ivchenko_bookreg

3.3 Electro-Optics of Nanostructures

151

Fig. 3.11. Sketches of the conductionand valence-band potential profiles for the GaAs/AlxGa1−xAs SL under a small (a), moderate (b), and high (c) electric field. From [3.62].

Bloch oscillations is another manifestation of the electric-field e ect on free charges where observable values are measured as a function of time, see [3.63] and references therein. This manifestation can be better visualized if one uses the gauge A = (0, 0, −F ct), where c is the light velocity, and considers nonstationary solutions of the Schr¨odinger equation

	i	∂ψ	= E K		+	|e|Az	ψ = E K	z −	|e|F t	ψ ,
		∂t		z		c

where the miniband dispersion E(Kz ) is a periodic function of the electron wave vector in a SL, see (3.199). The solution can be written as

152 3 Resonant Light Reflection and Transmission

ψ(t) = c exp −			E Kz − | |					dt .
	i	t				e F t
		0
The electron motion is periodic with the period
T		=	2π			.
			d		|eF |

Since the electron velocity is v = −1 ∂E/∂Kz = −(|e|F )−1 ∂E/∂t, the total distance passed from the minimum to maximum (or from left to right) at Kz = 0, F > 0 is

L = |e1|F		T	/2	∂t dt =	|eF |
		0
				∂E	∆

which coincides with the Wannier-Stark localization length defined by (3.207). Obviously, the Bloch oscillations can be also understood as quantum beats in a system of split levels (Chap. 5).

In addition to the clear physical interpretation, the tight-binding approximation can be used as an e ective tool in the quantum-mechanical engineering. The first example can be the Stark ladder states in a semi-infinite SL. In this case n = 0, 1, 2... and the boundary condition is C−1 = 0 because QWs with n = −1, −2... are taken away. The solution for Cn can be presented as Cn = D1Jn−λ(x). In order to have nontrivial solutions the following equation for λ needs to be satisfied

J−1−λ(x) = 0 .

We showed above that, for an infinite regular SL, the numbers λ ≡ n0 are integers and the Stark ladder is equidistant. For the semi-infinite SL, the values of λ di er from integers and the set of the first few levels is not equidistant.

Two other examples are a semi-infinite SL with a wider terminating QW [3.64, 3.65] and a non-periodic finite SL with a miniband formed at a certain electric field Fcr [3.66, 3.67]. In the latter structure the series an, bn of the

successive layer thicknesses (n = 1, 2 · · · N ) satisfy the relations
En0+1 = En0 − |e|Fcr	2	+ bn , In,n+1 = I ,			(3.208)
	an + an+1
where E0 is the diagonal energy in the n-th QW, I			n,n+1	is the transfer integral
n

between n-th and n + 1-th QWs. The values of a1 (or E10), Fcr and transfer integral I are chosen arbitrarily while other thicknesses (b1, a2, b2, · · · ) are found recursively from the above equations. At F = Fcr, all single-QW levels En = En0 + |e|F zn, are equalized (zn is the center of the n-th QW). Since all transfer integrals are chosen to be identical, the miniband of discrete levels,

E(k) = E10 + 2I cos	πm	, m = 1, 2	· · · N,	(3.209)
	N + 1

is formed just as in the case of a finite ideal SL in zero electric field.

3.3 Electro-Optics of Nanostructures

153

3.3.3 Quantum-Confined Pockels E ect

The electric field E [001] applied to a bulk zinc-blende semiconductor results in a linear-in-field birefringence with the principal axes of the dielectric

¯

permittivity tensor oriented along the three directions [110], [110], and [001]. This is the well-known Pockels e ect. Two mechanisms of the Pockels e ect in QW structures were discussed in [3.68]. One of them involves uniaxial strain uxy E (x [100], y [010]) induced by the electric field (piezoelectric e ect), and the mixing of heavyand light-hole states induced by this strain, i.e., a contribution to function G(z) in (3.116) and to the overlap integral I2 proportional to uxy . The second mechanism is purely electronic in nature, namely, one has to take into account that the interband matrix element of the momentum operator c, s, k|e · p|v, m, k in a semiconductor of the Td class contains terms linear in the electron wave vector k. For k z we have

c, 1/2, kz |e · p|v, 3/2, kz = −P e+ − Qkz e− ,	(3.210)
c, −1/2, kz |e · p|v, −3/2, kz = P e− − Qkz e+ ,

where e is the light-polarization unit vector, e± = ex ± iey , and P and Q are constants. When calculating the optical anisotropy by the envelope-function method, kz has to be replaced by the operator −id/dz. In the lowest order in the small parameters I2/I1 and Q/(aP ) we obtain instead of (3.122)

2 I2					2Q I3				(3.211)
ρ = √				−				,
			I1		aP		I1
	3

where I3 = a dz ϕe1(z) dF/dz, and the factor a is introduced for the sake of convenience so that the quantities I1, I2, I3 have the same dimension. In

L,R

an electric field, the envelope F (z) becomes asymmetrical even for tl-h = 0, and the integral I3 is nonzero.

Besides the above two bulk mechanisms, in QWs with tLl-h, tRl-h = 0, there is an additional mechanism of the Pockels e ect, which involves the interfaces [3.69]. Indeed, an electric field changes the values of the envelope function G(z) at the interfaces, and, as follows from (3.117), this results in a change in (3.116) of the extent of admixture of the |Γ8, 1/2 to |Γ8, ±3/2 states, and, hence, to a change in the overlap integral I2.

We have been assuming until now that the potential barriers at the C A /CA and CA/C A interfaces are the same. But in a general case for C=C and A=A the e ective dipole moments corresponding to the C-A and C -A bonds are di erent, so that the di erence between the band o sets VL and VR at the leftand right-hand interfaces may be as high as 50–100 meV [3.43]. At equilibrium, this di erence generates a built-in electric field which equalizes the electrochemical potential and leads to an electric-field- induced contribution to the anisotropy. Application of a positive or negative external field F increases or compensates the built-in field and the dependence of ρ on F becomes asymmetric [3.45].

154 3 Resonant Light Reflection and Transmission

It is known that in zinc-blende lattice heterostructures, at k = 0, the second heavy-hole hh2 and the lowest light-hole lh1 subbands at k = 0 lie close to each other and far from other quantum-confined states crossing at a particular thickness acr. Therefore even a small mixing perturbation can result in a remarkable modification of the interband optical matrix elements from these two valence subbands. We first consider this particular mixing in the absence of an external electric field. If the mixing is neglected, only one of the envelope functions ϕm in (3.114) is nonzero for the states (hh2, ±3/2) and (lh1, ±1/2) at k = 0. It is given by

ϕhh2(z) = Ch	± sin φh exp −æh |z| −
	sin khz ,

for the states (hh2, ±3/2), and

			cos φl exp æl	z 2
	ϕlh1	(z) = Cl	cos klz ,		a
			−	| | −

a	if	z	a ,
		| | ≤	a
	, if		2
2		|z| ≥ 2

if |z| ≤ a2 ,, if |z| ≥ a2

for the states (lh1, ±1/2). According to (2.15), the normalization coe cients are

Ch =

a

1 −

sin kha

1

cos k

a

−1/2

+

− h

,

2

kha

2æha

Cl =

a

1 +

sin k

a

1 + cos k a

−1/2

l

+

l

,

2

kla

2æla

φh = kha/2, φl = kla/2 and kh, kl, æh, æl are expressed via the hole quantumconfinement energies Ehh0 2, Elh0 1 and satisfy the standard equations

cot φh = −	mhhA	æh		, tan φl =	mlhA	æl		,
	mhhB		kh		mlhB		kl

see (2.14) and (2.12), respectively. The Γ8-hole Hamiltonian contains a specific contribution

2			{JxJy }s	! tl-h	δ(z − zR) − tl-h δ(z − zL)",	(3.212)
Hl-h = √3m0a0
				R	L

describing the heavy-light hole mixing. Then the coupling matrix element is derived as

2		hh2, ±3/2|Hl-h|lh1, 1/2 =		(3.213)
	L	R
	tl-h	ϕhh2(zL)ϕlh1(zL) − tl-h	ϕhh2(zR)ϕlh1	(zR) ,
2m0a0

where zL,R is the coordinate of the left or right interface. In the two-level approximation, the energies of the mixed states labelled as h+ and h− are given by

				3.3		Electro-Optics of Nanostructures					155
	Eh± =		1	Ehh0		2 + Elh0 1			± W ,		(3.214)
where			2
						1
	W = ∆¯2 + V¯ 2 , ∆¯ =							Ehh0 2 − Elh0 1
							2

¯

and V is the modulus of the coupling matrix element (3.213).

Since at zero electric field the optical transitions hh2 → e1 are forbidden the matrix element Me1,h± is proportional to the admixture of the lh1 state in the h± hybrid. It follows then that the relative squared matrix element Q(e1– h±) = |Me1,h± /Me1,lh1|2 can be written as

		¯				¯
Q(e1– h	) =	W + ∆	, Q(e1– h	) =	W −	∆	.	(3.215)
		2W			2W
−			+

Thus, in the vicinity of the crossing point the both transitions are allowed and the sum of Q(e1– h−) and Q(e1– h+) should be constant and equal to unity. However, in the considered two-band approximation and at zero electric field the both transition rates are uniaxially symmetrical.

Winkler [3.70] applied a multiband k · p Hamiltonian and reproduced the essential details of experimental absorption spectra. However a weak electric field in the growth direction was assumed in order to make e1- hh2(1s) excitons dipole-allowed and describe the e1-hh2 spectral peak observed by Reynolds et al. [3.71]. The above considerations remove the restrictions [3.72, 3.73] imposed on the selection rules for the exciton angular momentum. In particular, due to the hh2-lh1 mixing not only 2p but also 1s excitons e1-hh2 or e2-lh1 become dipole active and can contribute to the optical spectra [3.71, 3.74] even in the absence of electric fields. It should be mentioned that Schulman and Chang [3.75, 3.76] solved the tight-binding model and reported for the first time the mixing of heavyand light-hole states with k = 0 in the (001)-grown zinc-blende-lattice heterostructures.

In the presence of an electric field the envelopes ϕm(z) lose particular parity and both the hh2 → e1 and lh1 → e1 transitions from the unmixed valence states are allowed. In this case the mixing due to the Hamiltonian (3.212) leads to an in-plane anisotropy of the optical transitions from the split states h±. This is another manifestation of the quantum-confined Pockels e ect for resonating transitions e1-hh2 and e1-lh1. The above band-to-band description gives a reasonable understanding of the experimental observations, but misses the role of excitons which obviously a ect the shape of absorption spectra in high-quality samples. In particular, a significant discrepancy between band-to-band calculations and experiment was systematically observed in the vicinity of the nearly degenerated e1-lh1 and e1-hh2 transitions to the e1 conduction-electron subband [3.77]. It is shown in [3.69] that, for the resonating hh2 and lh1 subbands, the related excitonic states are also strongly coupled by the interface Hamiltonian Hl-h, which explains the discrepancy. A theoretical model developed in [3.69] in the perturbative scheme, see (3.212, 3.213), shows quantitative agreement with experimental results.

156 3 Resonant Light Reflection and Transmission

e1-hh1

e1-lh1

Fig. 3.12. Polarization-resolved absorption spectra of InxGa1−xAs/InP MQWs

˚

(a = 100 A) measured for di erent electric fields at 77 K. The incident light is

¯

polarized along the [110] (solid line) and [110] (dashed line) eigenaxes of the sample. Besides the usual features of the quantum confined Stark e ect (red-shift of the e1-hh1 transition, redistribution of the total oscillator strength), a large optical anisotropy develops in the e1-lh1 region. At high fields, one can distinguish a doublet of excitonic transitions, split by 6 meV and almost 100 % polarized. From [3.69].

3.3 Electro-Optics of Nanostructures 157
Absorption spectra of In Ga	˚	˚
	As(100 A)/InP(80	A) MQWs with x =
x 1−x		¯

0.83 measured for linear polarizations parallel to [110] and [110] directions and various applied voltages are displayed in Fig. 3.12. The most obvious part of the experimental results displayed in Fig. 3.12 is the classical scenario of the quantum-confined Stark e ect: redshift of the absorption edge, associated with decrease of its absorption strength, and appearance of the e1-hh2 transition which was parity forbidden at zero field. The red-shifts and weak broadening of the transitions when increasing the bias show that the field is quite homogeneous through the MQWs. The second e ect which can be observed in Fig. 3.12 is the variation of polarization properties related to the quantum-confined Pockels e ect . One can see that the e1-hh1 exciton is also weakly polarized at zero field and gains some polarization (up to 10 %) at large field. A strong polarization feature develops in the region of the e1- hh2 and e1-lh1 transitions, revealing two optical resonant transitions with orthogonal linear polarizations.

Figures 3.13(a-d) are confined to on the region of e1-hh2 and e1-lh1 excitons at fields larger than 19 kV/cm, where the quantum-confined Pockels e ect dominates over the zero-field contributions to the in-plane anisotropy. There are two observations allowing one to adjust the theoretical simulation to the experimental spectra. Firstly, the excitonic peak intensities are clearly redistributed as the electric field increases, so that the intensities of the two peaks equalize at 30 kV/cm. Secondly, the energy distance between the peaks decreases with increasing the field. According to (3.215) the oscillator strengths of the split excitonic transitions are equal at the field matching the crossing of the unmixed e1-hh2 and e1-lh1 excitonic levels. The respective transition energies are plotted in Fig. 3.13e by dotted curves. The theoretical absorption spectra were calculated as a sum of both excitonic and continuum contributions related to three transitions to the conduction subband e1 from the lowest valence subband hh1 and two higher valence subbands originating from the mixing of lh1 and hh2 states. The inhomogeneous spectral broadening was introduced by means of a Gaussian distribution function with the broadening factor taken the same for both of the split polarized excitonic transitions and their respective continuum contributions. As seen in Fig. 3.13, a satisfactory fit of the spectral shape is achieved, especially at large fields where the quantum-confined Pockels e ect is the dominant contribution to the in-plane anisotropy. The simulation results allowed to estimate the matrix element of the heavy-light-hole mixing operator (3.212) as a function of the electric field. In particular, this matrix element decreases from 2.9 meV at 19 kV/cm to 1.9 meV at 66 kV/cm. As a result, we can determine simultaneously both of the tLl-h and tRl-h parameters which enter into (3.212). By using a standard least-square optimization algorithm, the values of tLl-h = 1.7 ± 0.1 and tRl-h = 0.1 ± 0.1 were obtained. The left and right interfaces appear to be not symmetrical (D2d symmetry requires tLl-h and tRl-h to be equal) which is consistent with the small (but nonzero) anisotropy of the e1-hh1 transition

158 3 Resonant Light Reflection and Transmission

	0.5	19 kV/cm		Eex,-	Eex,+	(a)
	0.4
	0.3				I1=2.9 meV
	0.6
		33 kV/cm				(b)
	0.5
u.)	0.4
(arb.	0.3				I1=2.5 meV
Absorption	0.6				I1=2.3 meV
		48 kV/cm
	0.4
						(c)
	0.6	66 kV/cm			I1=1.9 meV
	0.4
						(d)
	0.2
		855	860	865	870	875

Energy (meV)

Fig. 3.13. Polarization-resolved absorption spectra of the InxGa1−xAs/InP sample in the region of e1-hh2 and e1-lh1 transitions. Figures (a)-(d) correspond to di erent fields. The incident light is polarized along the [110] (gray curves) and

¯

[110] (black curves) eigenaxes of the sample. Continuous lines display experimental data, whereas simulation results are represented by dashed lines. [3.69]

3.4 Magneto-Optics of Nanostructures

159

at zero-field. Yet, note that the estimated anisotropy of the e1-hh1 transition is small enough (< 4 ÷5%) to justify the disregard of this e ect as compared to the quantum-confined Pockels e ect observed in the vicinity of e1-hh2 and e1-lh1 excitons. Going back to the structural properties of the studied QWs, it has been found that the stronger interface potential is associated to the left interface which corresponds to the interface “InGaAs grown on InP”.

3.4 Magneto-Optics of Nanostructures

3.4.1 Magneto-Excitons in Quantum Well Structures

In the e ective mass approximation, the exciton Hamiltonian in an external magnetic field B can be presented as a sum, Hexc(0) + Hexc(1) + Hexc(2) , of the unperturbed Hamiltonian and terms linear and quadratic in B, respectively. Weak magnetic-field e ects are calculated by perturbation theory assuming the parameter

	β =	ωc	=	aB	(3.216)
		2EB		λB

to be small, where ωc = (|e|B/µ c) is the cyclotron frequency for the electronhole in-plane reduced e ective mass µ , EB and aB are the exciton binding energy and Bohr radius, λB = ( c/|e|B)1/2 is the magnetic length. For β 1, the first-order term Hexc(1) leads to the splitting of excitonic spin sublevels (the Zeeman e ect) while the second-order term gives a diamagnetic shift ∆Edia B2. In the opposite limiting case of very strong magnetic fields, β 1, the cyclotron motion of free carriers is quantized forming the Landau levels, the carrier motion is e ectively reduced from 3D to 1D and the Coulomb interaction is considered as a perturbation. Then the interband optical transitions are those with coinciding Landau quantum numbers, Nc = Nv ,

and the band gaps	N + 2		ωc ,	(3.217)
Ecv,N = Eg +
		1

where N = 0, 1, 2... Moreover, the magnetic field increases the exciton binding energy and oscillator strength and, as a result, the absorption spectrum contains a series of excitonic peaks linked to the energies Ecv,N . Such quasi-1D electron-hole excitations di er substantially from conventional 3D excitons and are called diamagnetic excitons [3.78]. The contribution of continuum electron-hole excitations to the absorption results in an asymmetrical shape of each absorption peak.

In QW structures the motion of carriers along the growth direction z is quantized and, therefore, in a strong magnetic field B z their states become localized in all three directions. The density of states in this case is a sum of δ-functions, each of them corresponding to a separate 0D excitation. For the same reason, the absorption spectrum consists of separate peaks whose width

160 3 Resonant Light Reflection and Transmission

is determined by the homogeneous and inhomogeneous broadening. Since the broadening occurs equally towards higher and lower energies, the interband magneto-optical absorption peaks in QWs have symmetric shape [3.79].

The Zeeman e ect will be considered in the next subsection and in Chap. 5. Here we ignore the exciton fine structure and concentrate on the diamagnetic e ect in a QW structure subjected to a magnetic field B z. In the approximate numerical calculation of the eν-hν exciton envelope function, the separation ansatz

Ψ exc = F (ρe, ρh)ϕe1(ze)ϕh1(zh)

(3.218)

is used where the envelope F (ρe, ρh) satisfies the Schr¨odinger equation with the Hamiltonian

Hexc = Eg + Eeν + Ehν + VC (ρe − ρh)

(3.219)

+ 2me

−i ∂ρe

− c A(ρe)

2

+ 2mh

−i ∂ρh

+ c A(ρh)

2

.

1

∂

e

1

∂

e

Here Eeν , Ehν are the quantum-confinement energies, VC is the e ective 2D Coulomb potential

VC (ρ) =	e2	dzedzh			ϕe21(ze)ϕ2			(zh)
							h1			,
	æ
					ρ2		+ (ze		zh)2
and we use the gauge			1				−
		A =		B	× ρ					(3.220)
			2

for the vector potential. The two-particle eigenfunctions can be sought in the form [3.80]

F (ρe, ρh) = exp

R ·

P − c

B × ρ

exp

i 2 ρ · P

f (ρ − ρ0) ,

i

e

γ

(3.221)

where ρ = ρe − ρh, R is the in-plane center of mass, P is the generalized 2D momentum of the exciton and

ρ	=	λB2		B		P , γ = mh − me .
			B		×
0						mh + me

For the eν-hν excitons, the envelope function of the relative motion satisfies the equation

Ef (ρ) = Eg(ν) +		P 2		2	∆ρ +	e2B2	(3.222)
			−			8µ c2 ρ2
		2M		2µ
− 2iµ c B · ρ ×					∂ρ + VC (ρ + ρ0) f (ρ) ,
	eγ				∂