ivchenko_bookreg

4.1 Intersubband Optical Transitions. Simple Band Structure

181

mission Fourier spectroscopy in the multiple-reflection waveguide geometry (see the inset). The points indicate values of the spin-galvanic current (see Sect. 8.2) observed under photoexcitation of the structure by a CO2 laser, which causes direct transitions within each QW. One can see from Fig. 4.5 that both spectra repeat each other in agreement with the theory because the photocurrent is proportional to K ( ω) and, according to (4.21), the coe cient K ( ω) is proportional to Kz ( ω).

°

Fig. 4.4. Illustration of light transmission in the multiple-reflection wave-guide geometry.

Intersubband optical transitions in n-doped QW structures can be appreciable in the in-plane polarization (e z) if the e ective mass in the conduction band of a compositional bulk semiconductor is anisotropic, and none of the principal axes of the reciprocal e ective-mass tensor m−αβ1 coincides with the growth axis. Such a situation arises when the nanostructure is grown from many-valley semiconductors along the direction di erent from the principal axis of a particular valley. This can be achieved, for example, in Ge/SixGe1−x QW structures grown in the direction [001]. In bulk Ge the

¯

electron dispersion in one of the L valleys, say the [111] valley, is

Ek =	2		k2	+ k2	k2	,	(4.24)
	2		1 mt		+ ml
				2	3

where mt, ml are the transverse and longitudinal e ective masses, k is the electron wave vector referred to the extremum point k0 = (π/a0)(1, −1, 1)

¯¯

¯

and k1, k2, k3 are the components of k along the axes [112], [110] and [111],

respectively. In the coordinate system x

¯

[110], z

[001] relevant

[110], y

for (001)-grown structure, the dispersion (4.24) will be

Ek = 2

mxx

+ myy + mzz

+ 2

mxz

,

(4.25)

2

k2

k2

k2

k

k

z

x

y

z

x

where

vx = kx

mxx

182 4 Intraband Optical Spectroscopy of Nanostructures

ω

Fig. 4.5. Absorption spectrum (full line) of an n-doped (001)-grown GaAs/AlGaAs

˚

MQW structure (a = 70 A) obtained from transmission in a multiple-reflection wave-guide geometry, see the inset. Points show the spectral dependence of the spin-galvanic current caused by the spin orientation due to direct optical transitions between e1 and e2 conduction subbands. [4.7]

1

cos2

θ

sin2

θ

1

sin2

θ

cos2

θ

=

+

,

=

+

,

mxx

mt

ml

mt

ml

mzz

,

myy = mt

, mxz

= cos θ sin θ

ml

− mt

1

1

1

√

¯

and θ = arccos (1/ 3) is the angle between the axes [111] and z. Notice that equation (4.25) is a particular case of the more general equation (2.21).

The x-component of the velocity, vx = −1∂Ek/∂kx, has a linear-in-kz contribution

+ kz ,

mxz

which allows nonzero e2-e1 intersubband matrix elements

		(e2	&	∂	& e1) .	(4.26)
v21,x = −i mxz				∂z
			&		&
			&		&
			&		&

For the intersubband transitions at the point kx = ky = 0 in a high-barrier QW, we have

	kz(2,1)		8		1		1
v21,x =		= i		cos θ sin θ			−		.
	mxz		3a			ml		mt

4.1 Intersubband Optical Transitions. Simple Band Structure

183

For the two lowest subbands in a Metal-Oxide-Semiconductor (MOS) structure, one sometimes chooses trial functions in the form

ϕ1(z) = C1ze−α1z , ϕ2(z) = C2z 1 −						α1 3			2		z e−α2z
						+ α
with the normalizing factors
	, C2 = 2				3α25				1/2
3/2
C1 = 2α1											.
					α12 − α1α2 + α22
In this case, one has
		α2			4t3			1/2
v21,x = −2i											,
		mxz		3(1 + t3)(1 + t)3

where t = α2/α1.

4.1.2 Interminiband Light Absorption in a Superlattice

Now we turn to interminiband optical properties of SLs in order to show how infrared spectroscopy can reveal their miniband structure. In contrast to a MQW structure, in a SL the electron energy depends on an additional quantum index, namely, the z-component of the wave vector kz . The transition rate in a unit 3D volume is given by (4.1) where the area S is replaced by the SL volume V and the 2D vector k by the 3D wave vector K. For the absorption coe cient due to the e1 → e2 transitions, one can write

KSL(e z)

(4.27)

=4π2e2 2 |v21,z (K)|2 (fe1K − fe2K ) δ(Ee2K − Ee1K − ω) . ωcnω V K

In the particular case of equal masses, mA = mB, the electron dispersion in a SL can be expressed in the form

2K2

EeνK = 2mA + Eν (Kz )

so that

Ee2K − Ee1K = E2(Kz ) − E1(Kz ) ,

the parabolic in-plane dispersions cancel in the argument of the δ-function in (4.27). One is left with a purely 1D reduced density of miniband states

ρ21( ω) =				2πz	δ [E2(Kz ) − E1(Kz ) − ω]		(4.28)
			dK
=		1			1	,	(4.29)
		π |ve2z (Kω ) − ve1z (Kω )|

184 4 Intraband Optical Spectroscopy of Nanostructures where the z component of the electron velocity is

1 ∂Eν

veνz (Kz ) =

∂Kz

and Kω satisfies the equation E2(Kω ) − E1(Kω ) = ω.

The function ρ21( ω) has a shape of a two-pronged fork and exhibits two square-root singularities at the highand low-energy edges, respectively,

ω

21

= E0

≡

E0

−

E0

and

ω

= E

(π/d)

−

E

(π/d) .

21

2

1

21

2

1

By using this function Eq. (4.27) can be rewritten as

KSL(e z)

S

K |v21,z (K)|2 (fe1K − fe2K )

(4.30)

=

ωcnω

ρ21( ω)

.

4π2e2

2

z

ω

K

=K

Equation (4.30) can be analyzed by using di erent approximations. As the first limiting case let us assume ω to be close to the minigap E210 at the Γ point. Then the parabolic approximation

Eν (Kz ) = Eν0

+

2K2

z

2Mν

is valid and, for the 3D Boltzmann electron gas, we come to

KSL(e z)

(4.31)

=

4πe2

N3D |v21,z |2

π 2µ21

1/2 exp

E210 − ω

µ21

.

kB T (E210 − ω)M1

cnω

ω

−

kB T M1

Here N3D is the volume electron concentration, v21,z is the interminiband matrix element of the velocity operator at K = 0, the reduced e ective mass µ21 = M1|M2|/(M1 + |M2|). Note that M1 is positive and M2 is negative. The thermal energy is assumed to be small compared to the width of the lower miniband.

In a wide-barrier SL, the electron dispersion can be approximated by

E1(Kz ) = E10 + 12 ∆1(1 − cos Kz d) , E2(Kz ) = E20

and one has

sin	Kω d	=		E210 − ω	1/2	,
	2
				∆1 + ∆2

1

− 2 ∆2(1 − cos Kz d)

|ve2z (Kω ) − ve1z (Kω )| = (2 )−1(∆1 + ∆2)d sin Kω d .

The reduced density of miniband states is symmetric around the energy E210 − (∆1 + ∆2)/2 [4.8]

	4.2 Intersubband Optical Transitions. Complicated Band Structure				185
				−1
	ρ21( ω) = πd (E210 − ω)[ ω − (E210 − ∆1 − ∆2)]				(4.32)
				.

These equations can also be derived in the tight-binding approximation, where one starts with the energy levels calculated for a single QW and takes into account the coupling between the states in the nearest neighboring wells. In the same approximation the matrix element v21,z (K) is independent of K and equal to that for intra-well intersubband transitions. The absorption spectrum has in this case the shape of an asymmetric two-pronged fork

	ρ21( ω)	exp −	∆1	E210		ω
KSL(e z)					−		.
	ω		∆1 + ∆2		kB T

The relative intensity of the low-energy peak at

ω = E210 − ∆1 − ∆2 ≡ ω21

decreases with decreasing temperature.

It is instructive to trace the transition of the absorption spectrum from 3D to 2D. With increasing the barrier thickness the miniband width ∆1,2 decreases and the function ρ21( ω) tends to d−1δ(E210 − ω) while the expression in the square brackets in (4.30) reduces to |v21,z |2N1, where N1 is the 2D electron density in a single QW. As a result (4.27) reduces to the absorption coe cient of a MQW structure given by (4.6).

4.2 Intersubband Optical Transitions. Complicated

Band Structure

In a bulk semiconductor with the degenerate band Γ8 (zinc-blende lattice) or Γ8+ (diamond lattice), direct optical transitions between the heavyand

light-hole subbands are possible under arbitrary light polarization. In the

√

isotropic approximation, D = 3B, the coe cient of such an absorption is expressed by

K =	e2k	[f (Ehh) − f (Elh)] ,	(4.33)
	cnω

where the initial, heavy, and final, light, hole energies are related with the photon energy by

	Ehh	=	mlh	ω , Elh =	mhh	ω = Ehh + ω ,
			mhh − mlh		mhh − mlh

and the wave vector of the holes involved in the transition is given by

k =	1/2
	mhh − mlh ω .
		2mlhmhh

186 4 Intraband Optical Spectroscopy of Nanostructures

In QW structures, each 3D subband, hh or lh, gives birth to its series of subbands of quantum-confined states, hhν or lhν. At the point kx = ky = 0 they are purely ±3/2 or ±1/2 states, provided the heavy-light hole mixing at the interfaces discussed in Sect. 3.3.3, see (3.212), is ignored. For nonzero k, the QW subband states are intermixed and the energy dispersion has a complicated nonparabolic character, see Chap. 2 for details . For illustration Fig. 4.6 depicts the hole energy spectrum of an In0.49Ga0.51P/GaAs QW. Because of the aforementioned admixture and nonparabolicity the subband dispersion cannot be described by a certain value of the angular momentum component and a certain e ective mass. Usually the valence subbands are labelled as jν with j standing for hh or lh in accordance with the corresponding value of m at k = 0, see (2.16). There are two important consequences of this kind of band structure.

π

π

Fig. 4.6. (a) Valence subbands in an InGaP/GaAs QW. (b) The energy spacing between the lowest valence subband hh1 and the first three higher subbands. From [4.9].

Firstly, in contrast to the conduction-band subbands, the hole intersubband energy spacing is k-dependent and this dependence for the particular pairs ν , ν can be nonmonotonous as one can see in Fig. 4.6b. In the spherical approximation the reduced density of states

2

ρj ν ,jν ( ω) = S k δ (Ej ν k − Ejνk)

can be transformed to

4.2 Intersubband Optical Transitions. Complicated Band Structure

187

	ρ	( ω) =	ki		d(Ej ν k − Ejνk)	−1	,	(4.34)
			π	&	dk	&k=ki
	j ν ,jν	i
				&		&
				&		&
				&		&

where ki are roots of the equation Ej ν k −Ejνk = ω. The absorbance ηj ν ,jν is connected with ρj ν ,jν by

		2π2e2				(i)
ηj ν ,jν ( ω) =					i	ρj ν ,jν	( ω)			(4.35)
		ωnω c
	[f h(Ejνki )			f h(Ej				*	vj ν p ,jνp(k) 2	+
×			−					| ·	|	k=ki

p p

where ρ(ji)ν ,jν is the i-th term in the sum (4.34) and the angular brackets mean averaging over the azimuthal angle, ϕ, of the vector k. The reduced density of

states as well as the absorbance have square-root singularities near the point k0 where the spacing ∆j ν ,jν (k) = Ej ν k − Ejνk exhibits an extremum, e.g., a maximum. Expanding this energy spacing in up to the second order in k − k0,

∆j ν ,jν (k) = ∆j ν ,jν (k0) + 2 (k − k0)2 ,

2µ

we obtain two roots
k1,2	= k0	±				.
				2 [ ω − ∆j ν ,jν (k0)]
			2µ

The corresponding density of states becomes

ρj ν ,jν ( ω) = π			ω − ∆j ν ,jν (k0)		.	(4.36)
	k0	2µ

Note that, for the structure with the spectrum shown in Fig. 2.2, squareroot singularities must manifest themselves at ω ≈ 25 meV (transitions

hh1 → lh1) and ω ≈ 150 meV (transitions hh1 → hh3). Allowance for

√

warping of the valence band, D = 3B, results in angular dependence of the roots ki(ϕ), broadening of the singularities and final values of ρj ν ,jν ( ω) and ηj ν ,jν ( ω). In order to get the final value of ρlh1,hh1( ω) at the point k0, we may notice that the function ∆lh1,hh1(k) has a minimum at the points (±k0, 0), (0, ±k0) of the 2D k-space. It can be expanded near one of these four 2D valleys, say, the valley (k0, 0), as

2			2k2
∆lh1,hh1(k) = ∆lh1,hh1(k0, 0) +		(kx − k0)2 +	y	,	(4.37)
	2µx		2µy

where µx coincides with µ and µy tends to ∞ as the warping is vanishing. The reduced density of states for four equivalent valleys is

188 4 Intraband Optical Spectroscopy of Nanostructures

4√µxµy

ρlh1,hh1(max) = π 2 .

Secondly, intersubband transitions in p-type QWs are more complicated than those in n-type QWs because, due to the k-dependent band-mixing effect, there are no simple selection rules at k = 0 so that both the normal (e z) and in-plane (e z) polarizations are allowed with the transition rates being comparable. In the e ective-mass approximation the intersubband matrix elements vj ν p ,jνp(k) are calculated starting from the velocity operator expressed as the gradient in k space of the Luttinger Hamiltonian

	e · v(k) = −1e · kHΓ8 (k) =				(4.38)
				eαQαβ kβ ,
				αβ
where		4 B + BJα2	δαβ + (1 − δαβ ) √3 {JαJβ }s .
Qαβ = A −
2		5		D

In terms of the four-component envelope functions expressed as columns

ˆ
Fjνpk(z), see (2.37), the matrix elements are written in the form
e · vj ν p ,jνp(k) =	αβ	eα	dz Fˆj† ν p k(z) Qαβ kˆβ Fˆjνpk(z) ,	(4.39)

ˆ −

where k = (kx, ky , i∂/∂z) and the index p takes into account the double degeneracy of the hole states.

For a symmetric QW, the two degenerate states are classified by a parity p under the mirror reflection, z → −z, which transforms the electron spinor wave function ψ(x, y, z) into σˆz ψ(x, y, −z). We remind us (Chap. 2) that, for symmetric states labelled with the index p = +, the m-th compo-

ˆ − 3/2−m

nent Fjν+k,m(z) of the column Fjν+k(z) has the parity pm = ( 1) . The envelope Fjν−k,m(z) of an antisymmetrical state is obtained from the corresponding symmetrical state by changing z into −z and m into −m,

complex conjugation and multiplication by a factor (−1)3/2−m relating the

Bloch function Γ8, m with the function ˆ Γ8, m , see (2.41). In the polar-

| K | −

ization e z, the parity is conserved under intersubband transitions, p = p, whereas, in the polarization e z, the parity is reversed. Moreover, the symmetry imposes additional conditions upon nonzero matrix elements

|e · vj ν +,jν+(k)|2 = |e · vj ν +,jν+(k)|2 ,

|vjz ν −,jν+(k)|2 = |vjz ν +,jν−(k)|2 .

The phenomenon inverse to the absorption is the spontaneous emission of hot carriers heated up, e.g., by an external electric field [4.10]. The heating results in occupation of higher-energy states. Direct optical transitions from

4.2 Intersubband Optical Transitions. Complicated Band Structure

189

these states to the lower subbands is accompanied by emission of light. We define the spectral intensity of the secondary radiation by

I( ω) =	ω ∆W	,	(4.40)
	∆( ω) ∆Ω

where ∆W is the rate of emission of photons in the frequency region ∆ω within a solid angle ∆Ω from a unit area of the illuminating QW. For the intersubband transitions, the spectral intensity of spontaneous emission can be presented in the form

	e2ω2nω 1
I( ω) =				j νk	fj ν k(1		− fjνk) δ(Ej ν k − Ejνk − ω) (4.41)
	πc3
			S
				jν
		×			|	e	·	vjνp,j ν p (k) 2 .
								|

p p

The emission matrix element e · vjνp,j ν p (k) equals the complex conjugate matrix element [e · vj ν p ,jνp(k)] for the intersubband absorption. Equation (4.41) can be derived by using the straightforward expression for the di erential of the emission rate due to the j , ν → j, ν transitions

∆Wjν,j ν =

2π V q2∆q∆Ω

e

2

k

fj ν k(1 − fjνk)

(4.42)

0 (2π)3

c A0

δ(Ej ν k

ω)

2

vjνp,j ν p (k)

×

−

Ejνk

−

e

·

2 .

S

|

|

p p

Here q = nω (ω/c) is the light wave vector, the di erential ∆q equals to nω ∆ω/c, A0 = (2π c2/n2ω V0 ω)1/2 is the amplitude of the vector potential related to a single electromagnetic quantum, see (3.28). We consider the radiation emitted in a small solid angle ∆Ω around the normal to the interfaces. In this case the transition probability rate is independent of the secondary photon polarization and e is any in-plane unit vector. An additional factor of 2 in (4.42) takes into account two polarization states of a transverse electromagnetic wave. After various simplifications the equations (4.40, 4.42) reduce to (4.41).

A spectrum of the spontaneous emission from a p-doped GaAs/AlGaAs QW structure in the electric field 450 V/cm is plotted in Fig. 4.7 with the solid curve. The dashed curve is calculated as a half-sum of two spectra obtained for e z and e z. To take into account the homogeneous and inhomogeneous broadening as well as the resolution of the detector, the theoretical spectrum is convoluted with a Gaussian of the width 2 meV. The hole energy distribution was described by the Fermi-Dirac function with the hot-hole e ective temperature of 80 K. One can see a satisfactory agreement between the theory and experiment.

190 4 Intraband Optical Spectroscopy of Nanostructures

ω

Fig. 4.7. Measured (solid) and calculated (dashed) emission spectra for the applied electric field F = 450 V/cm. Rectangles show estimated spectral widths expected from the analysis of the reduced density of states in an ideal homogeneous structure. [4.10]

4.3 Intrasubband Optical Transitions

4.3.1 Intrasubband Light Absorption in a Quantum Well

Considering free-carrier intrasubband absorption in a QW in the Drude model, we start from the classical equation for in-plane motion of an electron

	v˙ (t) + γv(t) =	e	E(t) ,	(4.43)
		m

where γ is the phenomenological damping constant. In a QW where an electron moves freely in the interface plane and is e ected only by the in-plane component of an external electric field, equation (4.43) reduces to

e

v˙ (t) + γv (t) = m E (t) .

The amplitudes of the in-plane current and the electric field are related by

e2N2D

j (ω) = eN2Dv (ω) = m(−iω + γ) E (ω) .

The absorbance