 4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Dipole approximation . . . . . . . . . . . . . . 4.5.2 EM field including Mie scattering . . . . . . . .

4

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

Conclusions

107

A Hole wavefunctions for different kz

112

B Polarization selection rules

116

C Interband matrix elements

119

D Reference articles

122

Bibliography

125

5

Introduction In the last several years semiconductor nanowires have attracted much interest, due to the tunability of their fundamental optical and electronic properties. Techniques for the growth of nanostructures have been developed and high quality III-V semiconductor nanowires with a length of several microns and a lateral size of only a few nanometers have been obtained. Recent experiments have shown a large polarization anisotropy in such wires [1][2]. For example, Figure 1 shows the photoluminescence and excitation spectra of an InP nanowire on a flat gold surface [2]. The radius of this wire was ∼ 15 nm and the measured polarization anisotropy is fully explained by the dielectric mismatch between the wire and the surrounding.

Figure 1: Experimental results for optical absorption [2]. a) Photoluminescence image (CCD camera, incident laser light polarized parallel to the wire axis) and b) Excitation spectra for parallel (k) and perpendicular (⊥) polarized incident light of an InP nanowire on a flat gold surface. The length of the wire is ∼ 2 µm, its radius ∼ 15 nm and the wavelength of the exciting laser beam is 457.9 nm. The emitted light was unpolarized in order to take only the polarization anisotropy in the absorption process into account.

However, next to this classical effect of dielectric contrast it is expected that quantum confinement starts to contribute significantly for decreasing 6

wire radius. This quantum effect has already been observed in the shift in the fundamental band gap [3], but based on atomistic theories[4][5] it is also predicted that quantum confinement causes drastic changes in the polarization anisotropy of nanostructures. In this paper we will analyze the optical absorption properties of III-V semiconductor cylindrical nanowires using effective mass theory. Within this approach it is possible to describe the optical and electronic properties for varying wire thickness and for a wide range of semiconductor materials. Contrary, ab initio methods using a fully atomistic description (tight-binding, pseudo-potential,...) are limited with respect to the dimensions of the nanosystem since with increasing number of atoms the calculations become more and more complex, or even impossible. Although the effective mass approach generally is less accurate, it thus provides a relative fast and flexible tool to simulate real nanowires, with dimensions which are technically feasible at the present day. Despite the large amount of papers on the subject of nanowires, or even nanostructures in general, little attention has been paid to the effects of classical scattering. Usually it is assumed that the wavelength of the incident light is sufficiently larger than the wire radius in order to neglect the spatial variance of the electromagnetic (EM) field within the wire, which justifies considering the response of the nanowire to the incident light in the dipole limit. For increasing wire radius, however, the wave behavior of the EM field cannot simply be neglected any more. Therefore, it is one of the main questions in this thesis if this so called Mie-scattering already starts to play a significant role in the quantum confinement regime. For this purpose we have to know the local response field inside the wire. But historically most of the work in classical scattering theory was dedicated to measurable quantities far from the scattering objects, driven by the large interest from application fields as astronomy and meteorology. In Part I, Chapters 1 and 2, we therefore start with a classical theory describing the scattering of light by an infinite cylindrical structure. In particular we derive explicit expressions for the EM field inside the wire by using a procedure originally developed by Mie [6]. In Part II we subsequently focus on the effects of quantum confinement by means of a corrected description of the dielectric function of cylindrical nanowires. In Chapter 3 the electronic properties of nanowires made from III-V compounds are discussed, Chapter 4 treats the EM matrix element for band-to-band transitions between the top Γ8 valence bands and the lowest lying Γ6 conduction band in III-V semiconductor nanowires and finally in Chapter 5 the dielectric function and polarization anisotropy of a nanowire are obtained including the quantum confinement corrections by the bandto-band transitions. 7

Part I

Classical theory of light scattering by a cylinder

8

Chapter 1

General solution In this chapter classical theory is treated which describes the scattering of light by an infinite cylinder at arbitrary angle of incidence and wire radius. In the first sections general theory is discussed and specified to the case of an infinite cylinder by using a procedure originally developed by Mie [6]. In section the theory will be put in an applicable form by deriving measurable quantities (cross sections, efficiency factors) in the far field region.

1.1

General theory

1.1.1

Maxwell equations

The scattering of light at oblique incidence by an infinite cylinder needs a full, formal treatment, in particular when the solution has to be expanded in the wire radius. The starting-point of the full problem is Maxwell’s theory. Assuming the light waves to be periodic with time dependence e−iωt , the charge density ρ equal to zero and the magnetic permeability µ equal to 1, the Maxwell equations are: ∇ × H = −ik0 m2 E,

(1.1)

∇ × E = ik0 H,

(1.2)

∇ · H = 0,

(1.3)

2

∇ · (m E) = 0,

(1.4)

where k0 =

ω c

=

2π , λ0

(1.5)

is the wave number in vacuum and m2 = ε +

4πiσ . ω

(1.6) 9

Chapter 1. General solution

The parameter m is the complex refractive index of the medium at the frequency ω of the light waves and consists of an optic part and an electric part. The former is associated with ε, the dielectric constant, the latter with the conductivity σ, which is taken to be zero since the electrical part is beyond the scope of this paper. Both parts are complex and depend on the circular frequency ω of the light waves. It should be noted that in general m is a tensor and moreover depends on the position in the medium. For the applications considered in this paper the medium is assumed to be homogeneous and in that case m is a constant. We will also assume here that m is a scalar. As a consequence, from (1.1)-(1.4), the field vectors E and H satisfy the vector wave equation: ∆A + k02 m2 A = 0.

(1.7)

As a consequence the rectangular components of E and H satisfy the scalar wave equation ∆ψ + k02 m2 ψ = 0,

(1.8)

which has plane wave solutions with the propagation constant equal to k0 m. This shows that the wave is damped if m has a negative imaginary part and in that case absorption takes place.

1.1.2

Boundary conditions

In case of a sharp boundary between two homogeneous media (1 and 2) the integral representation of the Maxwell equations (1.1) and (1.2) gives the boundary conditions on the tangential components of the fields, after a well known limiting process (Jackson [12], page 16): n × (H2 − H1 ) = 0,

(1.9)

n × (E2 − E1 ) = 0,

(1.10)

where n is the normal to the boundary. In the same way the Maxwell equations (1.3) and (1.4) lead to the boundary conditions on the normal components: n · (m2 2 E2 − m1 2 E1 ) = 0,

(1.11)

n · (H2 − H1 ) = 0.

(1.12)

The tangential and normal boundary conditions are not independent. For instance, boundary condition (1.10) can be derived from (1.12), Maxwell equation (1.3) and applying the limiting procedure. In the same way it can be shown that (1.9) and (1.11) are dependent on each other. Therefore it is sufficient to look only at the tangential components.

10

1.2. Mie’s formal solution for circular cylinders

1.2

Mie’s formal solution for circular cylinders

In order to solve the boundary value problem exactly the coordinate system should be the one in which the scalar wave equation is separable in the coordinates. In case of circular cylinders these coordinates are (ρ, φ, z), where the cylinder axis coincides with the z -axis (see Figure 1.1). As a condition for this separability the cylinder length L has to be assumed much larger then its diameter: L À 2R,

(1.13)

where R denotes the cylinder radius. In this case the cylinder can be seen as infinitely long and then it is possible to use the following formal solution developed by Mie [6]. If ψ satisfies the scalar wave equation (1.8), define M ψ and N ψ as M ψ = ∇ × (ˆ ez · ψ),

(1.14)

mk0 N ψ = ∇ × M ψ .

(1.15)

Then both M ψ and N ψ satisfy the vector wave equation (1.7), and the elementary solutions of Maxwell’s equations can be expressed as E = M v + iN u ,

(1.16)

H = mM u − imN v ,

(1.17)

where u and v are the two independent solutions of the scalar wave equation. The scalar wave equation (1.8) in cylindrical coordinates for a homogeneous medium with complex refractive index m is µ 2 ¶ 1 ∂2 ∂ 1 ∂2 ∂2 2 2 + (1.18) + + + m k0 ψ = 0, ∂ρ2 ρ ∂ρ ρ2 ∂φ2 ∂z 2 and its solutions can be found by separating the variables. The resulting differential equation for the ρ coordinate is the Bessel equation, which has two independent solutions: Jn , the integral order Bessel function and Nn , the integral order Neumann function. This means that the solutions of (1.8) can be found by an appropriate superposition of: q (1.19) ϕn = Zn (ρ m2 k02 − g 2 )ei(gz−ωt) einφ , with n an integer, Zn any Bessel function of order n and g arbitrary. In cylindrical coordinates M ϕn and N ϕn are then derived as:  M ϕn = 

in ρ ∂ − ∂ρ

0

  ϕn ,

  mk0 N ϕn = 



∂ ig ∂ρ

−ng ρ m2 k02 −

  ϕn , (1.20) g2 11


on the basis of cylindrical unit vectors e ˆρ , e ˆφ , e ˆz . Consequently, with un = An ϕn and vn = Bn ϕn for certain An , Bn and taking the sum over all n, the components of E and H are ∞ X in g ∂un Eρ = vn − , (1.21) ρ mk0 ∂ρ n=−∞ Hρ =

∞ X g ∂vn inm un + , ρ k ∂ρ 0 n=−∞

normal to the cylinder surface and ∞ X ∂vn ing Eφ = − un , − ∂ρ mk0 ρ n=−∞ Ez = Hφ = Hz =

∞ X i(m2 k02 − g 2 ) un , mk0 n=−∞ ∞ X n=−∞ ∞ X n=−∞

(1.23) (1.24)

∂un ing + vn , ∂ρ k0 ρ

(1.25)

i(m2 k02 − g 2 ) vn k0

(1.26)

−m −

(1.22)

tangential to the cylinder surface.

1.3

Scattering problem

With the above formal solution it is now possible to solve the general scattering problem of an arbitrary polarized plane electromagnetic wave incident obliquely on a circular cylinder of infinite length. For oblique incidence the direction of propagation of the incident wave makes an angle θ with the normal to the z -axis, see Figure 1.1. Furthermore the cylinder is assumed to be surrounded by vacuum and the refractive index of the cylinder is equal to m. In case of a surrounding homogeneous medium with refractive index m1 the solutions are of the same form if m is considered as the refractive index of the cylinder relative to the medim2 . With the above definitions the incident wave, depicted in um: m = m 1 Figure 1.1 is represented by the scalar wave function ˜0 e−i(k0 x cos θ+k0 z sin θ+ωt) ψ0 = E ˜0 e−i(hz+ωt) = E

∞ X

(−i)n Jn (lρ)eınφ ,

(1.27)

n=−∞

where h ≡ k0 sin θ, l ≡ k0 cos θ = 12

(1.28)

q k02 − h2 .

(1.29)

1.3. Scattering problem

H

O II

E O II k

E sca

H

OI

EO I

k

X

Z

E

int

Y

Figure 1.1: Definition of the coordinates for scattering by a circular cylinder. The incident waves are showed, including the corresponding incident fields: E 0 I , H 0 I in Case I and E 0 II , H 0 II in Case II. The angle of incidence is defined by θ .

Equation (1.27) represents a wave travelling in the −ˆ ex direction if θ equals zero. Note that the last expression in (1.27) is an expansion in Bessel functions and has the required form of (1.19). In this way also the scattered wave and internal wave (inside the cylinder) can be formed from a superpositionpof functions of the form (1.19). Finiteness at the origin requires that Jn (ρ m2 k02 − h2 ) is the radial function describing the internal wave, where h is given by (1.28) because of continuity at the boundary ( (1.9) and (1.10)). The last argument also holds forpthe scattered wave, which is described by (1) the first Hankel function Hn (ρ k02 − h2 ), describing an outgoing wave at large distances from the cylinder. Following the procedure of Van de Hulst [7] and Kerker [8], the polarized incident wave has to be resolved into two components: • Case I: a Transverse Magnetic (TM) mode. The magnetic field of the incident wave is perpendicular to the cylinder axis (Figure 1.1). ˜0 (−i)n ϕn (with This mode is described by choosing un = il1 E g = −h, Zn = Jn ) and vn = 0 in (1.21)- (1.26). This choice also ˜0 (cos θˆ fixes the orientation of the incident electric field: E 0 I = E ez − 1 −i(hz+lx+ωt) sin θˆ ex ) e . The factor il is just a normalization constant, for further details see Bohren and Huffman [11]. • Case II: a Transverse Electric (TE) mode.The electric field is perpen˜0 (−i)n ϕn and dicular to the cylinder axis. Now un = 0 and vn = il1 E −i(hz+lx+ωt) ˜0 e the incident field is given by E 0 II = E ˆy e . For an arbitrary elliptically polarized incident wave the solutions can be found by an appropriate superposition of Case I and Case II. The decompo13


sition of the incident wave does not necessarily mean that the scattered and internal waves resolve in the same way. This can be explained by looking closely at the general expressions of the scalar fields inside and outside the cylinder. These are: Case I ρ > R unI vnI ρ < R unI vnI

˜0 Fn {Jn (lρ) − bnI Hn(1) (lρ)} , =E ˜0 Fn {anI H (1) (lρ)} , =E

(1.30)

˜0 Fn {dnI Jn (jρ)} , =E ˜0 Fn {cnI Jn (jρ)} , =E

(1.32)

n

(1.31) (1.33)

Case II ρ > R unII vnII ρ < R unII vnII

˜0 Fn {bnII Hn(1) (lρ)} , =E ˜0 Fn {Jn (lρ) − anII H (1) (lρ)} , =E

(1.34)

˜0 Fn {dnII Jn (jρ)} =E ˜0 Fn {cnII Jn (jρ)} , =E

(1.36)

n

(1.35) (1.37)

where Fn ≡ and j ≡

1 −i(hz+ωt) e (−i)n einφ il q m2 k02 − h2 .

(1.38)

(1.39)

Unlike the incident waves, which are chosen to be TM or TE, the solutions for the scattered and internal scalar waves are in general decomposed into two components: • A solution with the same orientation as the incident wave (TM or TE), contained in unI (Case I) and vnII (Case II) respectively. • A ”cross mode” with an opposite orientation, TE (v1 ) in Case I and TM (uII ) in Case II. Only in case of normal incidence, θ = 0, the cross terms turn out to be zero and the scattered and internal waves resolve in the same way as the incident wave (see paragraph below). As stated in section 1.2, the scalar wave expressions (1.30)-(1.37) also determine the fields inside and outside the cylinder in the various cases. The incident, scattered and internal fields are denoted with E 0 , E sca and E int , respectively. As an example, combining the expression for the scattered scalar wave in Case I (the second term in (1.30)) with the equations for the field components (1.21)-(1.26) one finds for the scattered electric field E sca in cylinder coordinates: 14

1.3. Scattering problem

 E sca I

˜0 = E

∞ X



 Fn anI 

in ρ ∂ − ∂ρ





 Hn(1) (lρ) + i(−bnI )  

0

n=−∞

−ih ∂ k0 ∂ρ nh k0 ρ l2 k





 (1)   Hn (lρ) . (1.40)

The scattered electric field E sca II in Case II can be obtained from (1.40) by replacing anI by −anII and −bnI by bnII .

1.3.1

Scattering coefficients, general solution

The coefficients anI , bnI , cnI and dnI (anII , bnII , cnII and dnII ) are in general functions of the angle of incidence θ and the wire radius R. They can be determined by the fact that the boundary conditions (1.11)-(1.12) require continuity of the tangential components of E and H. As a consequence the equations (1.23)-(1.26) have to be continuous at R = ρ. These conditions lead in both cases to four linear algebraic equations which can be solved for the four coefficients: Case I anI (R, θ) =

ı sin θ n(m2 − 1){Nn−1 − On−1 } 2 2 lR{( kj0 )2 Ln − (m2 + 1) kj0 Dn + L−1 n (Cn − m Dn )} (1) 0

bnI (R, θ) =

cnI (R, θ) =

dnI (R, θ) =

Hn

,

(1)

(lR){( kj0 )2 Kn − m2 kj0 Dn } + Hn (lR){− kj0 Dn Kn + m2 Dn2 − Cn }

, (1) 2 2 Hn (lR)Mn {( kj0 )2 Ln − (m2 + 1) kj0 Dn + L−1 n (Cn − m Dn )} ( ) (1) anI (R, θ)Hn (lR) l2 , j2 Jn (jR) ( ) (1) ml2 Jn (lR) bnI (R, θ)Hn (lR) − , (1.41) j2 Jn (jR) Jn (jR)

Case II: (1) 0

anII (R, θ) =

Hn

(lR){( kj0 )2 Kn −

j k0 Dn }

(1)

+ Hn (lR){m2 kj0 Dn Kn + m2 Dn2 − Cn }

(1)

2 2 Hn (lR)Mn {( kj0 )2 Ln − (m2 + 1) kj0 Dn + L−1 n (m Dn − Cn )}

,

bnII (R, θ) = −anI (R, θ), cnII (R, θ) =

dnII (R, θ) =

l2 j2

(

(1)

Jn (lR) bnII (R, θ)Hn (lR) − Jn (jR) Jn (jR) ( ) (1) ml2 anII (R, θ)Hn (lR) , j2 Jn (jR)

) ,

(1.42) 15


where (m2 − 1)2 n2 tan2 θ , j 2 R2 0 J (jR) , ≡ cos θ n Jn (jR)

Cn ≡

(1.43)

Dn

(1.44)

(1) 0

0

Kn ≡

Jn (lR) Hn (lR) , , Ln ≡ (1) Jn (lR) Hn (lR)

Mn ≡

Hn (lR) Hn (lR) , Nn ≡ , Jn (lR) Jn (lR)

On ≡

Hn (lR) Jn0 (lR)

(1) 0

(1.45)

(1)

(1.46)

(1) 0

(1.47)

and the functions l (1.29) and j (1.39) depend on θ. Despite of the different form, equations (1.30)-(1.37) are the same as derived by Bohren [11]. They give the complete, formal solution for the scattering problem of a plane electromagnetic wave incident obliquely on a circular cylinder of infinite length. In principle the electromagnetic fields and the intensities can be obtained by calculating the full expansion of (1.30)-(1.33) for Case I ((1.34)(1.37) for Case II) and subsequently use these expressions to calculate the fields (1.21-1.26). However, in practice it is impossible to get an exact analytic solution and a numerical procedure is the only way to solve the full problem. In the special case of a normal incident wave (θ = 0), the scattering coefficients (1.30)-(1.37) reduce to: Case I anI (R, 0) = 0, 0

bnI (R, 0) =

0

mJn (k0 R)Jn (mk0 R) − Jn (k0 R)Jn (mk0 R) (1) 0

(1)

mHn (k0 R)Jn0 (mk0 R) − Hn cnI (R, 0) = 0, (1) 0

dnI (R, 0) =

Hn (1) 0

mHn

(k0 R)Jn (mk0 R)

(1)

,

0

(k0 R)Jn (k0 R) − Hn (k0 R)Jn (k0 R)

(1.48) , (1) (k0 R)Jn (mk0 R) − m2 Hn (k0 R)Jn0 (mk0 R)

Case II 0

anII (R, 0) =

0

Jn (k0 R)Jn (mk0 R) − mJn (k0 R)Jn (mk0 R) (1) 0

(1)

Hn (k0 R)Jn0 (mk0 R) − mHn bnII (R, 0) = 0, (1) 0

cnII (R, 0) =

Hn

(1) 0

m2 Hn dnII (R, 0) = 0. 16

(k0 R)Jn (mk0 R)

(1)

,

0

(k0 R)Jn (k0 R) − Hn (k0 R)Jn (k0 R)

, (1) (k0 R)Jn (mk0 R) − mHn (k0 R)Jn0 (mk0 R) (1.49)

1.4. Far field theory

As stated before, the cross terms disappear, which means that all waves in Case I are TM and all waves in Case II TE.

1.4

Far field theory

In principle the scattering theory derived in the previous sections is complete and everything one wants to know can be derived from it. However, in order to make predictions about measurable quantities, in this section the theory will be put in an applicable form. It is important to realize that usually the experimental measurements are done at a large distance from the scattering object(s), so in the first paragraph general expressions for the fields in this region are derived. Subsequently measurable quantities (cross sections, efficiency factors) are defined and applied to the situation of scattering by an infinite cylinder. The theory depicted here is derived in a detailed form by Bohren and Huffman [11]. More intuitive approaches are found in [7],[8].

1.4.1

Far field approximation

As stated in section 1.3 the scattered wave is associated with the first Hankel function, based on the fact that the wave has to be an outgoing wave. At large distances from the cylinder the first Hankel function can be approximated by its asymptotic expression: r 2 iz e (−i)n e−iπ/4 , | z | À n2 . (1.50) Hn(1) (z) ∼ πz This is the only ingredient needed to approximate the scattered part of the fields at large distances from the wire. Consider for this purpose equation (1.40) for the scattered electrical field. In the far field approximation (lρ À 1) the Hankel functions in this expression are approximated by (1.50). After elaboration of the derivatives and 1 , which fall of much faster then the terms ∼ √1lρ , neglecting all terms ∼ lρ√ lρ this results in: r ˜0 e E sca I ∼ −E

−iπ/4

∞ 2 i(lρ−hz−ωt) X e (−1)n einφ [anI e ˆφ + bnI (sin θˆ eρ + cos θ)ˆ ez ] . πlρ n=−∞

(1.51) This is the result for an incident wave with the magnetic field perpendicular to the wire axis (Case I). For Case II, when the incident field is TE (the electric field perpendicular to the wire axis) the asymptotic expression of the scattered field has the same form, apart from changing anI into −anII and −bnI into bnII . 17


Equation (1.51) shows that the surfaces of constant phase, or wavefronts, of the scattered wave obey ρ cos θ − z sin θ = C,

C ∈ R,

(1.52)

which represents cones of half-angle θ and apexes at z = −C/ sin θ. Including the e−iωt factor, the scattered wave can be visualized as a cone sliding down the cylinder [11].

1.4.2

Poynting vector and electromagnetic energy rates

One of the most important properties of electromagnetic (EM) waves is the flux of EM energy through a certain area. In the case of light scattering at a particle not only the magnitude of this flux has to be specified, but also c its direction. This is given by the Poynting vector S = 8π Re{E × H ∗ }, which defines the time-averaged flux of energy crossing a unit area. As a consequence the rate of EM R energy crossing a plane surface A, with normal unit vector n ˆ , is equal to S · n ˆ dA. For a surface A which encloses a volume V the net rate W at which EM energy crosses the boundary A is defined as I (1.53) W = − S·n ˆ dA, n ˆ ≡ unit normal outward to A. A

This is a definition in the sense that the minus sign ensures that W is positive if there is a net rate of EM energy flowing into the volume V (S · n ˆ < 0), so in the case of absorption of EM energy in the volume. Denoting the incident and scattered EM fields in the same way as before, the Poynting vector at any point outside the particle can be written in these fields as: c S = Re{(E 0 + E sca ) × (H ∗0 + H ∗sca )} = S 0 + S sca + S ext , 8π where S0 = S ext =

c c Re{E 0 × H ∗0 } , S sca = Re{E sca × H ∗sca } , (1.54) 8π 8π c Re{E 0 × H ∗sca + E sca × H ∗0 )} . 8π

The decomposition in (1.54) nicely shows that, next to the expected Poynting vectors of the incident (S 0 ) and scattered (S sca ) fields, a term S ext arises which describes the interaction between the incident and scattered waves. To be more precise, it turns out that S ext represents the removal of energy from the incident light waves, the extinction. Consider for this purpose an imaginary sphere of radius a and surface A around a particle of finite size . The rate of energy Wabs absorbed within the sphere equals the energy 18


rate absorbed by the particle because the surrounding medium is supposed to be non-absorbing. Wabs is given by equation (1.53), now with e â the outward unit normal to the sphere and may be decomposed in: Wabs = W0 − Wsca + Wext , where I

I W0 = − Wext = −

IA A

S0 · e â dA ,

Wsca =

A

S sca · e â dA ,

(1.55)

S ext · e â dA .

The choice of the minus signs here again ensures that all the energy rates are positive, note in particular S sca · e â > 0. Furthermore the energy rate W0 associated with the incident wave vanishes for a non-absorbing medium, so Wext = Wabs + Wsca ,

(1.56)

which shows that Wext indeed represents the extinction, namely the sum of the energy scattering rate and energy absorbing rate. In case of an infinite cylinder the imaginary sphere in the preceding argumentation has to be replaced by an imaginary surrounding cylinder of infinite length. Now it is convenient to look at the rate of EM energy flow per unit length , since this quantity is finite. Furthermore, infinite cylinders don’t exist except as an idealization, so the statements here have to be carefully applied to the situation of a cylinder long compared with is diameter, as used in the previous sections. This is possible if edge effects are negligible, such that there is no net contribution to Wabs from the ends of the imaginary cylinder. In that case, denoting r as the radius of the constructed cylinder and a length L equal to the length of the cylinder, the expressions for the scattering and extinction rates of EM energy per unit length become Z Wsca /L =

(S sca )ρ ρ dφ |ρ = r ,

0

Z Wext /L =

2π

0

2π

(S ext )ρ ρ dφ |ρ = r ,

(1.57)

with (S sca )ρ and (S ext )ρ the (positive) radial components of the expressions derived in (1.54). On physical grounds the absorption energy rate has to be independent of r if the medium outside the cylinder is non absorbing. Indeed, with the far field solution of (1.51), the r dependence in (1.57) drops out. 19


1.4.3

Cross sections and efficiencies

In stead of using the energy rates it is more convenient to take the normalized forms of them: cross sections, or, better, efficiency factors. The former are surfaces, defined as Wabs , I0

Cabs =

Csca =

Wsca , I0

Cext =

Wext , I0

(1.58)

c ˜ 2 |E0 | is the incident intensity. Dividing these optical cross where I0 = 8π sections by the geometrical cross section G, dimensionless efficiency factors are found:

Cabs , G

Qabs =

Qsca =

Csca , G

Qext =

Cext . G

(1.59)

Note that equation (1.56) has a synonym in terms of efficiency factors: Qext = Qsca + Qabs .

(1.60)

For a circular cylinder with radius R and length L the geometrical cross section equals 2RL. Note that the efficiency factors indeed are dimensionless. With the far field scattered electric field (1.51) and a similar expression for the scattered magnetic field now it is a matter of patience to derive:

Qsca I

= =

Z 2π 1 (|T11 (π − φ)|2 + |T12 (π − φ)|2 ) dφ πx 0 ( ) ∞ X 2 |b0I |2 + 2 (|bnI |2 + |anI |2 ) , x

(1.61)

n=1

Qext I

= =

2 Re{T11 (π = φ)} x ( ) ∞ X 2 Re b0I + 2 (bnI ) , x

(1.62)

n=1

Qsca II

= =

Z 2π 1 (|T22 (π − φ)|2 + |T21 (π − φ)|2 ) dφ πx 0 ) ( ∞ X 2 (|bnII |2 + |anII |2 ) , |a0II |2 + 2 x

(1.63)

n=1

Qext II

= =

2 Re{T22 (π = φ)} x ( ) ∞ X 2 Re a0II + 2 (anII ) , x n=1

20

(1.64)


where T11 (π − φ) ≡ T22 (π − φ) ≡ T12 (π − φ) ≡ T21 (π − φ) ≡

∞ X n=−∞ ∞ X n=−∞ ∞ X n=−∞ ∞ X

bnI e−in(π−φ) , anII e−in(π−φ) , anI e−in(π−φ) , bnII e−in(π−φ)

(1.65)

n=−∞

are the four components of the amplitude scattering matrix T, as defined by Kerker [8] and Bohren & Huffman [11]. 1 2 Two general features can be mentioned from (1.61)-(1.64), the efficiency factors for light falling obliquely on a cylinder long compared to its radius: • The efficiencies are expansions in the size parameter kR of the particle in question. • The extinction quantities only depend on the scattering amplitudes in the forward direction (φ = π), while it contains the effect of scattering in all directions by the particle. This is a particular form of the optical theorem and a intuitive explanation is given in [7], [11]. The efficiencies Qabs , Qsca and Qext are the main quantities which can be measured in optical experiments. If the resolution in a particular experiment is high enough, also the differential efficiencies dQsca /dφ can be estimated, which are given by dQsca I /dφ = dQsca II /dφ =

1 (|T11 (π − φ)|2 + |T12 (π − φ)|2 ), πx 1 (|T22 (π − φ)|2 + |T21 (π − φ)|2 ). πx

(1.66)

They specify the angular distribution of the scattered light. It is important to note that the efficiencies defined here in principle can take values larger then unity, contrary to what one should expect from the meaning of the word ”efficiencies”. In particular it can be shown that in the geometrical limit, i.e. if all the dimensions of the scattering object are much 1 The expressions for Qext are derived after quite a lot of algebraic work [11], it can be done faster by using the optical theorem in advance [7]. 2 The transformation of φ to π − φ comes from the definition of the incident wave: as in [11] the incident wave is in the −ˆ ex direction, while Van de Hulst [7] and Kerker [8] use the opposite and no transformation is needed.

21


larger then the wavelength, the extinction efficiency approach the limiting value two. This is rather peculiar, because it suggests that the object removes twice the energy that is incident on it. This so called extinction paradox is resolved by taking also diffraction into account: the edge deflects rays in its neighborhood which from a geometrical view would have passed undisturbed. In this way the incident wave is influenced beyond the geometrical size of the scattering particle.

22

Chapter 2

Small dielectric cylinders As stated in chapter 1, it is not possible to express the general solution of the scattering problem explicitly as a function of the material properties (dielectric constant, wire radius), geometric configuration (angle of incidence, radial distance from cylinder) and the wave number of the incident light. In this chapter the special case of cylindrical wires with radius small compared to the wavelength of the incident light will be treated. It will be shown that in this approximation it is possible to get an analytic solution. In section 2.4 numerical results are given for InP.

2.1

Coefficients in Rayleigh approximation

When the radius of the cylinder is sufficiently small compared to the wavelength of the incident light, the Bessel functions appearing in the scattering coefficients can be expanded in terms of kR. To be precise, sufficiently small means the following condition: |m|x ¿ 1,

(2.1)

where x ≡ k0 R

(2.2)

is defined as the size parameter of the circular cylinder. This condition is physically based on the two Rayleigh assumptions: • The wave behavior of the incident field can be neglected with respect to the size of the particle: x ¿ 1. This implies the external field can be considered as an homogeneous field. • The applied field should penetrate so fast into the particle that the static polarization is established in a time t short compared to the period T , so t/T ¿ 1. Since the velocity inside the cylinder is c/m and the wave period T = 1/ck this assumption is satisfied by (2.1). 23

Chapter 2. Small dielectric cylinders

With condition (2.1) the following expressions for the Bessel and first Hankel functions can be used: z 2 J0 (z) ' 1 − z4 , J00 (z) ' − , 2 1 3 0 0 J1 (z) ' −J0 (z) , J1 (z) ' − z 2 , 2 16 2i 1 0 2i z H0 (z) ' 1 + 2i π γ + π log 2 , H0 (z) ' π z , 2i 1 , (2.3) H1 (z) ' −H00 (z) , H10 (z) ' π z2 where | z | ¿ 1, H denotes the first Hankel function and γ is Euler’s constant. The Hankel functions in 2.3 are expanded to zeroth order in z, because this is sufficient for a second order approximation of the scattering coefficients. With these expansions it can be easily shown that the scattering coefficients (1.41) and (1.42) up to second order in the size parameter x are approximated by: Case I a0I (x, θ) = 0, πx2 (m2 − 1) a1I (x, θ) = sin θ + O(x4 ), 4 (m2 + 1) iπx2 2 (m − 1) cos2 θ + O(x4 ), b0I (x, θ) = − 4 iπx2 (m2 − 1) b1I (x, θ) = − sin2 θ + O(x4 ), 4 (m2 + 1) Case II a0II (R, θ) = O(x4 ), iπx2 (m2 − 1) + O(x4 ), a1II (R, θ) = − 4 (m2 + 1) b0II (R, θ) = 0, πx2 (m2 − 1) b1II (R, θ) = − sin θ + O(x4 ), 4 (m2 + 1)

(2.4)

(2.5)

This result is in agreement with the earlier work of Wait [9], [10] and also gives the expressions derived by Van de Hulst [7] and Kerker [8] for normal incidence (θ = 0) . The internal coefficients for the fields inside the wire, cnI , dnI , cnII and dnII are not explicitly shown here because they have a complex form. They follow directly from equations (1.41) and (1.42). In principle now it is possible to proceed further and use equations (2.4) and (2.5) for the approximation of the fields (equations (1.21)-(1.26)). However, it is really important to be careful, because an expansion of the fields to second order in the size parameter needs more and further expanded coefficients then showed above. 24

2.2. Fields inside the wire

2.2

Fields inside the wire

To our best knowledge the expressions for the fields inside a dielectric cylinder have only been derived in the dipole limit x → 0 [7][13]. This is a relatively small result compared to the the huge amount of research done in the far field region outside the scattering object, where a lot of interest in particular for applications in meteorology and astronomy worked as a driving force. In the dipole approximation, also used by Wang and Lieber [1], the incident field is really taken to be homogeneous. It is a special, stronger form of the Rayleigh approximation discussed above, since the second order corrections now are completely neglected. In this way the expressions for the fields are independent of the size parameter and are derived in terms of the incident fields as [7][13]: E int k = E 0 k , 2 E int ⊥ = E0 ⊥ . 1 + m2

(2.6) (2.7)

Here we will extend this solution to finite values of the size parameter x. Before doing this care has to be taken by expanding the coefficients, as noted before. This is because the internal fields also depend implicitly on the size parameter via ρ, apart from the explicit dependence via the coefficients. This implicit dependence can be split up in two parts: • The internal fields are expressed in terms of Jn (jρ), see (1.30)-(1.37). In second order this results in a ρ dependence by (2.3). • Some of the components of the fields (1.21)-(1.26) have an extra 1/ρ dependence. Taking this into account for a second order approximation of the internal fields one needs the internal coefficients cnI , dnI , cnII and dnII up to the following orders in x: n order

0 3

1 2

2 1

3 0

... ...

Now it is a matter of mathematics to get the solution of the fields inside the wire up to second order. Since the expressions for oblique incidence are too complex to show in an illuminating way, only the results for normal incidence are showed below. ˜int I ρ (x, 0) and E ˜int I φ (x, 0) Starting with Case I, where for normal incidence E are directly zero (see end of section 1.3), the z component of the electric field 25


becomes Eint k z (x, 0) =

© m2 k02 ρ2 cos2 φ + E0 e−iωt 1 − imk0 ρ cos φ − 2 1 2 ρ2 (m − 1)(1 − 2 ) x2 + (2.8) 4 R ¾ ³ 1 2 x´ 2 x + O(x3 ). (m − 1) −2γ + iπ − 2 log 4 2

As required, this solution reduce to (2.6) in the dipole limit x → 0 (note: mk0 ρ = x Rρ → 0). Also the limit m → 1 provides the desired result E int I = E 0 I : for m = 1 there is no optical difference between inside and outside any more. Furthermore, the part between brackets in equation (2.8) can be divided in three parts, each written on a different line here and each with a different physical background: ˜0 e−iωt term in front of the brackets) • The first part (including the E expresses the original wave behavior of the incident field: it is the expansion of e−imkρ cos φ up to second order. • As the cylinder radius increases, the effect of optical focusing gets a more important role. This is described by the second part: it has its maximum in the middle of the cylinder and falls off quadratically to zero at ρ = R. Note that this term is quadratic in the size parameter. • Also the third part is quadratic in x, but in contrast to the second term constant over the wire. It has its origin in the expansion of H0 (kρ), see (2.3). It is a rather striking expression: the iπ part in it can be seen as a constant phase shift of the field. Roughly speaking it is responsible for a correction on the absorption: taking the absolute value squared this iπ part mixes with the complex part of m and decreases the flux of energy crossing the cylinder. Note that the log x2 part together with the x2 after the brackets is finite: limx→0 x2 log x = 0. It gives positive contribution to Iint = ˜int I z (x, 0)|2 that can be large enough to get a value for Iint /I0 |E larger then unity. This is an example of the extinction paradox, as will be explained further in the next sections where a link will be made between internal quantities and the external efficiency factors. The components of the electric field in Case II show mainly the same features as mentioned above. Remember from section 1.3 that the internal electric field is perpendicular to the wire axis (TE) at normal incidence, so ˜int II z (x, 0) = 0. The other two components become E 26

2.3. Efficiency, polarization anisotropy and - contrast in Rayleigh approximation

˜int II ρ (x, 0) = E

˜int II φ (x, 0) = E

½ 2 2 2 2 ˜0 e−iωt 1 − imk0 ρ cos φ − m k0 ρ cos2 φ + E 2 (m + 1) 2 2 1 2 ρ (2.9) (m − 1)(1 − 2 ) x2 + 8 R µ ¶ ¾ 1 (m2 − 1) 1 x − 2γ + iπ − 2 log x2 + O(x3 ), 2 4 (m + 1) 2 2 ½ m2 k02 ρ2 2 −iωt ˜0 e E 1 − imk0 ρ cos φ − cos φ 2 cos2 φ + (m + 1) 2 ρ2 3 2 (m − 1)(1 − 2 ) x2 + (2.10) 8 R µ ¶ ¾ 1 (m2 − 1) 1 x −(m2 + ) − 2γ + iπ − 2 log x2 + O(x3 ). 2 4 (m + 1) 2 2 sin φ

Again, the terms on the first line describe the original wave behavior of the incident field. The incident TE field was taken to be in the positive e ˆy direction and decomposing this in cylindrical coordinates one gets indeed the first terms in the expressions (2.9) and (2.10).

2.3

Efficiency, polarization anisotropy and - contrast in Rayleigh approximation

Contrary to the quantities inside the wire, in the far field region it is possible to get simple analytic expressions in Rayleigh approximation at oblique incidence. For expansion of the scattering and extinction efficiencies (1.61)(1.64) to third order in the size parameter x, the coefficients have to be estimated to second and fourth order for Qsca and Qext , respectively. The third order term for Qext I and Qext II are too extended to show here, but are included in all calculations of the next section. With this in mind the efficiencies (1.61)- (1.64) are approximated by: ½ ¾ 2 sin2 θ(1 + sin2 θ) π 2 x3 Qsca I (x, θ) = |m2 − 1|2 cos4 θ + + O(x5 ) , |m2 + 1|2 8 (2.11) 2 2 2 3 ªπ x |m − 1| © Qsca II (x, θ) = (2.12) + O(x5 ) , 2 − cos2 θ 2 2 |m + 1| 4 ¾ ½ πx 4 sin2 θ 2 2 + O(x3 ), (2.13) Qext I (x, θ) = Im{m − 1} cos θ + 2 2 |m + 1| 2 Qext II (x, θ) =

Im{m2 − 1} 2πx + O(x3 ) . |m2 + 1|2

(2.14) 27


As required, in the limit m → 1 all efficiencies become zero: the refractive index is the same everywhere and there is no scattering any more. Also in the dipole limit x → 0 the efficiencies become zero: in the far field region the scattered field can be neglected with respect to the incident field. The solutions (2.11)-(2.14) depend in a specific way on the angle of incidence θ, which will be illustrated in the next paragraph. It has its origin in the boundary conditions (1.9)-(1.12) on the fields. Apart from this the limit θ → π2 gives an extra requirement: in this limit the difference between Case I and Case II has to vanish as can be argued with symmetry arguments. This can be seen by taking θ = π2 in Figure 1.1. In both cases the incident fields E0 and H0 become perpendicular to the wire axis and by rotational symmetry around this axis Case I and Case II describe the same situation. Indeed, taking the limit θ → π2 the efficiencies in Case I and Case II are equal: π π |m2 − 1|2 π 2 x3 Qsca I (x, ) = Qsca II (x, ) = , 2 2 |m2 + 1|2 2

(2.15)

π Im{m2 − 1} π Qext I (x, ) = Qext II (x, ) = 2πx + O(x3 ) . 2 2 |m2 + 1|2

(2.16)

It is a limit in the sense that an incident wave in the same direction as the wire axis (θ = π2 ) needs a special treatment. How to consider light incident on the endpoints of an (relatively) infinite cylinder? In fact this is the situation of wave guiding, which will not be treated in this paper. Furthermore, the solutions (2.11)-(2.14) can be compared to literature for θ = 0. At normal incidence they are in agreement with the efficiency factors derived by Van de Hulst [7] and Kerker [8]: π 2 x3 + O(x5 ) , 8

(2.17)

|m2 − 1|2 π 2 x3 + O(x5 ) , |m2 + 1|2 4

(2.18)

πx + O(x3 ), 2

(2.19)

Qsca I (x, 0) = |m2 − 1|2 Qsca II (x, 0) =

Qext I (x, 0) = Im{m2 − 1} Qext II (x, 0) =

2.3.1

Im{m2 − 1} 2πx + O(x3 ) . |m2 + 1|2

(2.20)

Polarization anisotropy, polarization contrast

In order to express the difference between incident TM (Case I) and TE (Case II) waves properly, it is common to define a polarization anisotropy ρ [1] [26]. Up to now, for dielectric cylinders this quantity has mainly been 28

2.3. Efficiency, polarization anisotropy and - contrast in Rayleigh approximation

estimated by looking at the internal fields at normal incidence and in the dipole limit. Denoting the polarization anisotropy in this special case by ρint , it is defined by: ρint ≡

|Eint I |2 − |Eint II |2 , |Eint I |2 + |Eint II |2

(2.21)

where |Eint |2 = Iint indicates the internal intensity in dipole approximation. By using the fields in dipole approximation (2.6) and (2.7) this yields ρint =

|m2 + 1|2 − 4 . |m2 + 1|2 + 4

(2.22)

In principle one could proceed further by using the expressions (2.8), (2.9) and (2.10) to get the expanded intensities and so an expansion of the polarization anisotropy ρint (x, 0) inside the wire , but in fact this last step requires far too much calculations: also the direction of the field has to be taken in to account properly. Even harder, the intensity (or Poynting vector) starts to depend on the position in the wire. Instead, it is much easier to calculate the polarization anisotropy by making use of the efficiency factors in the far field region. In terms of the extinction efficiencies, the extinction polarization anisotropy is defined by ρext ≡

Qext I − Qext II , Qext I + Qext II

(2.23)

A formal prove that this ratio in general equals the polarization anisotropy inside the wire, ρext = ρint , is complicated and will not be given here. Instead, the equality can be explained by the following argument: the internal fields are modified with respect to the incident field both by scattering and absorption, so ρint contains the relative difference of the total removal of energy. This is nothing else than the relative difference in extinction between the two cases, which is described by ρext . Contrary to the general case, it is easy to prove the equality at normal incidence in the dipole limit. Using equations (2.19) and (2.20), ρext is approximated by ρext (x, 0) =

|m2 + 1|2 − 4 + O(x2 ) = ρint + O(x2 ) , |m2 + 1|2 + 4

(2.24)

so taking the dipole limit x → 0 on both sides yields ρext = ρint . Next to the extinction polarization anisotropy defined above, it is also insightful to define scattering and absorption polarization anisotropies. Since scattering, absorption and extinction are related to each other by (1.60) only the scattering polarization anisotropy will be treated here. It is defined by ρsca

≡

Qsca I − Qsca II . Qsca I + Qsca II

(2.25) 29


Using (2.17) and (2.18) the scattering polarization anisotropy at normal incidence is expanded in the size parameter by ρsca (x, 0) =

|m2 + 1|2 − 2 + O(x2 ) , |m2 + 1|2 + 2

(2.26)

Although often used, the polarization anisotropy is not a quite useful quantity to work with in practice. This will be illustrated in the next section, but at this stage it is anticipated by introducing a new quantity that describes the difference between the case of incident TM waves and incident TE waves. It is called the polarization contrast and defined by Cext ≡

Qext I , Qext II

(2.27)

Csca ≡

Qsca I , Qsca II

(2.28)

for the total removal of EM energy and for scattering, respectively. Again, using the expanded efficiencies (2.17)-(2.20) the approximations at normal incidence are Cext (x, 0) =

|m2 + 1|2 + O(x2 ) , 4

(2.29)

Csca (x, 0) =

|m2 + 1|2 + O(x2 ) . 2

(2.30)

Note that the limit m → 1 does not work for the scattering polarization anisotropies and contrasts any more.

30

2.4. Results

2.4

Results

As an illustration of the results in the previous sections it is insightful to choose a particular material, InP for example. In particular it is interesting for which radius (or size parameter) the Rayleigh approximation is valid. Recall that the complex refractive index is the only material property appearing in the classical theory derived here, apart from R. It depends on the circular frequency ω of the incident light, see section 1.1 : the material responds to the incident periodic EM field and this response depends on the frequency and so on the wavelength of the incident light. For InP this is illustrated in Figure 2.1. Actually the figure shows the real and imaginary parts ²0 and ²00 of the complex dielectric function ² [11], which is related to the complex refractive index by m2 ≡ ² ≡ ²0 + i ²00 .

(2.31)

Absorption and scattering are more simply described by these optical ”constants”, so from now on all quantities are discussed in terms of ². Ε' , Ε''for InP 17.5

15

12.5

Ε' 10

7.5

Ε''

5

2.5

0 350

400

450

500

550

600

Λ

Figure 2.1: Bulk values of the real and imaginary part ²0 and ²00 of the complex dielectric function ² for InP, as a function of λ0 . It is calculated by interpolating between 32 (optic) experimental values in this interval.

From Figure 2.1 it becomes directly clear that showing the efficiencies as a function of the dimensionless size parameter x is misleading: it really matters if x is changed by varying the wave number or the radius. At different wave numbers also ² has changed, only over a narrow range at small values of k the optical constants can be considered as constant. However, in literature it is common to show the efficiency as a function of x at a fixed ², mainly because it is the most convenient way. This is also 31


the starting point in this paper, see Figure 2.2. For six fixed values of λ0 , so for six different values of ², the extinction efficiencies in Case I and Case II up to third order in x at normal incidence are plotted as a function of x.

608 508 422 395 376 354

nm nm nm nm nm nm

HbL: QIIext, Θ = 0 608 508 422 395 376 354

0.015 QIIext

QIext

HaL: QIext, Θ = 0 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0.01

nm nm nm nm nm nm

0.005 0 0

0.01 0.02 0.03 0.04 0.05 x

0

0.01 0.02 0.03 0.04 0.05 x

Figure 2.2: Extinction efficiencies Qext I and Qext II at normal incidence as a function of the size parameter x = kR. In every plot k, and so ², is fixed. The corresponding values for the wavelength are showed in the left corner. The efficiencies are expanded up to third order in x.

In the illustrated domain the linear terms (2.19) and (2.20) in the extinction efficiencies dominate and the total removal of EM energy from the incident beam increases with the size parameter. The slope of this relation depends on the wavelength and in a quite different way for the two cases. It is explained by looking closely to the dependence on the complex dielectric function: • In Case I the slope increases to a maximum around λ0 = 400 nm after which it decreases for increasing wavelength. This is caused by the factor Im{m2 − 1} = ²00 in (2.19), see Figure 2.1. • In Case II the slope decreases in the whole domain for increasing λ0 . 2 −1} 00 The increase of the denominator of Im{m ∝ (²0 )2²+(²00 )2 in (2.20) |m2 +1|2 dominates the increase of the numerator to ∼ 400 nm. Afterwards ²00 falls off so fast that the slope remains decreasing for increasing wavelength. It even seems from the figure that the third order terms can be neglected in the given domain. Nevertheless, a closer look gives the opposite: for increasing x the third order terms start to give significant corrections. This is illustrated in Figure 2.3: here the correction by the third order terms with respect to (2.19) and (2.20) are shown in percentages. The figure reveals that also the magnitude of the deviation depends on the wavelength: in both cases the influence of the third order correction becomes larger for increasing wavelength. This is explained with the same 32

2.4. Results

HaL: dev. QIext, Θ = 0

10

nm nm nm nm nm nm

608 508 422 395 376 354

6 %

608 508 422 395 376 354

15 %

HbL: dev. QIIext, Θ = 0 8

20

4

nm nm nm nm nm nm

2

5 0

0 0

0.01 0.02 0.03 0.04 0.05 x

0

0.01 0.02 0.03 0.04 0.05 x

Figure 2.3: Deviation of linear behavior in percentages of Qext I and Qext II at normal incidence, as a function of x.

kind of arguments as for the extinction factors itself, but it is omitted here since the third order corrections are not shown explicitly. In the same line as for the extinction efficiencies also the scattering and absorption efficiencies can be illustrated. Since these quantities are dependent of each other only Qsca I and Qsca II are showed here. Figure 2.4 shows that for small x the extinction is completely dominated by the absorption: the contribution of the scattering to the total extinction (Figure 2.2) is only about 1%. This is due to the absence of a first order term in the expansion HaL: QIsca , Θ = 0 608 508 422 395 376 354

QIsca

0.01 0.008 0.006

nm nm nm nm nm nm

608 508 422 395 376 354

0.00008 QIIsca

0.012

HbL: QIIsca, Θ = 0

0.004

0.00006 0.00004

nm nm nm nm nm nm

0.00002

0.002 0

0 0

0.01

0.02 x

0.03

0.04

0

0.01

0.02 x

Figure 2.4: Scattering efficiencies Qsca I and Qsca II at normal incidence as a function of x. In every plot k, and so ², is fixed. The corresponding values for the wavelength are showed in the left corner. The efficiencies are expanded to third order in x.

of Qsca compared to Qext , see equations (2.17)-(2.20). The correction to the third order terms in Figure 2.4 are depicted in Figure 2.5. This is the deviation caused by the x5 terms in percentages. The calculation of these x5 terms is only done for scattering, since in this case the coefficients are needed to x4 while for extinction one needs also the sixth order terms. 33

0.03

0.04


HaL: err . QIsca , Θ = 0 608 508 422 395 376 354

%

3 2

nm nm nm nm nm nm

608 508 422 395 376 354

4 3 %

4

HbL: err . QIIsca, Θ = 0 5

2

1

nm nm nm nm nm nm

1

0

0 0

0.01

0.02 x

0.03

0.04

0

0.05

0.1 x

0.15

Figure 2.5: Correction to the third order expansion of Qsca I and Qsca II by the x5 terms in percentages; again at normal incidence, as a function of x.

The above illustrations are useful for determining the range of the size parameter in which a certain approximation is valid. For instance, Figure 2.5 shows that for a maximal deviation of 5% the third order approximation holds to x ∼ 0.03 and x ∼ 0.2 for Qsca I and Qsca II , respectively. However, this way of displaying is quite awkward for investigating the wavelength dependence: the different curves belong to different values of k. Actually, keeping the wire radius fixed requires looking at a smaller x value by going to a curve at higher wavelength. Also the determination of the limiting R values will not work properly. A size parameter x ∼ 0.03 at λ0 = 400 nm gives R ∼ 2.2 nm, but it would be much more convenient to get these values as a function of the wavelength. Before doing this, the next paragraph will illustrate the dependence on the angle of incidence.

2.4.1

Efficiencies and polarization anisotropy at oblique incidence

The most dominant feature that will appear by displaying the efficiencies as a function of the angle of incidence θ is the symmetry requirement for θ = π2 , as explained in section 2.3. Starting with extinction, it is important to note that also the x3 terms are encountered in Figure 2.6. This means for instance that the first order term (2.14), which is independent of θ, is corrected a little bit by the third order term. The scattering efficiencies are shown in Figure 2.7. In both cases the cylinder radius is fixed, R = 2 nm. Figure 2.6 illustrates the θ dependence of Qext I and Qext II as given in equations (2.13) and (2.14). The difference between Qext I and Qext II at θ = 0 is explained by the large denominator in (2.14): |m2 + 1|2 ' 200. For increasing θ, Qext I decreases to a limiting value which is equal to Qext II at θ = π2 : the first term in (2.13) 34

0.2

2.4. Results

falls off to zero, while the second term increases leading to Qext I ' Qext II at θ = π2 . Also the scattering efficiencies, shown in Figure 2.7, are the same at θ = π2 . As discussed before this is due to the fact that the TM and TE case describe the same physical situation at θ = π2 . HaL: QIext, R = 2 nm

QIext

0.6 0.4

nm nm nm nm nm nm

QIIext

608 508 422 395 376 354

0.8

HbL: QIIext, R = 2 nm

0.2 0

Π 8

0

Π 4 Θ

Π 2

3Π 8

0.014 0.012 0.01 0.008 0.006 0.004 0.002 0

376 nm 354 nm

Π 8

0

422 nm 395 nm

608 nm 508 nm

Π 4 Θ

3Π 8

Π 2

Figure 2.6: Extinction efficiencies Qext I and Qext II as a function of the angle of incidence θ for a fixed cylinder radius R = 2 nm. In every plot k, and so ², is fixed. The efficiencies are expanded to third order in x.

608 508 422 395 376 354

HbL: QIIsca, R = 2 nm

nm nm nm nm nm nm

QIIsca

QIsca

HaL: QIsca , R = 2 nm 0.0175 0.015 0.0125 0.01 0.0075 0.005 0.0025 0

422 395 376 354

0.00015

nm nm nm nm

608 nm 508 nm

0.0001 0.00005

0

Π 8

0 Π 4 Θ

3Π 8

Π 2

0

Π 8

Π 4 Θ

3Π 8

Figure 2.7: Scattering efficiencies Qsca I and Qsca II as a function of θ for R = 2 nm. In every plot k, and so ², is fixed. The efficiencies are expanded to third order in x.

In order to look more closely to the difference between TM and TE waves, the extinction polarization anisotropy (2.23) as well as the scattering polarization anisotropy (2.25) corresponding to the expanded efficiencies at R = 2 nm are depicted in Figure 2.8. Indeed, ρsca and ρext become zero in the limit θ → π2 . At normal incidence the polarization anisotropies reach their maximum value, around 0.985 for extinction as well as for scattering. But the distinction between the curves for different wavelength is hard to extract from the figures. Also the difference between extinction and scattering is not illustrated clearly. As stated in section 2.3 it is more convenient to look at the polarization 35

Π 2


HaL: Ρ sca for R = 2 nm

HaL: Ρ extfor R = 2 nm 1

1

0.8 608 508 422 395 376 354

0.6 0.4 0.2 0

0

nm nm nm nm nm nm

Π 8

Ρ ext

Ρ sca

0.8

608 508 422 395 376 354

0.6 0.4 0.2 0

Π 4 Θ

Π 2

3Π 8

0

nm nm nm nm nm nm

Π 8

Π 4 Θ

Π 2

3Π 8

Figure 2.8: Polarization anisotropy ρext and scattering polarization anisotropy ρsca as a function of θ for R = 2 nm. In every plot k, and so ², is fixed. The factors are expanded up to second order in x.

contrast, equations (2.27) and (2.27). For R = 2 nm this is displayed in Figure 2.9. Now the difference between scattering and extinction becomes clear: at normal incidence the depolarization for scattering is significant larger then for extinction. By rotating the angle of incidence to θ = π2 the scattering polarization anisotropy also decreases faster to the limiting value zero. This effect is also visible in Figure 2.8, but less clearly. C sca for R = 2 nm

C sca

200 150 100

C extfor R = 2 nm

nm nm nm nm nm nm

C ext

608 508 422 395 376 354

250

50 0

0

Π 8

Π 4 Θ

3Π 8

Π 2

140 120 100 80 60 40 20 0

608 508 422 395 376 354

0

Π 8

Π 4 Θ

3Π 8

Figure 2.9: Polarization contrast Cext and scattering polarization contrast Csca as a function of θ for R = 2 nm. In every plot k, and so ², is fixed. The factors are expanded up to second order in x.

2.4.2

Efficiencies and polarization anisotropy as a function of wavelength

As stated in section 2.4, the most physically accurate picture of the scattering process is obtained by showing the efficiencies and polarization anisotropies as a function of the wavelength (or wave number). Apart from the practical points made in the previous sections, this kind 36

nm nm nm nm nm nm

Π 2

2.4. Results

of figures also contain far more information: for every wavelength a set of optical constants has to be used. This is done by interpolating between measured data points for the complex dielectric function, see Figure 2.1. In this way the response of the dielectric cylinder as a function of the frequency of the incident EM field is obtained. HaL: QIext, Θ = 0

HbL: QIIext, Θ = 0

2.5

5 nm 4 nm 3 nm 2 nm 1 nm

1.5 1


0.03 QIIext

QIext

2

0.02 0.01

0.5 0 350

400

450

500

550

0 350

600

400

450

Λ

500

550

600

Λ

Figure 2.10: Extinction efficiencies Qext I and Qext II at normal incidence as a function of the wavelength at constant cylinder radii. The corresponding five R values are showed in the right corner. The efficiencies are expanded to third order in x. The plots are calculated by interpolating between thirty-two (optic) experimental values of ² in this interval. HaL: QIsca , Θ = 0

HbL: QIIsca, Θ = 0

0.3

0.2 0.15


0.1


0.001 0.00075 0.0005

0.05 0

0.0015 0.00125 QIIsca

QIsca

0.25

0.00025 350 375 400 425 450 475 500 Λ

0 350

400

450

500 Λ

Figure 2.11: Scattering efficiencies Qsca I and Qsca II at normal incidence as a function of the wavelength at constant cylinder radii. The efficiencies are expanded to third order in x.

For the extinction and scattering efficiencies at normal incidence this is depicted in Figure 2.10 and Figure 2.11, respectively. The domain of the wavelength is limited: the high and low frequency regions are omitted. The figures show the dependence of the wavelength at five fixed values of R. The dependence of the cylinder radius is visible, especially for extinction: the extinction efficiencies increase linearly with increasing radius by going from one curve to the next in Figure 2.10. 37

550

600


The shape of the curves has been explained above: they where already visible in Figure 2.2 and Figure 2.4, but not so clearly. Now the particular dependence of (2.19) and (2.20) on the complex dielectric function is really visible. For instance, Figure 2.10 nicely shows that the behavior of the imaginary part ²0 is completely reflected in the extinction efficiency. HaL: err . QIsca ,Θ=0

%

15 10

HbL: err . QIIsca ,Θ=0 1.5



1.25 1 %

20

0.75 0.5

5 0.25 0 350

400

450

500

550

0 350

600

400

450

Λ

500

550

600

Λ

Figure 2.12: Correction to the third order expansion of Qsca I and Qsca II by the x5 terms in percentages; again at normal incidence, as a function of λ0 at constant R.

In the same way as in Figure 2.5, the correction to the third order terms in Figure 2.11 is illustrated in Figure 2.12. This is the deviation caused by the x5 terms given as a percentage. For a cylinder radius below 2 nm the used expansion is accurate to 5%. For larger radii it really depends on the wavelength if the approximation is acceptable. HaL: Ρ sca , Θ = 0 0.992

HbL: Ρ ext, Θ = 0 0.985

1 - 5 nm


0.98

0.988

Ρ ext

Ρ sca

0.99 0.986

0.975 0.97

0.984 0.965

0.982 0.98 350

0.96 400

450

500 Λ

550

600

350

400

450

500 Λ

Figure 2.13: Polarization anisotropy ρext and scattering polarization anisotropy ρsca at normal incidence as a function of the wavelength at constant cylinder radii. The corresponding five R values are showed in the right corners. Figure (a) shows that ρsca is independent of the wire radius. The polarization anisotropy ρext is expanded including the second order term, ρsca up to second order in x.

The polarization ratio and scattering polarization ratio corresponding to Figure 2.10 and Figure 2.11 are shown in Figure 2.13. Figure 2.13 (a) shows 38

550

600

2.4. Results

that ρsca expanded up to second order in x is independent of the wire radius, see (2.26). For extinction also the second order is included. Figure 2.13 (b) shows that in that case ρext depends on the wire radius: in particular for wavelengths larger then 400 nm the polarization ratio becomes larger. In other words, increasing the wire radius implies a bigger difference between the case of an incident wave with the magnetic field perpendicular to the wire axis (TM) and the case where the electric field is perpendicular (TE). Remember that this result only applies for small R, results for larger radius or size parameter are showed in [7] [8] [11]. It is really interesting to compare the obtained polarization anisotropy ρext with the depolarization in the dipole limit ρint (2.22), also used in [1] [26]. This is shown in Figure 2.14. The blue curve shows the case when for ρint in the dipole limit only the real part of the bulk ² is taken into account. The difference with the the red curve, indicating ρint for the complete ², becomes dramatically large for λ below 400 nm, and remains significant for the other wavelengths. HaL: Ρ ext, Ρ int; , Θ = 0

HbL: C ext, C int; Θ = 0 C ext C int

0.98 0.96 Ρ

0.94 0.92 0.9 0.88 0.86 350

Ρ int, dip.app., Ε' Ρ int, dip.app. Ρ ext,R = 5 nm Ρ ext,R = 2 nm 400

450

500 Λ

550

600

140 120 100 80 60 40 20 0 350

C int, dip.app., Ε'

C int, dip.app. C ext,R = 5 nm C ext,R = 2 nm 400

450

500 Λ

Figure 2.14: Comparison of the internal polarization anisotropy/contrast in the dipole limit with ρext / Cext . In blue curve, indicating ρint , only the real part of ² is taken into account. The red curve shows ρint for the complete ² .

The yellow and green curve show that increasing the wire radius to R = 5 nm already gives a significant difference between the solution of the dipole limit and the expanded polarization anisotropy ρext . Figure 2.14 (b) shows the same results in terms of the contrasts.

39

550

600

Part II

Absorption

40

Chapter 3

Electronic properties In this chapter the electronic properties of nanowires made from III-V compounds are discussed. The results are based on more detailed studies which can be found in basic semiconductor books [14] [15] and articles by Luttinger [16], Sercel [17] and Marechal [18]. The electronic band structure and wave functions in a nanowire are calculated using the effective mass approximation. This method is in particular convenient to study the optical properties of a semiconductor structure, because analytic expressions for the band dispersion, effective mass and electron/hole wavefunctions around high symmetry points can be obtained. Before turning to the nanowire, in section 3.1 first the band dispersion in bulk material will be derived. Next to general theory, the specific situation of the degenerate top valence band in III-V semiconductor materials will be treated. It has its specific importance in the next chapters and will therefore also be the guideline in the other sections of this chapter: in section 3.2 the effective mass theory for bulk systems is treated, sections 3.3 and 3.4 summarize the envelope description in case of an infinite nanowire and explicit results for InP and InAs are found in section 3.5.

3.1

The k · p method

There are various ways to determine the electronic bands of a semiconductor. Global dispersion relations of bulk materials are available (pseudo-potential techniques, tight binding) but in a lot of cases they are unnecessary. In particular, for describing the optical properties of a semiconductor structure it is often sufficient to know the band dispersion in a small range around the band extremes. This is achieved by the k · p method, which differs from the procedures mentioned above in the fact that, next to the band gaps, also the oscillator strengths of the transitions are used as input. In the k · p method the band dispersion around any point ka is obtained by extrapolation from the k = ka energy gaps and optical matrix elements, using either de41

Chapter 3. Electronic properties

generate or non-degenerate perturbation theory . The input data at k = ka can be obtained from experimental results, typically at the high symmetry points of the crystal. Starting point is the one-electron Schr¨ odinger equation describing the motion of an electron in an averaged potential V (r), which is obtained from the Hamiltonian of a perfect crystal containing N unit cells after usual assumptions such as the Born-Oppenheimer and mean field approximation. The potential V (r) is assumed to reflect the periodicity of the perfect crystal: V (r + R) = V (r),

(3.1)

where R are the lattice vectors. Including the spin-orbit interaction the Hamiltonian describing the unperturbed semiconductor becomes H0 =

p2 ~ + V (r) + 2 2 (σ × ∇V ) · p, 2m0 4c m0

(3.2)

where m0 denotes the free electron mass and σ are the Pauli spin matrices. The relativistic character of the spin-orbit term is reflected by the c12 dependence. Note that the total Hamiltonian, including the spin-orbit interaction, is invariant under a translation by R. H0 thus commutes with the translation operator of the crystal and has Bloch functions as solutions. After normalizing over the whole crystal, containing N unit cells, these are defined as: 1

ψnk (r) = N − 2 eik·r unk (r),

(3.3)

where the unk ’s have the periodicity of the lattice, are normalized over one unit cell and k lies in the first Brillouin zone. The Bloch functions (3.3) form a complete and orthonormal set. Next 1 to this, the Bloch solutions ψn0 = N − 2 un0 at k = 0 are also periodic. Once these, or to be more precise, the corresponding interband matrix elements and energies ²n ≡ ²n (0) are known, the energy dispersion around the zone center (k = 0) can be derived using perturbation theory. In principle, this argumentation can be extended to any point k = ka , provided the transition matrix elements and energies at k = ka are known. This result has been widely discussed in literature [14] [15] [16] [17] [18], here only the results around the zone center are summarized. Assuming the band structure has an extremum (almost) at the zone center and taking k sufficiently small 1 , the dispersion relation for a nondegenerate band (apart from spin) is given by ²n (k) = ²n + 1

~2 k 2 , 2m∗n

(3.4)

Sufficiently small means that the corresponding energy difference ²n (k) − ²n remains much smaller then the band edge differences ²n − ²n0 and that the terms linear k are small enough to be neglected.

42

3.1. The k · p method

where m∗n is the effective mass of the band, 1 m∗n

=

1 2 X | π nn0 · k |2 + 2 2 m0 m0 k 0 ²n − ²n0

(3.5)

n 6=n

and π nn0 are the the interband matrix elements at the zone center: π nn0

~ σ × ∇V | un0 0 i ≡ hun0 | p + 4m0 c2 µ ¶ Z ~ 3 ∗ = d r un0 (r) p + σ × ∇V un0 0 (r). 4m0 c2

(3.6) (3.7)

With the same assumptions, for a degenerate band relation (3.4) is replaced by hj,j 0 (k) = ²j δj,j 0 +

~2 k 2 1 2 m∗jj 0

(3.8)

with 1 m∗jj 0

=

X (k · π jm )(k · π mj 0 ) 1 2 δj,j 0 + 2 2 . m0 ²j − ²n0 m0 k

(3.9)

²n0 6=²j

Here j denotes the degeneracy and the summation over n0 describes the coupling between the group of degenerate states and the other bands. Contrary to the case of a non-degenerate band, one is left with a matrix hj,j 0 (k) which has to be diagonalized in order to get the dispersion relation(s). It should be noted that within the notation used here the tensor behavior of the effective mass is neglected. In general, the coupling between k and the interband matrix elements causes the effective mass to be non isotropic and inclusion of this effect is achieved by the substitution X 1 1 2 k −→ k k , (3.10) ∗ αβ α β m∗n αβ mn 1 m∗n αβ

=

β 1 2 X πnα0 n πnn0 δαβ + 2 m0 ²n − ²n0 m0 0

(3.11)

n 6=n

in (3.4) and a similar one in case of a degenerate band. The surfaces of constant energy belonging to this effective mass tensor are not spheres any more, but warped in certain directions, depending on the symmetry properties of the band under consideration. Furthermore, there are three remarks important to be made at this stage. First, in most of the cases the summation over the bands n0 in (3.5) and (3.9) can be executed over a limited number of values. For large energy differences ²n − ²n0 the contribution of n0 to the effective mass becomes 43


relatively unimportant. Also the interband matrix elements in the numerator reduces the number of bands that contribute to the effective mass. As will be explicitly shown in subsection 4.4.1, the matrix elements are subject to selection rules which are determined by the symmetry properties of the bands in question. Most of the matrix elements become zero by this kind of symmetry arguments. Secondly, including the spin-orbit interaction in case of a non-degenerate band makes little practical difference, since its effect is absorbed in the interband matrix elements which are determined by experiment. For a degenerate band this is different, because the spin-orbit interaction in general lifts the degeneracy and will cause small splitting between the bands. A last important remark has to be made concerning the limitations of the k·p method as depicted here, up to second order in k. The above results rely on the assumption that ²n (k) − ²n remains much smaller then the band edge differences ²n − ²n0 (and a similar assumption in case of degenerate bands), which is not necessarily satisfied, e.g. in semiconductor compounds with a narrow band gap. Instead of expanding beyond second order in the same framework, a commonly used approach [14] [15] [17] initiated by Kane [19] solves this problem by diagonalizing the group of neighboring bands exactly and afterwards treating the coupling with the well separated other bands in a second order perturbation. However, in the remaining part of this paper it is assumed that the bands under investigation are well separated from the others, i.e. splitting terms as the band gap Eg and spin-orbit splitting ∆0 are assumed to be sufficiently large.

3.1.1

Top valence bands in III-V semiconductors

In principle the above theoretical statements now can be applied to any band, or group of bands, once the energy and the interband matrix elements are known. Here the band structure of the six fold degenerate (including spin) top valence band at the Γ point (k = 0) in III-V semiconductor compounds will be summarized. However, the explicit diagonalization of the matrix (3.8) will be performed further on in the envelope function framework since this is the most convenient way when the nanowire structure is anticipated. Starting with symmetry considerations, it is well known [14][15] that the top valence bands in III-V materials have Γ4 like symmetry, apart from spin. The corresponding spatial parts of the valence band wavefunctions at k = 0 are p-like, which means that they are triply degenerate and transform under rotations like the three components of a vector. Including spin this leads to six band edge Bloch functions, which are denoted by |Xi|σi, |Y i|σi, |Zi|σi, with σ =↑,↓. The one-electron Hamiltonian H0 is diagonalized by linear combinations of these band edge Bloch functions. Rewriting the spin orbit term in (3.2) 44

3.1. The k · p method

as Hs.o. = λL0 · σ,

(3.12)

with L0 the angular momentum of the atomic states and treating Hs.o. as a small perturbation2 , this term is diagonalized by the eigenfunctions of the total angular momentum J = L0 + σ of the atomic states. Subsequently the total Hamiltonian H0 can be expressed in the transformed zeroth order eigenfunctions |j, jz i, with j the eigenvalues of J and jz the eigenvalues of its projection Jz along the z axis. These are defined as ¯3 3® ¯ , = − √12 |X + iY i| ↑ i, 2 2 q ¯3 1® 2 ¯ , √1 |X + iY i| ↓ i + = − 2 2 3 |Zi| ↑ i, 6 q ¯3 1® ¯ ,− = √16 |X − iY i| ↑ i + 23 |Zi| ↓ i, 2 2 ¯3 3® ¯ ,− = √12 |X − iY i| ↓ i 2 2 for the j =

3 2

(3.13) (3.14) (3.15) (3.16)

quadruplet and

¯1 1® ¯ , = − √13 |X + iY i| ↓ i + |Zi| ↑ i, 2 2 ¯1 1® ¯ ,− = − √13 |X − iY i| ↑ i − |Zi| ↓ i, 2 2

(3.17) (3.18)

for the two j = 12 states. The last ones are split from the j = 32 states by the spin-orbit interaction, with a magnitude ∆0 = 23 λ. For ∆0 sufficiently large, such that the matrix elements which couple the j = 32 and j = 12 bands are negligible compared to ∆0 , the 6 × 6 Hamiltonian can be decoupled into a 4 × 4 and a 2 × 2 matrix. In most III-V semiconductors, the 4 × 4 matrix of the j = 32 states corresponds to the top most valence band. Assuming the spin-orbit coupling large enough, in this paper the valence band dispersion will be derived by diagonalizing the Γ8 Hamiltonian of these j = 32 states. This is achieved in the same framework as used by Sercel [17] and as in [18]; the explicit results are given in section 3.2. As stated above, corrections to this approach can be found by including the split-off (Γ7 ) band of the j = 12 states and possibly also the lowest conduction band, which usually has Γ6 symmetry. The last one in general is less important since in most III-V semiconductors the spin-orbit splitting is much smaller than the band gap Eg . Including more bands will improve the results, but makes the calculations harder. Focussing on the dispersion for small k around the zone center, it is assumed that these corrections can be neglected in first instance. 2

λ is small because of the relativistic character of the spin-orbit interaction

45


3.2

Effective mass approximation

Suppose an infinite system which is built from the perfect crystal, and a disturbance δV which has to be restricted by specific properties, as will be explained below. The Schrödinger equation (S.E.) of the system is given by (H0 + δV ) |Ψi = E |Ψi.

(3.19)

In principle the solutions of the S.E. can be found by expanding Ψ in terms of the complete orthonormal set of Bloch functions, but without making any further approximation this requires an extensive job since the disturbance δV breaks the translational symmetry of the crystal. The problem is solved much easier by assuming δV to be slowly varying over one unit cell and making use of the band parameters of the unperturbed system, equations (3.4) and (3.8). This approach is known as the effective mass approximation. It can be derived either by utilizing Bloch functions, or in the context of the more localized Wannier functions. Here the Wannier functions are used. They are related to the Bloch functions by Fourier transformation and defined by 1 X e−ik·R ψnk (r). (3.20) anR (r) = N − 2 k

Note that the Wannier functions are indexed by the lattice vector R, reflecting the localized character. They form a complete, orthonormal set just as the Bloch functions and depend on the difference between r and R: anR (r) = an (r − R). Using the complete and orthonormal set of Wannier functions, the solution Ψ(r) of (3.19) is expanded as: X Ψ(r) = Fj (R)ajR (r), (3.21) jR

where j sums over the j degenerate bands and thus includes only one band n in the non degenerate case. The functions Fj (R) are known as the envelope wave functions: as will be shown below, they describe wave packets, extended over (a part of) the crystal and are the envelopes of the atomistic variations caused by the Wannier functions. In order to convert the total Hamiltonian H0 +δV in (3.19) into operators acting on the Wannier functions, it is stated here that k and R are conjugate operators in the sense that R ←→ i∇k

and

k ←→ −i∇R ,

(3.22)

in the limit of large N . Note that R now is treated as a continuous variable, which is justified by the large N limit, i.e. the size of the semiconductor compound is much larger than the distance between the atoms. 46

3.2. Effective mass approximation

Using this result and assuming that δV is a slowly varying function with respect to a lattice vector, it can be shown [14][18] that the S.E. (3.19) reduces to a Schrödinger equation for the envelope functions: {²n (−i∇R ) + δV (R)}Fn (R) = EFn (R)

(3.23)

in case of a non degenerate band n and X {hj,j 0 (−i∇R ) + δV (R)}Fj 0 (R) = EFj (R)

(3.24)

j0

for a degenerate band. For a given band, equation (3.23) ((3.24)) describes the motion of a particle with effective mass m∗n (m∗jj 0 ) in a potential δV . Note that the total wave function of this particle, moving in the perturbed crystal, is obtained from the solutions of (3.23)/(3.24) by multiplying with the Wannier functions as in (3.21).

3.2.1

Crystal Hamiltonian in envelope representation

The above envelope framework initially was derived in the context of impurity states, but as stated by Sercel [17], the procedure can also be used to develop a representation of the unperturbed Hamiltonian H0 which anticipates a centrosymmetric or cylindrical heterostructure. Instead of the Wannier representation (3.21), the solution is expanded in the zone center Bloch functions |uj i by the ansatz X (3.25) |Ψi = |Fj i |uj i, j

which is justified if the energy difference ²j − ²n0 between the degenerate bands and al others is sufficiently large such that unk ' un0 . The notation used in (3.25) stresses the fact that the envelope functions Fj act in a different space as the zone center Bloch functions, this is shown in more detail in chapter 4 considering the transition matrix element. The Bloch functions are defined within a unit cell, while the envelopes are extended over a sufficiently large group of lattice points. It should be mentioned again that the assumption unk ' un0 is essential in this context. In addition to the assumptions in section 3.13 , an extra approximation has to be made here concerning the anisotropy, in order to profit fully from the envelope representation. Conform the situation in most of the III-V semiconductor materials, it is assumed that the anisotropic terms in the Hamiltonian can be neglected, at least as a first order approximation. In this spherical approximation the lower cubic terms causing the warping of 3

I.e. k sufficiently small and energy gaps such as ∆0 and Eg large enough to neglect the coupling of the band(s) under investigation with the others.

47


the bands are set to zero by a restriction on the involved Kohn-Luttinger parameters: γ2 = γ3 [17][18]. As an upshot, adopting the spherical approximation amounts to replace the space group Td of the crystal with the full rotational group. Now the crystal Hamiltonian is invariant under rotations and additional operators can be found which share the same basis of eigenstates. In a cylindrical representation these operators are Pz and Fz , where Pz is the envelope momentum along the z-axis and Fz denotes the total angular momentum along the z axis: Fz = Jz + Lz ,

(3.26)

with Lz the z component of the envelope angular momentum L. The z component of the total angular momentum is a conserved operator, or in other words, Fz commutes with the crystal Hamiltonian. Consequently, the eigenvalue fz of Fz is a good quantum number and the Hamiltonian is diagonal with respect to Fz .

3.2.2

Top valence bands in III-V semiconductors

This is illustrated in more detail by narrowing the focus again to the situation of the top Γ8 valence bands in III-V semiconductors. Following the same approach as in section 3.1, j in (3.25) sums over the jz values of the j = 32 quadruplet and |uj i = | 23 , jz i. The envelope functions |Fj i now are represented as |kz ; k, mi, where kz is the eigenvalue of Pz , k denotes the radial wavenumber and m ² Z are the eigenvalues of the envelope angular momentum Lz . Making use of Lz = Fz − Jz , the envelope functions in the cylindrical representation are of the form Jfz −jz (kρ)ei(fz −jz )φ eikz z , where Jn (z) is a Bessel function. With hρ φ z|kz ; k, fz − jz i the envelope functions in cylindrical coordinates, this results in hρ φ z|kz ; k, fz − jz i | 32 , jz i ∝ Jfz −jz (kρ)ei(fz −jz )φ eikz z | 23 , jz i. (3.27) as a basis for the solution (3.25), which is is orthogonal in fz , jz , k and kz . The Hamiltonian HFΓz8 of the the top Γ8 valence band in III-V semiconductors now is expressed in this basis by [17][18] 

HFΓz8

48

p+q  s =   r 0

s p−q 0 r

r 0 p−q −s

 0 r  , −s  p+q

(3.28)

3.3. Envelope description for infinite cylinders

where the basis is ordered with respect to jz as { 23 , 12 , − 21 , − 32 } and p, q, r and s are given by ~2 ((γ1 + γ2 )k 2 + (γ1 − 2γ2 )kz2 ) , 2m0 ~2 ((γ1 − γ2 )k 2 + (γ1 + 2γ2 )kz2 ) , p−q = − 2m0 ~2 √ r = 3γ2 k 2 , 2m0 ~2 √ 2 3γ2 kkz . s = 2m0 p+q = −

(3.29) (3.30) (3.31) (3.32)

This Hamiltonian has two different eigenvalues, corresponding to a heavy hole (HH) and a light hole (LH) band which are degenerate at the zone center: ~2 2 (γ1 − 2γ2 )(kHH + kz2 ), 2m0 ~2 2 = − (γ1 + 2γ2 )(kLH + kz2 ). 2m0

²HH = −

(3.33)

²LH

(3.34)

Both bands are doubly degenerate and the unnormalized eigenvectors are given by  k2   |HH1i =  



2 HH +4kz √ 2 3kHH  2kz  kHH  ,

 1 0  √  − 3  2kz  kLH  |LH1i =   1 , 0



0 1



    |HH2i =  − 2kz  ,  2 kHH 2  

(3.35)

kHH +4kz √ 2 3kHH

 0  1   |LH2i =  − 2kz  , k√ LH − 3

(3.36)

with respect to the basis given in (3.27), ordered as { 32 , 21 , − 21 , − 23 } with respect to jz .

3.3

Envelope description for infinite cylinders

In principle it is possible to apply the effective mass approximation in the context of the geometrical configuration of a nanowire. However, as pointed out in [14][18], care has to be taken concerning the foundation of the theoretical framework developed in the previous sections.

49


In the first place, the effective mass approximation relies on the assumption that the potential is a slowly varying function over a unit cell. Imposing the wire configuration by taking δV (r) = −V0 Θ(R − ρ),

(3.37)

with Θ the Heaviside function, this requires the wire radius R to be sufficiently large. Intuitively this makes sense directly, for if there are just a few atoms within the wire, the potential change at the boundary of the wire cannot be neglected any more with respect to the interatomic distances a. To be more precise, by rewriting the potential (3.37) in Fourier space, it can be seen [14][18] that the entire concept of an effective mass is only useful if a R ¿ 1, the limit in which only the Fourier components δV (k) around the zone center contribute significantly. Secondly, in the theory of section 3.2 the atomic wavefunctions are assumed to be the same everywhere. If the effective mass approximation is not treated in a suitable form, it thus fails to describe in a proper way the heterostructure situation with two or more completely different environments (e.g. a semiconductor compound in vacuum) and consequently drastic changes in the atomic wavefunctions. This problem is solved in a general way by assuming that every different environment can be described in a large part independently of the other(s)[18]. The bands in the different systems are subsequently related to each other by matching bands with the same symmetry. Instead of equation (3.21), the corresponding wavefunction is assumed to be of the form X (s) (3.38) Ψ(r) = Fj (R)ajR (r), jR

where s indicates that for the atomic functions, in this case in Wannier representation, the solutions are taken far in the corresponding system s. Even if the environments are large enough to approximate them mainly as bulk systems in this way, it still remains a problem to match the wavefunctions of the different systems near the boundaries. For example, it cannot be expected that the atoms around the interface of different systems simply are positioned at the lattice points of a perfect crystal. The atomic functions on both sides of the boundary are not orthogonal to each other and in case of a semiconductor structure in vacuum, there are even no atoms outside the structure any more. Another practical point is the oxidation of the structure, resulting in a system probably better described as a core shell structure. However, the neglect of these effects concerning changes in the atomic wavefunctions and matching of the boundary can be justified by the spatial 50


extent of the envelope functions: by taking an infinite potential well for the nanowire geometry, the envelope functions fall off to zero at the boundary. In this model the atomic functions outside the wire are of no importance and possible fluctuations around the boundary are neglected because the overlapping envelope is almost zero. More problems are expected when V0 is finite. In this case the envelope function leaks with a certain extent into the region outside the wire and the change in atomic wavefunctions plays a more important role. In the present paper the potential V0 in (3.37) is assumed to be large enough to consider it as representing an infinite potential well. In case of an infinite cylinder structure, this leads to a boundary condition on the envelopes of bulk wavefunctions: Fj (ρ = R, φ, z) = 0, ∀ j, φ, z.

(3.39)

The solutions for this boundary condition can be labeled with a set of quantum numbers, say λ, where λ will be specified for the valence band in paragraph 3.4.1 and for the conduction band in paragraph 3.3.2. In addition, for a given band λ the wavefunctions depend on the wavenumber kz . Denoting the complete labeling with λ kz one obtains a normalization condition forPthe envelope functions Fλ kz ,j if the total wavefunction Ψλ kz (r) = Cλ kz jR Fλ kz ,j (R)ajR (r) is normalized to unity: Z Z X X dr |Ψλ kz (r)|2 = Cλ2 kz Fλ∗ kz ,j 0 (R0 )Fλ kz ,j (R) dr a∗j 0 R0 (r)ajR (r) = Cλ2 kz

X X

j, j 0 R, R0

Fλ∗ kz , j 0 (R0 )Fλ kz , j (R)δR0 ,R δj 0 ,j

j, j 0 R, R0

= Cλ2 kz

XX j

|Fλ kz , j (R)|2 = 1,

(3.40)

R

from which the normalization constant Cλ kz is obtained. In the third step in equation (3.40) the orthonormality of the Wannier functions is used. The summation over R can be replaced by an integral in the same context as equation (3.22). The same normalization condition is obtained using the Bloch representation (3.25).

3.3.1

Hole in III-V semiconductor nanowires

With the above remarks in mind consider again the top valence bands of IIIV semiconductors. The wire geometry is imposed by the infinite potential well: ½ ∞, ρ > R, δV (ρ) = (3.41) 0, ρ ≤ R. 51


This leads to the boundary condition on the envelopes (3.39), with j = jz = { 32 , 12 , − 12 , − 32 }. Since the bulk heavy- and light hole solutions (3.35) and (3.36) have four components which cannot be zero simultaneously, this requirement (3.39) can be satisfied only if the total wavefunction is a superposition of the four bulk heavy- and light hole eigenstates for a given fz . Consequently, apart from normalization constant (3.40) the envelope wavefunctions are determined by Fλ kz ,jz (ρ, φ, z) = {(vHH1 |HH1ijz + vHH2 |HH2ijz )Jfz −jz (kHH ρ) +

(3.42)

(vLH1 |LH1ijz + vLH2 |LH2ijz )Jfz −jz (kLH ρ)} ei(fz −jz )φ eikz z , where |HH1i-|LH2i are the bulk eigenstates given in (3.35) and (3.36) and vHH1 , vHH2 , vLH1 , vLH2 are the coefficients which satisfy Fλ kz ,jz (ρ = R, φ, z) = 0. The boundary condition for jz = { 32 , 21 , − 21 , − 32 } results in the determinant equation n 0 = Jf − 3 (kLH )Jf − 1 (kLH )Jf + 1 (kHH )Jf + 3 (kHH ) 2 2 2 2 o + Jf − 3 (kHH )Jf − 1 (kHH )Jf + 1 (kLH )Jf + 3 (kLH ) 2

2

2

2

+ 3Jf − 3 (kLH )Jf − 1 (kHH )Jf + 1 (kHH )Jf + 3 (kLH ) 2 2 2 2 n 4kz2 + kLH kHH Jf − 3 (kLH )Jf − 1 (kHH )Jf + 1 (kLH )Jf + 3 (kHH ) 2 2 2 2 o +Jf − 3 (kHH )Jf − 1 (kLH )Jf + 1 (kHH )Jf + 3 (kLH ) 2

+

2

2

(3.43)

2

2 +4k 2 )(k 2 2 (kLH z HH +4kz ) Jf − 3 (kHH )Jf − 1 (kLH )Jf + 1 (kLH )Jf + 3 (kHH ), 2 k2 3kL 2 2 2 2 HH

which is a relation for the allowed energies. Here the wire radius R is absorbed in the wave numbers by kHH → kHH R, kLH → kLH R and kz → kz R. From now on kHH , kLH and kz denote these dimensionless ”wavenumbers”, unless stated otherwise. Together with the constraint obtained from the equation for the energy, ²HH = ²LH = E,

(3.44)

where the bulk energies ²HH and ²LH are given by (3.33), the determinant equation (3.43) fixes the radial wavenumbers kHH (kz ) and kLH (kz ) for a given kz . Using these solutions of kHH (kz ) and kLH (kz ), also the coefficients vHH1 -vLH2 are obtained from the boundary equation (3.39). Before turning to more explicit results in the next sections, the following general remarks are important to keep in mind. First, the determinant equation (3.43) is invariant under the inversion fz → −fz , which reflects the time-reversal symmetry of the Hamiltonian HFΓz8 given in (3.28): the total angular momentum reverses direction if t → −t. Consequently, the energy solutions E are doubly degenerate in fz and the corresponding wavefunctions turn into each other under fz → −fz . 52


Secondly it should be noted that the wavenumber kz , giving the dispersion in the z direction where the electron (hole) is still free to move, cannot be separated from the lateral terms in the envelope wavefunction (3.42). The radial wavenumbers kHH and kLH are functions of kz , so the dispersion in the z direction in general depends on the lateral distribution of the wavefunction. Furthermore, for a given kz the set of equations (3.43), (3.44) has to be solved numerically. Only in special cases the energy E and hole wavefunction reduce to relative simple analytical expressions. In the next section, first some analytical results at kz = 0 are summarized. Subsequently an expression for the effective mass of a hole in a III-V nanowires will be derived by expansion around kz = 0, the wire zone center.

3.3.2

Electron in III-V semiconductor nanowires

Up till now only the situation of the degenerate top valence bands in III-V semiconductors was discussed. Since also the conduction band properties are needed in the remaining of this paper and because it is also illustrative to consider a nondegenerate example which is much easier to handle, here the electron dispersion and wavefunctions of the lowest lying conduction band in III-V semiconductors are treated shortly. Again an infinite confinement is assumed, as given in equation (3.41). Since the conduction band is non-degenerate, the S.E. for the electron is given by (3.23) and the envelope function for ρ < R is given by Fλ kz (ρ, φ, z) = Cλ Jlz (klz ρ)eilz φ eikz z ,

(3.45)

with Cλ the normalization constant. Assuming also the warping sufficiently small, i.e. adopting the spherical approximation by taking an uniform effective mass m∗c for the conduction band, the energy dispersion becomes ~2 (k 2 + kz2 ) = E. 2m∗c lz

(3.46) j

z ,n , the alloThe boundary condition (3.39) now simply gives klz ,n = lR wed values of klz which are independent of kz . Here jlz ,n is the nth zero of the Bessel function Jlz (x)

It is convenient to introduce a notation which summarizes the labeling of the conduction subbands, as derived in the current framework. In the present case, the total Hamiltonian already is diagonal in the envelope angular momentum, so the conduction subbands are labeled with |lz |. 53


Following the notation of [4], the irreducible representation of the conduction subbands in cylinder configuration is characterized by (±)

C|lz |, n ,

(3.47)

where (±) denotes the parity, n the nth solution at this parity and the absolute value of the envelope angular momentum is taken because of the degeneracy in lz .

3.4

Hole dispersion around kz = 0

As discussed in [17] [18], the top valence band Hamiltonian HFΓz8 , given by (3.28), decouples into two 2 × 2 blocks at the wire zone center, kz = 0. Both blocks have solutions which are characterized by parity: the corresponding Bessel functions are only even or only odd under ρ → −ρ. Apart from a general discussion, in this section the focus will be narrowed to an exceptional case: the odd solutions for |fz | = 12 . In this case it is possible to derive a transparent equation for the effective mass in the z direction by Taylor expansion around kz = 0.

3.4.1

Solutions at the wire zone center

At the wire zone center, kz = 0, one obtains from the energy equation (3.44): s r m∗HH γ1 + 2γ2 kHH = βkLH , β ≡ = , (3.48) γ1 − 2γ2 m∗LH where m∗HH and m∗LH are the effective masses of the heavy and light hole bulk bands, respectively. Also the boundary condition simplifies at kz = 0. By block diagonalizing (3.28) it is found that the four heavy- and light hole wavefunctions decouples into two groups: either vHH2 = vLH2 = 0 , vLH1 = α1 vHH1 ,

(3.49)

vHH1 = vLH1 = 0 , vLH2 = α2 vHH2 .

(3.50)

or

Here α1 and α2 are determined by the determinant equation (3.43) at the wire zone center, which decouple into two mutually excluding determinants 1 Jfz − 32 (kHH ) 3 Jfz − 3 (kLH )

= −

1 Jfz + 32 (kHH ) 3 Jfz + 3 (kLH )

= −

2

Jfz + 1 (kHH ) 2

Jfz + 1 (kLH )

≡ α1 ,

(3.51)

≡ α2 ,

(3.52)

2

or

2

54

Jfz − 1 (kHH ) 2

Jfz − 1 (kLH ) 2

3.4. Hole dispersion around kz = 0

so the only possible solutions indeed are given by (3.49) and (3.50). Note that the inversion symmetry of fz is revealed by the two determinants: using J−n (z) = (−1)n Jn (z),

(3.53)

it easy to show that (3.49) turns into (3.50) under fz → −fz . Energy equality (3.48), together with either (3.49) or (3.50) determines the energy at the zone center. The different energy bands and corresponding wavefunctions are characterized by parity at the wire zone center: for a given fz ² Z + 21 the solution corresponding to (3.49)/(3.50) contains only even/odd (odd/even) Bessel functions. Note that Bessel functions transform under inversion in ρ in the same way as their label (i.e. under z → −z, Jn (z) → Jn (z) if n is even, Jn (z) → −Jn (z) if n is odd). As long as kz = 0, parity thus is a good quantum number and this is still approximately the case for kz close to 0. Consequently, the energy bands and corresponding valence subbands are labelled with +/− for respectively even/odd solutions at kz = 0. Note that for a given parity there are different solutions labelled by n. At this point it is convenient to specify the labeling of the valence subbands further. As stated in paragraph 3.2.1, in a cylindrical representation the total Hamiltonian H Γ8 is diagonal in Fz , which implies that the subbands are also labeled with |fz | (the absolute value is taken because of the degeneracy in fz → −fz ). Consequently, the complete set of solutions for the Γ8 valence band in the infinite cylinder configuration is characterized by the quantum numbers fz , (±), nth solution at this parity. This irreducible representation of the valence subbands is indicated with (±)

(3.54)

E|fz |, n ,

where, contrary to the notation in [4], n denotes the nth solution at a particular parity. In general, even at kz = 0 the original bulk heavy- and light hole solutions are coupled to each other in a nanowire. However, it turns out that the odd solutions at |fz | = 21 form an exceptional group. The determinant equation (either (3.49) or (3.50)) in this case reduces to J1 (kHH )J1 (kLH ) = 0,

(3.55)

with Bessel zeros j1,n as solutions: kHH =

j1,n j1,n , kLH = R βR

(3.56) 55


or kHH = β

j1,n j1,n , kLH = , R R

(3.57)

where kHH , kLH are the original wavenumbers, so R is written out explicitely. As can be seen from (3.51) or (3.52), the relevant coefficient α1 /α2 is zero for these solutions. Since the other heavy -, light hole pair already is excluded (equation (3.49) or (3.50)) the odd wavefunctions at kz = 0, |fz | = 21 consequently are pure heavy - or light hole like. Note that for the light hole solutions both α1 and α2 should be inverted.

3.4.2

Hole dispersion around kz = 0 for |fz | = 12 , (−)

The confinement by the infinite wire geometry, resulting in the determinant equation (3.43), reduces the dimensions in which the electron (hole) is free to move to one. The dispersion relation in this direction (z) becomes more complex than the quadratic dispersion of the two original bulk bands, due to the fact that kz cannot be separated from the lateral terms in the envelope wavefunction in case of the degenerate III-V top valence band. However, for the odd solutions at |fz | = 21 it is possible to approximate the dispersion around the wire zone center with an effective mass. For this purpose, the first step is to expand kHH (kz ) and kLH (kz ) up to second order in kz . Recall that the Γ8 band minimum is assumed to be at at the zone center, so ¯ ∂E ¯¯ = 0. (3.58) ∂kz ¯kz =0 Utilizing this assumption, one finds for the lateral wavenumbers, by Taylor expansion around kz = 0, 2 2 kHH (kz ) ' a2HH + b2HH kz2 , kLH (kz ) ' a2LH + b2LH kz2 .

(3.59)

Consequently, expanding up to second order in kz , kHH and the corresponding Bessel function are approximated by kHH (kz ) ' aHH +

b2HH 2 k , 2aHH z

Jn (kHH (kz )) ' Jn (aHH ) +

(3.60)

b2HH 0 J (aHH )kz2 2aHH n

and the expressions for kLH are similar.

56

(3.61)

3.4. Hole dispersion around kz = 0

Before expanding (3.43) in this way, first it can be simplified for |fz | = by using the Bessel function property (3.53):

1 2

4 2 2 k k J1 (kLH )J1 (kHH ) {J0 (kLH )J2 (kHH ) + 3J0 (kHH )J2 (kLH )} + 3 LH HH © ª 4kLH kHH J12 (kLH )J0 (kHH )J2 (kHH ) + J12 (kHH )J0 (kLH )J2 (kLH ) kz2 + 4 2 2 (3.62) (k + kHH )J1 (kLH )J1 (kHH )J0 (kLH )J2 (kHH )kz2 , 3 LH where it should be mentioned that kHH = 0 (or kLH = 0) is not a solution [18]. Expanding the determinant equation up to second order with (3.59)(3.61), the zeroth order part of the first line in (3.62) gives the solutions at kz = 0: 0 =

0 = J1 (aLH )J1 (aHH ) {J0 (aLH )J2 (aHH ) + 3J0 (aHH )J2 (aLH )} , (3.63) where the term between the brackets in (3.63) corresponds to the even solutions. In order to simplify the quadratic term in the expansion of (3.62), aHH and aLH should be fixed by either the even or the odd solutions in (3.63). For the odd solutions this results in a transparent equation for the dispersion around the wire zone center. Imposing J1 (aHH )J1 (aLH ) = 0 and utilizing a property of the Bessel functions, n (3.64) Jn0 (z) = ∓Jn±1 (z) ± Jn (z), z the quadratic term becomes n 2 h i b 1 0 = aLH aHH 13 2aLH J (a ) ∓J (a ) ± J (a ) J1 (aHH )J2 (aHH )+ 0 LH 1±1 LH aLH 1 LH LH h i 2 1 bHH 1 3 2aHH J0 (aLH )J1 (aLH ) ∓J1±1 (aHH ) ± aHH J1 (aHH ) J2 (aHH ) + h i b2HH 1 J (a ) ∓J (a ) ± J (a ) J1 (aLH )J2 (aLH ) + 0 HH 1±1 HH 1 HH 2aHH aHH h i b2LH 1 J (a )J (a ) ∓J (a ) ± J (a ) J2 (aLH ) + (3.65) 0 HH 1 HH 1±1 LH 1 LH 2aLH aLH o ¡ 2 ¢ 2 1 2 kz . aLH aHH J1 (aLH )J0 (aHH ) J2 (aHH ) + J1 (aHH )J0 (aLH ) J2 (aLH ) The final step is to specify the odd solution further by choosing either J1 (aLH ) = 0 or J1 (aHH ) = 0. Here the discussion is restricted to aHH = j1,n , which corresponds to the lowest, heavy hole like energy bands.4 After choosing the convenient signs in (3.65), the determinant equation in this case reduces to an expression for the expansion factor bHH of the heavy hole lateral wavenumber: 2

bHH = − 1

2 aLH J1 (aLH )

3 J0 (aLH )

4

− J2 (aLH )

=−

j1,n 2β j1,n J1 ( β ) , j1,n j1,n 1 3 J0 ( β ) − J2 ( β )

(3.66)

Note again that for the odd solutions there is no heavy -, light hole mixing any more.

57


where the second expression follows from aLH = β1 aHH , see (3.48). 2 (k ) + k 2 ) Expanding also the energy equation UEh = −(γ1 − 2γ2 )(kHH z z 1 (equation (3.44)) using (3.59)-(3.61), the odd |fz | = 2 heavy hole bands are approximately given by

E Uh

= −(γ1 − 2γ2 )j1,n − (γ1 − 2γ2 )(1 + b2HH )(kz R)2 ,

(3.67)

where bHH is given in (3.66) and the dependence on the wire radius is explicitly shown by defining an energy unit Uh : Uh ≡

~2 . 2m0 R2

(3.68)

This results in an expression for the effective mass of a heavy hole in the odd |fz | = 12 energy bands of an infinite wire: m∗HH , z

= m0 (γ1 − 2γ2 )−1 (1 + b2HH )−1 ,

where z is the only direction in which the hole is still free to move.

58

(3.69)

3.5. Results

3.5

Results

As an illustration of the above theory, in this section numerical results are given on the basis of specific examples. In particular it is insightful to compare III-V materials with different properties, in this case different KohnLuttinger parameters. Hence, the III-V compounds InP and InAs are investigated, their Kohn-Luttinger parameters given in Table 3.1 are taken from [20]. Note that the values of γ3 are not needed here: the theoretical InP InAs

γ1 5.08 20.0

γ2 1.60 8.5

Table 3.1: Kohn-Lutinger parameters for InP and InAs

framework rests on the assumption γ3 = γ2 .

3.5.1

Hole energy bands of III-V material nanowires

The first seven hole energy bands of InP and InAs nanowires are shown in figure 3.1. They are calculated from equations (3.43) and (3.44). The black line in the graphs corresponds to the reference band −(γ1 + 2γ2 )kz2 with kLH = 0, where all bands end because there are no solutions if kLH ≤ 0. The R dependence is absorbed in the units along the axes, kz R and E R 2 , with R in nm, kz R dimensionless and E R2 in eV nm2 . InP -1 E R2 HeV nm2 L

InAs

fz = 12, H+L 1 fz = 12, H-L 1 fz = 32, H+L 1 fz = 32, H-L 1 fz = 32, H+L 2 fz = 12, H+L 2 fz = 12, H-L 2

-2 -3 -4 -5 -6

0

fz = 12, H+L 1 fz = 12, H-L 1 fz = 32, H+L 1 fz = 32, H-L 1 fz = 32, H+L 2 fz = 12, H+L 2 fz = 12, H-L 2

-1 E R2 HeV nm2 L

0

-2 -3 -4 -5 -6

0

1

2

4 3 kz R

5

6

0

0.5

1

1.5 2 kz R

Figure 3.1: Hole energy bands for InP and InAs. The bands are labeled by absolute total angular momentum in the z-direction, |fz |, and parity, denoted with (+) nth (nth even solution) and (−) nth (odd). The black line in the graphs corresponds to the reference band (γ1 + 2γ2 )kz2 with kLH = 0. The R dependence is absorbed in the units along the axes, with wire radius R in nm.

Using the notation given in (3.54) and (3.47), the representation of the first seven hole subbands of InP and InAs nanowires are shown in Table 3.2. 59

2.5

3

3.5


The subbands vi are ordered with respect to their zone center offset, see Figure 3.1. For convenience, also the representation of first two electron subbands c1 and c2 are given, within the framework of Subsection 3.3.2. Subband

InP

v1

E 1 ,1

v2

E 1 ,1

v3

E 3 ,1

v4

E 3 ,1

v5

E 3 ,2

v6

E 1 ,2

v7

E 1 ,2

c1

C0, 1

c2

C1, 1

(+)

2

(−)

2

(+)

2

(−)

2

(+)

2

(+)

2

(−)

2

InAs

(−)

E 1 ,1 2

(+)

E 1 ,1 2

(+)

E 3 ,1 2

(+)

E 1 ,2 2

(−)

E 3 ,1 2

(−)

E 1 ,2 2

(+)

E 3 ,2 2

(+)

C0, 1

(+)

(−)

C1, 1

(−)

Table 3.2: Irreducible representation of the first seven hole subbands vi and first two electron subbands cj for InP and InAs nanowires. The characterization is also valid for conduction subbands calculated in a finite potential well.

Around kz = 0 the hole band dispersion can be approximated by the quadratic expressions given in Table 3.3. In general, for |fz | = 12 and odd parity these numerical results are in good agreement with the analytical expansion given in (3.67). For instance, for InP the effective mass in the z direction m∗HH , z corresponding to the values in Table 3.3 are 3.45 m0 and 11.49 m0 for the first even and first odd subband, while the analytical expression (3.69) gives 3.39 m0 and 10.20 m0 , respectively. Comparing the two materials, the following remarks are supported by Figure 3.1, Table 3.2 and Table 3.3: 60

3.5. Results

hole state |fz | = 12 , (+) 1 |fz | = 12 , (−) 1 |fz | = 12 , (+) 2 |fz | = 12 , (−) 2 |fz | = 32 , (+) 1 |fz | = 32 , (−) 1 |fz | = 32 , (+) 2

InP −0.75 − 0.14(kz R)2 −1.05 − 0.29(kz R)2 −2.16 − 0.07(kz R)2 −3.53 − 0.087(kz R)2 −1.38 + 0.31(kz R)2 −1.79 − 0.59(kz R)2 −2.08 − 0.16(kz R)2

InAs −2.14 − 1.87(kz R)2 −1.68 + 0.77(kz R)2 −3.91 − 0.72(kz R)2 −5.62 − 0.35(kz R)2 −2.95 + 0.44(kz R)2 −4.07 − 0.45(kz R)2 −6.52 + 1.29(kz R)2

Table 3.3: Numerical results for the hole energy E R2 (eV nm2 ), fitted to kz2 R2 around the wire zone center

• The shape of a particular band, including its zone center energy, is material dependent. It depends on the magnitude of the gamma’s by γ1 − 2γ2 , but also their ratio γγ12 is a deciding quantity. Consequently, the corrections to the band gap Eg caused by the confinement are material dependent. For the present two examples the zone center band gaps of InAs are more shifted by the infinite wire configuration. • Next to the shape of the individual bands, also their mutual ordering is material dependent. This means that the parity of the lowest lying band (and the others) can differ depending on the material. For example, the lowest lying band is even for InP and odd for InAs. Furthermore, it should be noted that the results are valid for all R, with the only requirement that R should be sufficient large in order to consider the confinement potential (3.41) as a slowly varying function with respect to the unit cell dimensions. This means that in the limit R → ∞ the confinement correction on the band gap has to disappear, which is indeed the case as can be concluded from the 1/R2 dependence.

3.5.2

Hole wave functions of III-V material nanowires

As pointed out in section 3.3, the wavenumber along the cylinder axis kz is not independent of the lateral part of the hole wavefunction. As a consequence, the total wavefunction for a particular band depends in a non trivial way on kz and should be calculated from (3.42) for every value of kz separately. Here the results of this procedure are summarized by focussing, next to the kz dependence, on three other subjects: the parity of the wavefunctions, the invariance under total z-angular momentum reversion and the influence of material properties (Kohn-Luttinger parameters). Starting with parity and the invariance under fz → −fz , Figure 3.2 shows the φ = 0 radial part of the envelope wavefunction, decomposed into the different jz components at the same value of kz and for the same 61

Chapter 3. Electronic properties (+)

material. The graphs in the first row correspond to the two E 1 ,2 solutions, 2

(−)

those in the second row to the solutions with representation E 3 ,1 . The total 2

envelope function is normalized using equation (3.40) and the wire radius R is absorbed in the dimensionless unit Rρ . fz = 12 , kz R = 2.66667 , H+L 2

fz = -12 , kz R = 2.66667 , H+L 2

1.5

1

1

0.5

0.5 Χ jz

Χ jz

1.5

0 -0.5

0 -0.5

jz = 3 2 jz = 1 2

-1 0

0.2

0.4

0.6

jz = -1 2 jz = -3 2

0.8

jz = 3 2 jz = 1 2

-1 1

0

0.2

ΡR

0.8

1

1

0.5

0.5

0 -0.5

0 -0.5

jz = 3 2 jz = 1 2

-1 0

0.2

0.4

0.6

jz = -1 2 jz = -3 2

0.8

jz = 3 2 jz = 1 2

-1 1

0

0.2

ΡR

0.4

0.6

jz = -1 2 jz = -3 2

0.8

ΡR

Figure 3.2: Radial (φ = 0, eikz z omitted) part of the normalized hole envelope functions, decomposed in the different jz components: the jz = 3 1 1 2 components are given in yellow, jz = 2 in green, jz = − 2 in blue and 3 jz = − 2 in red. The first row gives the solutions for |fz | = 12 , + (2), the second row for |fz | = 32 , −(1). The Kohn-Luttinger parameters are taken from InP as given in Table 3.1 and kz R = 2.67 is fixed. (±)

Figure 3.2 illustrates that for a given subband E|fz |,n the solution at fz for a particular jz is the same (apart from minus sign) as at −fz for −jz . Moreover, as expected the two total wavefunctions turn into each other under time reversal, because under fz = lz + jz → −fz = −lz − jz any component Jl (kρ) |j, jz i → J−l (kρ) |j, −jz i, so besides jz → −jz the odd solutions reverse sign, as shown in Figure 3.2. The wavefunctions in Figure 3.2 are calculated away from the wire zone center, at kz R = 1.5 10−9 with R in nm. As a consequence, next to the jz components with the parity of the zone center, also other jz components appear which have the opposite parity. For example, in the first graph the dominant jz = 12 component (green curve) and the jz = − 23 component (red curve) are the evolved even components which are present at the zone center, while the jz = 32 (orange) and jz = − 21 (blue) curves are odd in ρ → −ρ. 62

1

fz = -32 , kz R = 2.66667 , H-L 1

1.5

Χ jz

Χ jz

0.6

ΡR

fz = 32 , kz R = 2.66667 , H-L 1

1.5

0.4

jz = -1 2 jz = -3 2

1

3.5. Results

fz = 12 , kz R = 0. , H+L 1

fz = 12 , kz R = 0.125 , H+L 1

1.5

1

1

0.5

0.5 Χ jz

Χ jz

1.5

0 -0.5

0 -0.5

jz = 3 2 jz = 1 2

-1 0

0.2

0.4

0.6

jz = -1 2 jz = -3 2

0.8

jz = 3 2 jz = 1 2

-1 1

0

0.2

ΡR

1.5

1

1

0.5

0.5 Χ jz

Χ jz

0.8

1

fz = 12 , kz R = 0.5 , H+L 1

1.5

0 -0.5

0 -0.5

jz = 3 2 jz = 1 2

-1 0

0.2

0.4

0.6

jz = -1 2 jz = -3 2

0.8

jz = 3 2 jz = 1 2

-1 1

0

0.2

ΡR

0.4

0.6

jz = -1 2 jz = -3 2

0.8

1

ΡR

fz = 12 , kz R = 0.75 , H+L 1

1.5

fz = 12 , kz R = 1. , H+L 1

1.5

1

1

0.5

0.5 Χ jz

Χ jz

0.6

ΡR

fz = 12 , kz R = 0.25 , H+L 1

0 -0.5

0 -0.5

jz = 3 2 jz = 1 2

-1 0

0.2

0.4

0.6

jz = -1 2 jz = -3 2

0.8

jz = 3 2 jz = 1 2

-1 1

0

0.2

ΡR

0.4

0.6

jz = -1 2 jz = -3 2

0.8

1

ΡR

fz = 12 , kz R = 1.25 , H+L 1

1.5

fz = 12 , kz R = 1.5 , H+L 1

1.5

1

1

0.5

0.5 Χ jz

Χ jz

0.4

jz = -1 2 jz = -3 2

0 -0.5

0 -0.5

jz = 3 2 jz = 1 2

-1 0

0.2

0.4

0.6

ΡR

jz = -1 2 jz = -3 2

0.8

jz = 3 2 jz = 1 2

-1 1

0

0.2

0.4

0.6

jz = -1 2 jz = -3 2

0.8

ΡR

Figure 3.3: Radial part of the |fz | = 12 , + (1) hole envelope wavefunctions for InAs. The value of kz R changes from 0 in the first picture to the maximum value 1.5 (at the end of the band) in the last graph.

This is a general property: at the wire zone center the total wavefunction consist only of either even or odd components, while away from kz = 0 also significant contributions with the other parity arise. 63

1


The variation of the wavefunction as a function of kz is illustrated in more (+) detail in Figure 3.3. It shows the radial part of the E 1 ,1 hole wavefunctions 2

for InAs for different values of kz , given at the top of each graph. (−) In Appendix A also the first odd solution, E 1 ,1 , for InAs is shown in 2

this way; the illustrations can be compared to the results in case of InP, Figure 10 and Figure 11. The following remarks are revealed by Figure 3.3 and the figures in Appendix A: • As noted before, at kz = 0 the total hole wavefunction is either even or odd under inversion ρ → −ρ. Remarkable is that the shape of the functions depends only a little on the choice of material. • Increasing kz slowly, the shapes of the different jz components change in a continuous way: an extra graph between the first and the second would give a result in between. • Comparing the results for InP and InAs, it can be seen that for a given band the altering of the hole wavefunctions by increasing kz is material dependent. Actually, the amount of change depends on the shape of the corresponding band: a smaller effective mass corresponds with a faster change in the hole wavefunctions around kz = 0, which can be checked for the present examples with the help of Table 3.3. • At the end of a particular band, one of the jz components disappears, so in general there are three or fewer jz components which contribute to the total wavefunction at the end of a band. The above results have some important consequences, in particular concerning the calculation of the absorption matrix elements over the entire band, see next chapter. Since the different jz components of the hole wavefunction change just slightly over a particular band, it suffices to choose a suitable small number of kz points by which the hole wavefunctions at neighboring points are approximated.

3.5.3

Band gap in III-V material nanowires

Finally, it is illustrative to estimate the effect of the infinite confinement in the present model by comparing the bulk band gap Eg with the confinement energy Evconf 1 →c1 of the fundamental transition v1 → c1 between the highest lying hole state v1 and lowest electron state c1 at the wire zone center. The values of effective mass of the Γ6 conduction band and the bulk band gap Eg for InP and InAs are given in Table 3.4. Using the values 64

3.5. Results

m∗c /m0 Eg (eV ) V0 (eV )

InP 0.0795 1.4236 4.28

InAs 0.026 0.417 4.93

Table 3.4: The band gap Eg , potential well V0 and effective masses of the Γ6 bulk conduction band for InP and InAs

of the effective mass, the first row in Table 3.5 gives the energy of the lowest conduction subband c1 at the wire zone center. The second row

Ec1 (eV ) Evconf 1 →e1 (eV )

InP

InAs

2.77 R−2 3.52 R−2

8.48 R−2 10.16 R−2

Table 3.5: The energy in eV at kz = 0 of the lowest conduction subband c1 and the confinement energy Evconf of the fundamental transition 1 →c1 v1 → c1 for InP and InAs in an infinite wire confinement, as derived with the model described in this paper. R denotes the wire radius.

in Table 3.5 shows the confinement energy Evconf 1 →c1 (R) of the fundamental conf transition v1 → c1 . Note that Ev1 →c1 (R) does not include the bulk Eg , it is defined as Evtrans (R) = Eg + Eci (R) − Evi (R) ≡ Eg + Evconf (R). i →cj i →cj

(3.70)

However, the assumption of an infinite potential well is too strong. The difference between the vacuum level and the conduction band edge (electron affinity) is in the order of electron volts for III-V materials and taking this finiteness into account leads to significant corrections, in particular for the conduction subbands[18][21]. Here the discussion will be restricted to correcting the conduction subband c1 for InP and InAs, given in Table 3.5, with a reduction factor due to the finite potential well. For the valence subbands it is expected that the correction is less crucial, in the first place because the correction is smaller for bulk bands with a higher effective mass. Another reason is that, apart from the extra energy difference by the band gap, the difference with the vacuum level becomes larger for deeper lying subbands, in contrast to the conduction band states. In the finite potential well model, the dependence on the wire radius becomes more complicated then the simple R−2 dependence in the infinite case. Actually, the electronic properties depend on the dimensionless 65


quantity lV

R lV

, where lV is defined as r ~2 = . 2m∗ V

(3.71)

For the conduction band, the potential V equals the vacuum level offset V0 which is given for InP and InAs in Table 3.4.

Ec1 (eV ) Evconf 1 →c1 (eV )

InP

InAs

0.07 0.11

0.13 0.20

Table 3.6: The energy in (eV ) at kz = 0 of the lowest conduction subband c1 and the confinement energy Evconf of the fundamental transi1 →c1 tion v1 → c1 for InP and InAs at R = 4.83 nm and R = 4.85 nm, respectively. These are the corrected results of Table 3.5: the conduction subbands are calculated in the finite potential wells given in Table 3.4.

The corresponding energies of the lowest conduction subband c1 in the finite wire configuration are shown in Table 3.6. These values of Ec1 can be compared to those for the infinite potential well case at R = 4.8 nm: 0.2 eV and 0.36 eV for InP and InAs, respectively. Similarly, for the confinement energy Evconf 1 →c1 , the values in the infinite potential well case are 0.15 eV and 0.43 eV for InP and InAs, respectively. The difference between the two models is larger for InAs due to the smaller effective mass of the conduction band.

66

Chapter 4

EM transition matrix In this chapter the EM matrix element for band-to-band transitions between the top Γ8 valence bands and the lowest lying Γ6 conduction band in III-V semiconductor nanowires will be derived. Section 4.1 contains some general theory, in section 4.2 the matrix element is developed further in the Bloch representation, section 4.3 gives explicit expressions for the band-band matrix elements and section 4.4 treats the selection rules on the intersubband transitions. Explicit results for InP and InAs are shown in section 4.5.

4.1 4.1.1

General theory Radiation matter interaction

In this paper the interaction between the external EM field and the electrons within the semiconductor system is described using a macroscopic, semiclassical approach. In this method the EM field is treated classically, while the semiconductor material is described quantum mechanically in the spirit of the previous chapter. Next to the assumption that this semi-classical picture approximates the more realistic QED model, it is also assumed that the semiconductor heterostructure can be described using macroscopic Maxwell equations, i.e. where the different parts in the system are characterized by macroscopically averaged quantities, such as the dielectric function. As will be discussed in more detail in Chapter 5, it is unclear if this macroscopic framework still holds if the size of the system is reduced to nanoscale, when the dimensions of the system are not large any more compared to the microscopic (atomic) distances. Then a microscopic semiclassical theory would be a more realistic approach [22]. However, proceeding with the macroscopic semiclassical approach, the gauge freedom in choosing the scalar potential φ and vector field A is used 67

Chapter 4. EM transition matrix

by taking the Coulomb gauge, ∇ · A = 0.

(4.1)

Then the transverse part of the electric field equals − 1c ∂A ∂t . Note that the longitudinal part −∇φ is absorbed in the matter part of the semiconductor Hamiltonian: the averaged potential V (r) in equation (3.1) contains the full Coulomb interaction between the particles. Further details are found in [22]. In this scheme the Hamiltonian representing the radiation-matter interaction is given by e X Hr−m = − A(r i ) · pi , (4.2) m0 c i

where i labels the N electrons in the material and m0 the free electron mass. Taking m0 instead of an effective mass m∗i is justified for interband transitions [15]. Furthermore, since the EM field is typically is small, the term of order |A|2 is neglected in (4.2).

4.1.2

EM transition matrix

Treating the time dependent EM interaction (4.2) as a small perturbation, it induces a transition between initial state |Ψi i and final state |Ψf i, which are eigenstates of the semiconductor system in the absence of the EM field. The probability that the unperturbed system |Ψi i transforms under the absorption/emission of light to |Ψf i is proportional to the transition matrix element Mf i = hΨf |Hr−m |Ψi i.

(4.3)

In case of a degenerate initial and/or final state, with |Ψim i and |Ψfn i the m- and n-fold degenerate initial and final states, this degeneracy is taken into account by X |Mf i |2 = |hΨfn |Hr−m |Ψim i|2 , (4.4) m,n

i.e. the different possible transitions at the same energy are summed as probabilities. If the degeneracy is lifted by a perturbation which breaks a particular symmetry, say an external magnetic field, then |Ψim i and |Ψim0 i are separated in energy and their matrix elements can be distinguished as belonging to different transitions |Mf i | and |Mf i0 |. In this paper the absorption process is considered, in the specific case of direct band-to-band transitions in nanowires. As initial state the many 68

4.1. General theory

body groundstate Ψ0 of the undoped semiconductor nanowire is taken, with all valence bands completely filled and the conduction bands empty: Ã ! Y Ψ0 (r 1 , r 2 , .., r N ) = A Ψλi kzi (r i ), , (4.5) i

where the product of one particle states is anti-symmetrized by the operator A and λi denote all valence bands. As final state an excited state is constructed by taking an electron out of one of the top most valence bands,   Y Ψexcited (r 1 , r 2 , .., r N ) = A Ψλj = c kzj (r j ) Ψλi kzi (r i ) , (4.6) i 6=j

where j = c indicates that the electron is placed into one of the empty conduction bands. In the remainder this notation will be used to distinguish wavefunctions belonging to conduction bands from valence band wavefunctions Ψλi kzi (r i ). One can proceed further by forming the final exciton state from a linear combination of the excited states (4.6). However, in this paper only the band-to-band transitions from the groundstate () to a particular excited state (4.6) are considered. By writing down the antisymmetrization operator P explicitly in terms of the permutation operator P , A = √1N ! P (−1)P P , the transition matrix (4.3) for these specific initial and final states becomes Z X e 1 X e P +P 0 hΨexcited | A(r i ) · pi |Ψ0 i = − Mc v = − (−1) dr 1 . . . dr N × m0 c m0 c N ! 0 i PP ´ ´P ³Q ³ Q 0 (r ) . A(r ) · p P Ψ (4.7) P Ψ∗λl = c kzl (r l ) k 6=l Ψ∗λk kzk (r k ) j i i i j λj kzj In order to simplify this expression, we recall the orthonormality relations Z dr Ψ∗λi = c kzi (r i ) Ψλj kzj (r j ) = 0, (4.8) Z dr Ψ∗λi kzi (r i ) Ψλj kzj (r j ) = δλi ,λj δkzi ,kzj , (4.9) where (4.8) is due to the orthogonality of the zone center atomic functions, s-like and p-like for conduction and valence band wavefunctions, respectively. Equation (4.9) is explained by the orthogonality of (3.27) and the normalization (3.40) of the total wavefunction. It can be shown [15] that, utilizing (4.8) and (4.9), the transition matrix element (4.7) simplifies to Z e Mc v = − dr Ψ∗λc kzc (r) A(r) · p Ψλv kzv (r), (4.10) m0 c 69


in other words, Mc v is the transition matrix for an electron in one particular state Ψλv kzv in a valence band λv which is excited to a conduction band state Ψλc kzc by the EM radiation. It is easy to check the simplification (4.10) of equation (4.7) if N = 2: from the 8 terms only two identical terms are nonzero and they cancel the 2! in the denominator.

4.2

Bloch representation

The transition matrix (4.10) is developed further by making use of either the Wannier or the Bloch representation for the wavefunctions. Here the Bloch scheme is chosen, for the practical reason that zone center Bloch functions are much better known. An other disadvantage of expanding in Wannier functions lies in the fact that Wannier functions are extended over a region larger then a unit cell, which makes it less defensible to approximate the EM field as constant over the relevant intervals of integration [18].

4.2.1

Total wavefunction in Bloch functions

Instead of expanding the wavefunction in zone center Bloch functions by applying ansatz (3.25), the Bloch representation is generalized by expanding the wavefunction in k space using the Fourier transform of the Wannier functions, equation (3.20). Defining the Fourier transform of the envelope function Fˆλkz ,j (k0 ) as 1 P 0 Fˆλkz ,j (k0 ) ≡ N − 2 R Fλkz ,j (R) e−ik ·R , the total wavefunction Ψλ kz (r) in this way ecomes X 0 1 XX Ψλ kz (r) = Fλkz ,j (R) ajR (r) = N − 2 Fλkz ,j (R) e−ik ·R ψnk0 (r) jR

= N

− 21

X jk

= N

− 21

0

X jk

jR k0

( N

− 12

X

Fλkz ,j (R) e

−ik0 ·R

) eik

0

·r

ujk0 (r)

R 0 Fˆλkz ,j (k0 ) eik ·r ujk0 (r),

(4.11)

0

where in the second line the normalized Bloch functions (3.3) are used. This expression is specified further in the infinite wire configuration by decomposing the envelope Fλ kz , j in its radial part and the plane wave along the cylinder axis, eikz Z Fλ kz , j (R) = χλ kz , j (R⊥ ) √ , M 70

(4.12)

4.2. Bloch representation

where the envelope is normalized in the Z direction by denoting the number of atoms in this direction with M . Note that the not normalized lateral part χλ kz , j (R⊥ ) in general depends on kz , as concluded in the previous section. Utilizing equation (4.12), the Fourier transform of the envelope function Fˆλkz ,j (k0 ) is decomposed as 1 Fˆλkz ,j (k0 ) = N − 2

=

X

0

χλ kz , j (R⊥ ) e−ik⊥ ·R⊥

R⊥ 0 ξλkz ,j (k⊥ ) δkz ,kz0 ,

X ei(kz −kz0 )Z √ M Z (4.13)

where µ 0 ξλkz ,j (k⊥ )

≡

N M

¶− 1 X 2

0

χλ kz , j (R⊥ ) e−ik⊥ ·R⊥

(4.14)

R⊥

is the Fourier transform of the radial part χλ kz , j (R⊥ ). Inserting (4.13) in equation (4.11), the total wavefunction expanded in 0 Bloch functions uj (k⊥ ,kz ) reads 0 1 X 0 0 Ψλ kz (r) = N − 2 ξλkz ,j (k⊥ ) ei(k⊥ ·r⊥ +kz z) uj (k⊥ (4.15) ,kz ) (r). 0 jk⊥

Within the framework developed in the previous chapter, this expression is valid around the zone center of any bulk band. This general form thus also describes the total wavefunction in a confined conduction band, where the sum over j usually disappears because this band is non degenerate in most of the III-V materials. Furthermore, equation (4.15) is also valid for an arbitrary strength of the confinement V0 (as in equation (3.37)).

4.2.2

EM transition matrix in Bloch functions

Utilizing the general expression (4.15) both for the valence and conduction band wavefunction, the transition matrix (4.10) is expanded in Bloch functions by Z e Mc v = − dr Ψλ∗c kzc (r) A(r) · p Ψλv kzv (r) m0 c e X X ∗ ξλc kzc ,jc (k0⊥c ) ξλv kzv ,jv (k0⊥v ) × = − (4.16) m0 c jc jv k0⊥c k0⊥v Z 0 0 1 dr e−i(k⊥c ·r⊥ +kzc z) ujc (k0⊥c ,kzc ) (r) A(r) · p ei(k⊥v ·r⊥ +kzv z) ujv (k0⊥v ,kzv ) (r), N where the labeling ⊥ indicates that the corresponding vector lies in the plane perpendicular to the wire axis, as in section 4.2.1. 71


For the moment, we narrow the focus to the last line in equation (4.16). The integral entire r can be split up into N integrals over a R over theP R space 0 unit cell, drf (r) = R Ω0 dr f (R + r 0 ). Using the commutation relation [p, eik·r ] = −~k eik·r , the third line in (4.16) becomes 1 X −i(k0⊥c −k0⊥v )·R⊥ −i(kzc −kzv )Z e e × (4.17) N Z R 0 0 dr e−i(k⊥c −k⊥v )·r⊥ e−i(kzc −kzv )z ujc (k0⊥c ,kzc ) (r) A(R + r) · (p + ~k) ujv (k0⊥v ,kzv ) (r). Ω0

Further progress is made by assuming the variation of the EM waves to be small over a unit cell, i.e. A(R + r) ' A(R),

(4.18)

This approximation is justified by the fact that the wavelength λ0 of the EM radiation typically is much larger then the interatomic distances a0 . As in the case of the effective mass approximation, where the restriction on δV (slowly varying over one unit cell) leads to a separation between atomic wavefunctions and ”macroscopic” envelope functions, the envelope and Bloch parts of the transition matrix elements will factor into separate integrals due to assumption (4.18). In order to show this explicitly, reconsider expression (4.17) and include the following: • Assuming the EM field constant over a unit cell leads to the elimination of the ~k term, because in this case the integral contains just two orthogonal Bloch functions. • In the case of an infinite cylinder the EM field is of the form A(R) = A(R⊥ )eiqz Z

(4.19)

To be more precise, in Section 1.3 it was derived that qz = −k0 sin θ, with θ the angle of incidence compared to a plane perpendicular to the wire axis. • It is more convenient to rewrite kc and kv in terms of the total momentum K of the system by defining K ≡ kc − kv ; k ≡ kv . 72

(4.20) (4.21)

4.2. Bloch representation

Now (4.17) is simplified by X 0 1 X A(R⊥ )e−iK ⊥ ·R⊥ e−i(Kz −qz )Z · (4.22) N Z R⊥ Z 0 dr e−iK ⊥ ·r⊥ e−iKz z ujc (K 0⊥ +k0⊥ ,Kz +kz ) (r) p ujv (k0⊥ ,kz ) (r), Ω0

With the notion that under assumption (4.18) qz is much smaller then a reciprocal lattice vector Giz and that Kz lies in the first Brillouin zone, the summation over M lattice points results in a momentum conservation relation in the Z direction: X e−i(Kz −qz )Z = M δqz ,Kz . (4.23) Z

This simplifies (4.22) further into Z 0 0 ˆ δqz ,Kz A(K ) · dr e−iK ⊥ ·r⊥ e−iKz z ujc (K 0⊥ +k0⊥ ,Kz +kz ) (r) p ujv (k0⊥ ,kz ) (r), ⊥ Ω0

ˆ where A(K ⊥ ) is defined as the Fourier transform of A(R⊥ ), MX ˆ A(R⊥ )e−iK ⊥ ·R⊥ . A(K ⊥) ≡ N

(4.24)

R⊥

Furthermore, since the envelope functions are also assumed to be slowly varying over a unit cell, now it is possible to factor the envelope and Bloch parts of the transition matrix elements into separate integrals. Any wavevector κ in (4.16) is much smaller then a reciprocal lattice vector Gi because it is assumed that R À a0 and κ ∼ R1 and Gi ∼ a10 . Under this condition 0 ujκ ∼ uj0 and e−iK ⊥ ·r⊥ e−iKz z ∼ 1. Inserting (4.24) into (4.16) and using P P 0 0 0 ˆ 0 ∗ ∗ ∗ R⊥ χλc kzc ,jc (R⊥ )A(R⊥ )χλv kzv ,jv (R⊥ ) k0 K 0 ξλc kzc ,jc (k⊥ + K ⊥ )A(K ⊥ )ξλv kzv ,jv (k⊥ ) = ⊥

⊥

the final form of the transition matrix element in Bloch functions is obtained: XX e χ∗λc kzc ,jc (R⊥ )A(R⊥ )χ∗λv kzv ,jv (R⊥ ) · Mc v = − δkzc ,kzv m0 c jc jv R⊥ Z dr ujc 0 (r) p ujv 0 (r). (4.25) Ω0

This result is also obtained by using ansatz (3.25), a simplified expansion of the wavefunction in Bloch states which is justified if the wire radius R is sufficiently large, i.e. the Fourier related k values are so small that the corresponding energy difference ²j (k) − ²j remains much smaller then the band edge differences ²j − ²n0 . It should be noted that the degeneracy of the hole and electron energy bands is not taken into account in (4.25). It can be included in the same way as by replacing (4.3) with (4.4), as will be done further on when the matrix elements are calculated explicitly.

73


4.3

Reformulation of transition matrix element

In this section the transition matrix element (4.25) is reconsidered by including the degeneracy of the conduction and valence subbands in case of transitions between the top Γ8 valence bands and the lowest lying Γ6 conduction band in III-V semiconductors. Apart from the trivial degeneracy in ± kz , the transitions are degenerate in the quantum numbers ± lz c and ± σ of the conduction subbands and in ± fz v of the valence subbands, see Chapter 3. The EM field is specified further by considering two approximations with respect to the wire radius as derived in Part I. First, the spatial variation of the EM field across the wire diameter is entirely neglected, i.e. the transition matrix element is formulated in the dipole limit, where λ0 À R so A(R⊥ ) ' A. Secondly, a spatial variation of the EM field is taken into account by applying the scattering fields (2.8), (2.9) and (2.10), which are expansions up to second order in 2π λ0 R for normal incident light. Recall that the Coulomb gauge is chosen by deriving the transition matrix element, so the vector potential is related to the transverse electric field by E = − 1c ∂A ∂t . In case of absorption this yields E = −

iω A, c

(4.26)

with ω the frequency of the EM field.

4.3.1

EM field in dipole approximation

In Part I it was shown that the strength of the EM field inside the wire depends on the polarization of the incident light. In the dipole approximation this resulted in (2.6) and (2.7) for polarization parallel and perpendicular to the wire axis, respectively. In order to separate this polarization anisotropy from the dielectric mismatch, a matrix element Tc v is defined by |Mc v (kz )|2 ≡ (

e 2 E 2 ) | | |Tc v (kz )|2 , m0 ω 2

(4.27)

where kz = kzc = kzv and E is the strength of the electric field inside the wire in the dipole approximation, E ≡ E εˆ. Including the degeneracy, from equation (4.4) and (4.25) one obtains for the transition matrix Tc v between valence and conduction subband v and c: ¯ ¯2 ¯ X ¯¯X ¯ 2 3 ¯ ¯ , (4.28) hχ |χ (k ) ihSσ| ε ˆ · p | |Tc v (kz )| = j i z z l f ;j zc zv z 2 ¯ ¯ ¯ ¯ jz σ ld f d zc zv

74

4.3. Reformulation of transition matrix element

d denotes l where lzc zc = {|lzc | , −|lzc |}, the degeneracy at a given |lzc |. A sid . The notation of the atomic part is explained milar definition applies to fzv R ∗ in (4.34). Furthermore, hχlzc |χfzv ;jz (kz ) i = P dR⊥ χlzc (R⊥ )χfzv kz ;jz (R⊥ ), where the replacement of the summation R⊥ by an integral is justified since in the effective mass approximation the dimensions of the wire are assumed to be much larger then interatomic distances. The additional step size by going from summation to integration is eliminated by the normalization of the wavefunctions. Note that indeed the strength of the internal field is separated from Tcv . This classical penetration effect is absorbed in E, as shown in equation (4.28).

4.3.2

EM field including Mie scattering

In case of a spatially varying EM field the separation in (4.27) is not possible any more. Instead, the EM field has to be integrated between the conduction and valence subband envelope functions over the wire cross section area πR2 , as shown in (4.25). However, from equations (2.8), (2.9) and (2.10) one finds that the reduction factor caused by the penetration into the wire still can be divided out, just as in the dipole limit. Thus, by separating out the factor E = E0 in case of polarization parallel to the wire axis and 2 E0 at perpendicular polarization, the matrix element Tc v contains E = 1+ε only the scattering (optical focusing) and expansion terms due to the wave behavior of the EM field inside the wire. Denoting E 0 (R⊥ ) as the electric field without the penetration strength, E E 0 (R⊥ ) ≡ E(R⊥ ), the right hand side of (4.28) now is replaced with ¯2 ¯ ¯ ¯ X ¯X ¯ 2 0 3 ¯ hχl ,n |E |χf ,n;j (kz ) i · hSσ| p | jz i¯ (4.29) |Tc v (kz ; ε, R)| = zc zv z 2 ¯ ¯ . ¯ ¯ d d jz σ l ,f zc

zv

Note that, contrary to the dipole limit, the direction of the internal EM field is in general different from that of the incident field . Furthermore, due to the scattering field the matrix elements now depend on the dielectric function and the wire radius.

4.3.3

Polarization anisotropy of the transition matrix

At this point it is instructive to define a polarization anisotropy purely originating from the matrix elements. As demonstrated by equation (4.28), in the dipole limit it is possible to separate the anisotropy caused by the dielectric mismatch from that which is due to the polarization in the transition matrix. In other words, in principle one is able to determine the polarization anisotropy caused by the transition matrix elements alone once it is (experimentally) possible to eliminate the polarization anisotropy of the 75


dielectric mismatch, for instance by a sufficient increase of the intensity of the incident field in the perpendicular case or by changing the surrounding (for instance if the nanowire is covered by an oxide). In analogy with (2.21), the polarization anisosotropy ρTcv of the matrix element alone is defined as ρTcv

≡

|Tcv, k |2 − |Tcv, ⊥ |2 , |Tcv, k |2 + |Tcv, ⊥ |2

(4.30)

where Tcv, k = Tcv, z is the matrix element corresponding to a polarization parallel to the wire and Tcv, ⊥ denotes the perpendicular case.

4.4

Selection rules

The interband transition matrix (4.25) is investigated in more detail by considering the different kind of selection rules it imposes. A selection rule originates from an underlying symmetry of the system under consideration and generally disappears if the symmetry on which it relies is broken. By a selection rule some transitions are ”selected” to be allowed, while others are said to be forbidden. As stated in section 4.3, a spatially varying EM field has to be treated differently then the more common dipole approximation. The dipole approximation is crucial for the selection rules originating from the envelope part of the transition matrix element (4.25). Away from this limit the variation of the EM field starts to break the symmetry of the matrix element between the envelope parts of the electron and hole wavefunctions. On the other hand, the selection rules originating from the atomic like matrix element of the momentum operator p are independent of E, provided that the field can be considered as constant over a unit cell, an approximation which was made earlier. Leaving the discussion of a spatially varying EM field to paragraph 4.5.2, the selection rules in the dipole approximation fall into three different classes: • Polarization selection rules • Selection rules on the lz -angular momentum of the envelope wavefunction • Parity selection rules. The polarization rules are caused by the atomic like matrix element of the momentum operator p, while the selection rules on the lz -angular momentum of the envelope wavefunction and the related parity selection rules are 76

4.4. Selection rules

due to the envelope part of the transition matrix (4.25). Explicit investigation of the transition matrix requires a restriction to a more specific case. The selection rules depend on which particular system is considered and subsequently which symmetry properties are valid. Therefore the focus will be on the band-to-band transitions between the top most valence bands and the lowest lying conduction band in III-V materials. In this case the matrix element is given by (4.28), or by (4.29) in case of a spatially varying EM field.

4.4.1

Polarization selection rules

In this paragraph the polarization selection rules in case of transitions between the top Γ8 valence bands and the lowest lying Γ6 conduction band in III-V semiconductors are derived. However, since the theory strongly relies on the results of atomic physics, it is instructive to summarize these shortly in advance. Suppose an atomic system which is built from the orthonormal base {|η j mi}, where the quantum numbers j and m come from an angular momentum operator J and η refers to other possible quantum numbers which complete the basis of the system in consideration. Then for any vector operator V applies hη j m0 |V+ |η j mi = 0

if m0 − m 6= 1,

hη j m0 |V− |η j mi = 0

if m0 − m 6= −1,

hη j m0 |Vz |η j mi = 0

if m0 − m 6= 0,

(4.31)

where V+ ≡ Vx + iVy and V− ≡ Vx − iVy , as usual. In case of the absorption of light, where p is the vector operator of interest, this result gets its physical interpretation if one realizes that a photon carries spin 1, so m = {1, 0, −1}: in this case (4.31) is just a consequence of the conservation of angular momentum. Turning to the optical transitions in III-V semiconductors, remember that an analogy was made between the band edge Bloch states and atomic functions, see paragraph 3.1.1. Now it becomes more clear what is actually meant with ”atomic-like”: in the k · p method the optical matrix elements are used as input. Without specifying the precise form of a particular band edge Bloch function, it can be argued that its symmetry properties are the same as a particular atomic function. All what remains is to determine (experimentally) the optical matrix elements. Concerning transitions between the Γ8 valence bands and the Γ6 con77


duction band, the only nonzero matrix elements are given by −

i i i hS|px |Xi = − hS|py |Y i = − hS|pz |Zi = P, m0 m0 m0

(4.32)

where |Si denotes the band edge Bloch function of the conduction band which is s-like. The magnitude P of the matrix elements in (4.32) is related to the Kane matrix element Ep by EP

= 2m0 P 2 ,

(4.33)

which can be determined experimentally for each particular III-V bulk semiconductor material. As an upshot, the polarization selection rules in semiconductor materials result from restrictions imposed on the matrix element of the momentum operator by the symmetry properties of the atomic-like Bloch states. Geˆ 2 i is nerally, from group theory it is known that a matrix element hψ1 |O|ψ only nonzero if the symmetry S1 of ψ1 is the same as one of the irreducible representations of the direct product O ⊗ S2 , where O denotes the symmeˆ Indeed, analyzing the symmetry properties of the try of the operator O. momentum operator, one finds that p-like and s-like states lead to the only nonzero matrix elements given in (4.32)[14]. Narrowing the focus to band-to-band transitions between the top Γ8 valence bands and the lowest lying Γ6 conduction band in III-V semiconductors, the the atomic part of the transition matrix (4.25) is specified further by Z (4.34) dr ujc 0 (r) p ujv 0 (r) = hS σ | p | 32 jz i, Ω0

again with σ = ↑, ↓ and jz ² { 32 , 12 , − 12 , − 32 }. This matrix can be calculated in terms of the only nonzero matrix elements (4.32) by using the decomposition of | 23 jz i in the states |Xi,|Y i and |Zi, as given in (3.13). Table 4.1 shows the result for the unpolarized interband matrix element hS σ | px + py + pz | 32 jz i. Here the matrix elements Pu are introduced for convenience, hS|pu |U i ≡ Pu ,

u = {x, y, z}, U = {X, Y, Z}.

(4.35)

In terms of the Kane matrix elements, equations (4.32) and (4.33), Table 4.2 gives the quantitative results of the polarization dependence of the 1 3 matrix element 1 hS σ | pu | 2 jz i. This result plays a dominant role (m0 Ep ) 2

by determining the polarization anisotropy in the EM transition matrix elements, see paragraph 4.3.3. 78


| 32

px + py + pz

hS ↑ |

3 2

− √12 Px −

hS ↓ |

| 32

i

1 2

| 32 − 21 i

i

√ √2 P z 3

√i Py 2

− √16 Px −

0

√1 Px 6 √i Py 6

−

| 23 − 32 i

√i Py 6

√ √2 Pz 3

0 √1 Px 2

−

√i Py 2

Table 4.1: Result for the unpolarized interband matrix elements hS σ | px + py + pz | 32 jz i in terms of Pu ≡ hS|pu |U i. For a particular transition, the spin σ of the conduction band electron is shown in the left column and the Bloch angular momentum jz belonging to the valence band in the first row.

P For instance, Table 4.2 clearly demonstrates that the ratio of σ |hS σ | pz | 32 jz = P ± 12 i|2 and σ |hS σ | px | 23 jz = ± 12 i|2 equals 4. The polarization selection rule has an even stronger effect on the valence subband states which are dominated by terms with jz = ± 32 : in this case the matrix element of pz is strictly zero. A strictly zero matrix element of a particular transition and direction of the momentum operator is said to be polarization forbidden, or polF in short notation. This qualitative result is summarized in Table 4.3, which shows the allowed polarizations. Here x, y denotes the allowed polarizations perpendicular to the wire axis and z the allowed parallel 1

1

(m0 Ep ) 2

hS ↑ |

hS ↓ |

pu

| 32

3 2

i

| 32

1 2

i

| 32 − 12 i

| 23 − 23 i

px

− 2i

0

i √ 2 3

0

py

1 2

0

1 √ 2 3

0

pz

0

√i 3

0

0

px

0

− 2√i 3

0

i 2

py

0

1 √ 2 3

0

1 2

pz

0

0

√i 3

0

Table 4.2: Selection rules on the atomic-like interband matrix elements 1 3 1 hS σ | pu | 2 jz i. (m0 Ep ) 2

79


polarization. Another feature which will be extracted from Table 4.2 further εˆ · p

| 32

3 2

i

| 32

1 2

i

| 32 − 12 i

| 23 − 23 i

hS ↑ |

x, y

z

x, y

polF

hS ↓ |

polF

x, y

z

x, y

Table 4.3: Selection rules on the atomic-like interband matrix elements hS σ | ε · p | 32 jz i. The allowed polarizations are denoted with x, y and z, where x, y are perpendicular to the wire axis and z gives the parallel polarization. Polarization forbidden transitions are denoted with polF .

on is that the matrix elements corresponding to polarizations perpendicular to the wire axis are the same, i.e. |Tcv, x |2 = |Tcv, y |2 , which is a consequence of the rotational symmetry around the wire axis in the dipole approximation. Finally, it is instructive to come back to the selection rules in the atomic case. In fact, the results in Table 4.2 lead to the same selection rules as given in (4.31). For this purpose, first note that Table 4.2 can be formulated in an algabraic way by r 1 1 hSσ|px | 32 jz i = i(m0 Ep ) 2 |jz | (δσ−jz ,1 − δσ−jz ,−1 ), 6 r 1 1 hSσ|py | 23 jz i = (m0 Ep ) 2 |jz | (δσ−jz ,1 + δσ−jz ,−1 ), 6 r 1 2 3 hSσ|pz | 2 jz i = i(m0 Ep ) 2 |jz | δσ,jz . (4.36) 3 In terms of p+ ≡ px + ipy and p− ≡ px − ipy the matrix elements of px and py yield r 1 2 hSσ|p+ | 23 jz i = i(m0 Ep ) 2 |jz | δσ−jz ,1 , 3 r 1 2 hSσ|p− | 23 jz i = (m0 Ep ) 2 |jz | δσ−jz ,−1 . (4.37) 3 Together with the matrix element of pz in (4.36), these are the same selection rules as in the atomic case, now originating from the conservation of angular momentum on the Bloch part of the transition matrix.

4.4.2

Selection rules on the envelope wavefunctions

The selection rules on the envelope part of the transition matrix are not as general as the polarization selection rules derived in paragraph 4.4.1. While 80


the polarization selection rules originate from the bulk, atomic like matrix element of the momentum operator, the selection rules on the envelope part are dependent on the configuration of the system (wire radius, length) and also depend on whether the EM field can be considered in the dipole limit. T ransition

P olarization

Class

|jz |

C0, 1 → E 1 ,n

x, y, z

MP

1 2

C0, 1 → E 3 ,n

x, y

SP

3 2

C0, 1 → E 5 ,n

−

lF

−

C1, 1 → E 1 ,n

x, y, z

MP

1 2

C1, 1 → E 3 ,n

x, y, z

MP

1 2

C1, 1 → E 5 ,n

x, y

SP

3 2

2

2

2

2

2

2

Table 4.4: Summary of the polarization and class for the lowest band-toband transitions in C∞ nanowires with a constant EM field. Polarization perpendicular to the wire axis is denoted with x, y, parallel polarization with z. The envelope angular momentum forbidden transitions are denoted with lF while SP and M P denote the polarization class, single and mixed polarization respectively. The last column gives the allowed values of |jz |.

Proceeding with the transitions between the Γ8 valence bands and the Γ6 conduction band in III-V semiconductors in the dipole limit, equation (4.28), the φ part of hχlzc ,n |χfzv ,n;jz (kz ) i gives 1 2π

Z 0

2π

dφ e−i(lzc −(fzv −jz ))φ = δ lzc ,fzv −jz ,

(4.38)

which can be considered as a selection rule on the envelope angular momentum, since lzv = fzv − jz . As a direct consequence, transitions for which lzc − fzv 6= {− 32 , − 12 , 12 , 32 } are l-angular momentum forbidden (lF ). This result can be found in Table 4.4, which shows a combination of the polarization and l-angular momentum selection rules. The last row gives the allowed values of |jz |.

81


The constraint (4.38) also leads to a selection rule which is a consequence of the parity of the subbands at the wire zone center. An even (odd) wavefunction corresponds to an even (odd) Bessel function Jlz and by (4.38) lzc and lzv = fzv − jz has to be the same, i.e. the parity of the valence subband wavefunction has to match with the parity of the conduction subband. In other words, transitions E (±) → C (∓)

(4.39)

are parity forbidden (pF) at kz = 0. Away from the zone center this selection rule generally is broken: the valence subband wavefunctions are not characterized by parity any more.

82

4.5. Results

4.5

Results

In this section the above theoretical framework is applied to specific examples. Again InP and InAs are chosen since those III-V materials are two kind of extremes regarding their electronic properties. As a matter of fact, although this is also the case for the Kane matrix element Ep , the effect of this difference will be small since Ep is hardly material dependent, see Table 4.5.

Ep (eV )

InP 20.7

InAs 21.5

Table 4.5: The Kane matrix element Ep for InP and InAs

Since a spatially varying EM field requires a different approach compared to the theory derived in the dipole limit, this case is treated separately in paragraph 4.3.2.

4.5.1

Dipole approximation

The topmost illustration in Figure 4.1 (a) and Table 4.6 show the nume-

ÈΡ T v c È

v7 -> c1

v6 -> c1

v4 -> c1 v5 -> c1

v7 -> c1

v3 -> c1

0.80

v2 -> c1

1.00

0.80

v1 -> c1

1.00

v4 -> c1

0.65 0.68 0.72 0.75 0.78 0.82 0.62 0.65 0.68 0.72 0.75 0.78 0.82 0.14 aL : kzR = 0. bL : kzR = 0.45 0.12 0.10 0.08 z - pol. 0.06 y - pol. 0.04 0.02

v3 -> c1

0.14 0.12 0.10 0.08 0.06 0.04 0.02

v2 -> c1

ÈTv c È2 Harb. unitsL

ETrans HeVL, R = 4.85 nm

0.60

0.60 Ρ>0 Ρ c1

z - pol. y - pol.

v5 -> c1 v6 -> c1

aL : kzR = 0.

1.6

v4 -> c1 v5 -> c1 v6 -> c1

1.58

v1 -> c1 v3 -> c1 v2 -> c1

1.55

v3 -> c1

1.53 0.14 0.12 0.10 0.08 0.06 0.04 0.02

v1 -> c1



0.14 0.12 0.10 0.08 0.06 0.04 0.02

1.00

1.00

0.80

0.80

0.60

0.60 Ρ>0 Ρ c1

v6 -> c1

v4 -> c1 v5 -> c1

v3 -> c1

v1 -> c1

v2 -> c1

v7 -> c1

v6 -> c1

v4 -> c1 v5 -> c1

v3 -> c1

v2 -> c1

0.62 0.65 0.68 0.72 0.75 0.78 0.82 0.62 0.65 0.68 0.72 0.75 0.78 0.82 0.14 0.14 aL : kzR = 0. bL : kzR = 0.45 0.12 0.12 z - pol. 0.10 0.10 y - pol. 0.08 0.08 0.06 0.06 0.04 0.04 0.02 0.02

v1 -> c1



1.00 ÈΡ T v c È

0.80 0.60

1.00 Ρ>0 Ρ

Optical properties of cylindrical nanowires

Optical properties of cylindrical nanowires

Suggest Documents

Optical properties of cylindrical nanowires

The Optical Properties of Bismuth Nanowires

Elastic Properties of Nanowires

Study of Electronic and Optical Properties of CuInSe2 Nanowires

Magnetization Ratchet in Cylindrical Nanowires

Structure and optical properties of ZnO nanowires fabricated by pulsed

Optical properties of serrated GaN nanowires - OSA Publishing

Optical Properties of Nanoparticles and Nanowires - OhioLINK ETD

Magnetization Ratchet in Cylindrical Nanowires

Optical Investigation of ZnO Nanowires

Phase Transition and Optical Properties for Ultrathin KNbO3 Nanowires

The Mechanical Properties of Nanowires

Spatial distribution of spin-wave modes in cylindrical nanowires of ...

Optical vibrational modes of Ge nanowires: A

Endface reflectivities of optical nanowires - OSA Publishing

Optical characterization of wurtzite gallium nitride nanowires

Optical Anisotropy of Semiconductor Nanowires - Springer Link

Probing domain walls in cylindrical magnetic nanowires with electron ...

Electrical Properties of Cu Nanowires - CiteSeerX

Magnetotransport properties of individual InAs nanowires

Novel Structures and Properties of Gold Nanowires

Quantum transport properties of ultrathin silver nanowires

Electrical and mechanical properties of metallic nanowires ...

Electronic Properties of Semiconductor Nanowires - Semantic Scholar