ivchenko_bookreg

3.4 Magneto-Optics of Nanostructures

161

where M = me + mh, Eg(ν) = Eg + Eeν + Ehν , and

	∂2		∂2
∆ρ =		+			.
	2		∂y	2
	∂x

In the following we consider only axially-symmetric excitonic states excited under normal incidence, set their in-plane momentum P and the P - dependent shift ρ0 to zero and seek solutions independent of the azimuth angle of the 2D vector ρ, f (ρ) ≡ f (ρ). This also allows to throw away the term in (3.222) proportional to γ and reduce ∆ρ to

ρ ∂ρ	ρ ∂ρ .
1 ∂		∂

In the weak-field limit, the diamagnetic shift for the s-excitonic states with angle-independent f (ρ) is given by

∆Edia =	π e2B2		f 2(ρ)ρ3 dρ	(3.223)
	4 µ c2

with f (ρ) calculated at zero field.

In the very strong magnetic field, the function of the relative motion for axially-symmetric excitons can be approximated by

	1		ρ2		−	ρ2
	√2π N ! λB		2λB2			4λB2
fN (ρ) =		LN		exp			,	(3.224)

where LN (x) ≡ L0N (x) is Laguerre polynomial

LN (x) = ex dN e−xxN . dxN

The exciton excitation energy is written as

E = Eg(ν) +	N + 2	ωc − ε ,	(3.225)
	1

where ε is the exciton binding energy which in this approximation is just the energy of the Coulomb interaction in the state (3.224) given by the triple integral

∞∞ ∞

εN,ν =	e2		2πρdρ		f 2(ρ)ϕeν2 (ze)ϕhν2 (zh)
				dzedzh							.
	æ
						ρ2	+ (ze	−	zh)2
		0	−∞ −∞

Analytical results are allowed in the limit of infinitely high barriers, narrow QWs and ultra-high magnetic field such as

162	3 Resonant Light Reflection and Transmission
	a < λB < aB .
Then one can obtain [3.80, 3.81]
	e2
	εN,ν = DN æλB ,	(3.226)

where the dimensionless parameters DN tend to constant values with decreasing the ratio a/λB , particularly D0 → (π/2)1/2 [3.80]. Thus, the energy

(3.225) increases linearly with B due to the Landau quantization and has a

√

negative contribution proportional to B due to the Coulomb interaction.

Å Å

Fig. 3.14. Absorption spectra of periodic GaAs/AlxGa1−xAs heterostructure with

˚ ˚

a = 52 A, b = 140 A calculated for the magnetic field B = 10 T for σ+ polarization with (solid) and without inclusion (dotted) of the exciton e ects. The upper and lower curves describe the contributions to absorption coming from the hh1 → e1 and lh1 → e1 transitions, respectively. From [3.86].

3.4 Magneto-Optics of Nanostructures

163

Three comments concerning the above consideration are relevant. For intermediate magnetic fields where the parameter β in (3.216) is comparable with unity the exciton states can be found only numerically, see, e.g., [3.82–3.85]. Secondly, the separation (3.218) neglects any electron-hole correlation along the z axis. This correlation can be approximately taken into account keeping the separation (3.218) where the single-particle envelopes are found self-consistently from the Schr¨odinger equations where the confinement potential has a contribution dependent on f (ρ) [3.81]. Thirdly, the exciton states can be substantially modified because of the heavy-light-hole mixing described by the Luttinger Hamiltonian. Figure 3.14 illustrates a sophisticated calculation of magnetoabsorption spectra in MQWs performed neglecting and taking into account the Coulomb interaction [3.86]. The inclusion of this interaction shifts the absorption peaks to lower energies and results in a redistribution of oscillator strength between the peaks. The lower peaks are labelled by Xh and Xl rather than by 0hh and 0lh, since the corresponding states contain a remarkable admixture of other Landau levels. The anomalous peak Xl arises due to a strong admixture of the 0lh Landau level to heavy-hole states with a nonzero component of the orbital moment.

3.4.2 Polarized Reflection Spectra in an External Magnetic Field

Before we go into details of polarized optical reflection let us remind some characteristics of the polarized electromagnetic wave. This information will be useful in other chapters as well. We consider a transverse quasi-monochromatic electromagnetic wave with the electric field

E(t) = E0(t)e−iωt + E0 (t)eiωt ,

where E0(t) is the amplitude slowly varying in time, the vector E is assumed to be perpendicular to the propagation axis 3. We define the polarization density matrix as the time-averaged products

dαβ = E0,αE0,β (α, β = 1, 2) .

If the light intensity is defined as I = |E1|2 + |E2|2 then the density matrix can be presented in the form

	2			Pl + iPc		1 − Pl
	dˆ =	I		1 + Pl		Pl − iPc .
Here the Stokes parameters
Pc =	I+ − I−		, Pl =		I1 − I2		, Pl =	I1 − I2
								I1 + I2
	I+ + I−				I1 + I2

(3.227)

(3.228)

are, respectively, the degrees of circular polarization, of linear polarization in the axes 1, 2 and of linear polarization in the axes 1 , 2 rotated around the axis 3 by 45o with respect to the axes 1, 2.

164 3 Resonant Light Reflection and Transmission

The above density matrix is written in the basis of the linearly polarized components E = E1e1 + E2e2, where e1, e2 are the unit vectors along 1 and 2. Note that, in the basis of circular polarized components

E = E

e

+ E

e , e

±

= e1

±

ie2

+

+

− −

√2

the density matrix has the form

1 − Pc

2 Pl

+ iPl

dˆ(c) =

I

1 + Pc Pl

− iPl .

This is demonstrated by applying the relations

1

i

e1 = √

(e+ + e−) , e2 = −√

(e+ − e−) ,

2

2

e

=

1

(e

+ e

) =

1 − i

e +

1 + i

e

.

−

1

√2

1

2

2

+

2

Thus, say, for E

e1 or E

e1 we have respectively

2

1 1

2

i

1

dˆ(c)

=

I

1 1

and dˆ(c) =

I

1

−i .

(3.229)

(3.230)

(3.231)

A magnetic field applied in the Faraday configuration, B z, induces a di erence between the amplitude reflection coe cients, r+ and r−, for the σ+ and σ− circularly-polarized light normally incident on a (001)-grown QW structure. As a result, under reflection the normally-incident and linearlypolarized light exhibits a rotation of the polarization plane and gains an ellipticity. The first e ect, the Faraday rotation in reflection geometry, is also known as the polar Kerr e ect. As will be shown below, in the exciton resonance frequency region this di erence appears mainly due to the Zeeman splitting of the exciton radiative level into two circularly-polarized sublevels. If the initial light E0 is polarized along the axis 1 then, according to (3.231),

the amplitude of the reflected light is equal to

e+ + r+ e−

Er = √2 (r+e+ + r−e−) = √2 r+

E0

E0

r−

The Faraday angle under reflection is readily calculated as

1

r−

θ =

arg

(3.232)

2

r+

and the acquired circular polarization is given by

Pc =

R(σ+) − R(σ−)

,

(3.233)

R(σ+) + R(σ−)

3.4 Magneto-Optics of Nanostructures

165

where R(σ±) = |r±|2.

In the linear-in-field approximation, one can neglect the diamagnetic shift of the exciton excitation energy and take into account the Zeeman splitting of the exciton levels. In symmetric zinc-blende-lattice (001)-grown QWs, the degenerate radiative doublet of the e1-hh1 (or e1-lh1) exciton splits into the sublevels ω0,± = ω0 ± (ΩB /2) with angular momentum component M =

±1, which are optically active in the σ+ and σ− polarizations, respectively. The splitting ΩB equals to gzz µB Bz , where gzz is the exciton g-factor, see for details Chap. 5. The amplitude reflection coe cients, r±(ω), from a single QW structure or MQWs are expressed via the corresponding zero-field reflection coe cient r(ω) where the variable ω is shifted by ΩB /2, namely,

r±(ω) = r	ω 2B		,	(3.234)
		Ω

where r is found from (3.4) and (3.21, 3.55) or (3.90).

In the case where the Zeeman splitting ΩB exceeds the exciton damping Γ , it can be readily measured spectroscopically. In the other limiting case, |ΩB | Γ , the splitting can be found by studying normal-incidence reflection of unpolarized or linearly polarized light and measuring the spectral behavior

of the degree of circular polarization of reflected light
Pc = −	ΩB		1 ∂R(ω)		,	(3.235)
	2		R(ω)	∂ω

where R is given by (3.50).

In the same way, for small magnetic fields the rotation angle θ is given by

			θ = 2r01		Im	∂ω	.
				ΩB		∂r(ω)
In particular, for a single-QW structure with Γ0 Γ one has
θ =	4nb	ΩB Γ0	[Γ 2 − (ω0 − ω)2] cos 2ϕ0a + 2(ω0 − ω)Γ sin 2ϕ0a					,
	nb2 − 1
				[(ω0 − ω)2 + Γ 2]2

where the phase ϕ0a is defined according to (3.48).

3.4.3 Giant Magneto-Optical E ects in Semimagnetic

Nanostructures

(3.236)

(3.237)

Semimagnetic or diluted magnetic semiconductors are semiconducting compounds in which a fraction x of the cations is randomly replaced by transition metal ions, e.g., 3dn ions. Most often, they are solid solutions A1−xMxB with A= Cd, Zn, Hg, B = Te, Se, S, and M = Mn2+, Fe2+, Co2+. The magnetic subsystem interacts e ectively with charge carriers, electrons and

166 3 Resonant Light Reflection and Transmission

holes, due to the strong sp-d exchange interaction between the carrier spins and magnetic moments of the ions. The semimagnetic bulk semiconductors and nanostructures are known for giant magneto-optical properties and magnetic polarons. Particularly, in this systems the Zeeman splitting of the carrier or exciton spin states are described by the e ective g-factors being in excess of a hundred.

˚

Fig. 3.15. Zeeman splitting of the heavy-hole exciton in an 18-A-thick single QW

˚

(a) and a 40-A-period SL (b) with x = 0.1. Curves are calculated with use of (3.238) and parameters TAF = 2.4 K, x˜ = 0.05, or 0.07 for the single QW, or SL, respectively. [3.87]

Usually, the sp-d interaction of magnetic ions with free carriers is described by two constants, α and β, of the exchange interaction. For example, the normal magnetic field, B z, changes the carrier energy by

		3.4 Magneto-Optics of Nanostructures			167
	α		β
∆Ee,s =		xs Sz , ∆Eh,j =		xj Sz ,	(3.238)
	v0		3v0

where s = ±1/2, j = ±3/2, ±1/2 are the electron and hole spin indices, x is the content of magnetic ions, v0 is the unit cell volume (for the zinc-blende lattice, v0 = a30/4), and Sz is the average spin component of a magnetic ion along z. In the thermal equilibrium one has

Sz = −S BS (SgµB Bz /2kB T ) ,

(3.239)

where S is the atomic spin of a magnetic ion (5/2 for Mn2+, 3/2 for Co2+), g = 2 is the ion g-factor, and BS (X) is the Brillouin function

	2S + 1		2S + 1				1	X
BS (X) =		coth			X −		2S coth		.	(3.240)
	2S			2S				2S
Note that at small X this function is approximated by
	BS (X) ≈			S + 1
						X .
				3

In order to take into account the antiferromagnetic Mn-Mn spin coupling two additional fitting parameters are introduced, namely, the content x in (3.238) is replaced by the e ective content x˜ and T in (3.239) is shifted by some value TAF. In the paramagnetic phase x˜ = x and TAF = 0 whereas in the spin-glass phase x˜ < x and TAF > 0. In the Cd1−xMnxTe system, α/v0 = 220 meV and β/v0 = 880 meV. Opportunities of resonance reflection spectroscopy to study the magneto-optical properties in CdTe/CdMnTe QWs and SLs are demonstrated in [3.87]. Figure 3.15 shows the Zeeman splitting of the heavy-hole exciton states | ± 1 measured at magnetic fields parallel

˚ ˚

to the growth axis in a 18-A-thick single QW and 40-A-period SL. Since in the studied structures only barriers are semimagnetic, there exists a certain magnetic field Bcr at which the valence band o set for one of the heavyhole states vanishes resulting in the magnetic-field induced type-I–type-II transition. According to (3.238) the value of Bcr satisfies the condition

β

2v0

x˜ Sz + Vhh = 0 ,

where Vhh is the heavy-hole subband o set at zero field. In CdTe/Cd0.9Mn0.1Te QWs, Bcr equals 1.3 T. At this particular field one of the heavy-hole exciton can be considered approximately as formed by a 2D electron and a 3D hole. Reduction of the e ective oscillator strength of the low-energy σ+-polarized exciton observed in [3.87] is a convincing demonstration of the type-I–type-II transition.

The large Faraday rotation in bulk diluted magnetic semiconductors is further enhanced in semimagnetic QWs with larger excitonic oscillator strength. Particularly, this has opened the way for using CdMnTe heterostructures as

168 3 Resonant Light Reflection and Transmission

Fig. 3.16. Schematic representation of the optical cavity with an embedded QW.

magneto-optic layers for high resolution Faraday microscopy at low temperature in order to make a cartography of the magnetic field at the surface of a superconductor [3.88, 3.89]. The structure is schematically depicted in Fig. 3.16. It consists of a metal, an inner barrier layer, a QW and a top barrier layer with CdMnTe and CdMgTe used, respectively, as the QW and barrier compositional materials. In fact it is an optical cavity with an embedded QW. The metal is illuminated through the magneto-optical layers by linearly polarized light. The plane of light polarization is rotated in the areas that carry magnetic flux. After passing through a crossed (or slightly uncrossed) analyser the reflected light is focused on a detector. Since the magnetic field lines diverge out of the surface plane of the superconducting film, the semimagnetic QW must be as close as possible to the superconducting film to achieve a good spatial resolution.

The reflection coe cients r± from the structure shown in Fig. 3.16 can be presented in the form

3.4			Magneto-Optics of Nanostructures		169
r = r(0) +			iΓ0C		(3.241)
				.
	ω0
		− ω − i(Γ + Γ0B)

Here the first term describes the reflection when the exciton contribution is ignored

		r(0) =	r0b + rme2iϕm0
					,
			1 − rb0rme2iϕm0			2
			A =	1 + rme2iϕmw
	C = tb0t0be2iϕw0 A ,					,		(3.242)
				1 − rb0rme2iϕm0
	rb0(2rme2iϕm0 + e2iϕw0 ) + rme2iϕmw
	B = 1 +						.
			1 − rb0rme2iϕm0

Other notations are as follows: Γ0 and Γ are the exciton radiative and nonradiative damping rates in a single QW,

r		= r	=	nb − 1	, t	t		= 1	−	r2	,
				nb + 1
	b0	− 0b				0b	b0			b0

nb is the refractive index of the barrier assumed to coincide with that of the QW, the phases are given by

ϕw0 = (ω/c)nbLw0 , ϕmw = (ω/c)nbLmw ,

the thicknesses Lw0, Lmw and the reflection coe cient, rm, from the metal to the barrier are defined in Fig. 3.16. In the following we use ρm and ϕm for the modulus and phase of rm.

The amplification of rotation in the optical cavity occurs mostly due to the factor A defined in (3.242). This factor exhibits a maximum when two conditions are fulfilled. Firstly, the total dephasing 2ϕm0 + ϕm must be an integer number of 2π. The reflection coe cient r(0) then exhibits a minimum value of (ρm − rb0)/(1 − ρmrb0). For a perfect metal (ϕm = π) the optical thickness of the cavity is equal to (2p + 1)λ/4 with p an integer and λ the wavelength of light. Secondly, the QW must be located at an anti-node of the electric field which means that 2ϕmw + ϕm = 2lπ with l being an integer. In this particular case the amplification factor equals to

A =		1 − rb0	2
			ρm .
		1 + ρm

If the contribution of the QW to the reflected signal is small even at the minimum of reflectivity, i.e.,

&	&	(3.243)
&r(0)& |tb0t0bΓ0A/Γ | ,
&	&

then the rotation angle is given by

170 3 Resonant Light Reflection and Transmission

	θ =	tb0t0bA		Γ0ΩB	F (Ω, ΩB ) ,
		2r(0) Γ 2
	where Ω = ω − ω0 and
	F (Ω, ∆) =	Γ 2[Γ 2 + (∆/2)2 − Ω2]				.
		[Γ 2 − (∆/2)2 + Ω2]2 + Γ 2∆2

For small magnetic fields one can set ∆ = 0 in the expression for F . Note that

the function F (Ω, 0) has a maximum value of 1 at Ω = 0, changes sign at

√

the points Ω = ±Γ and reaches minima −1/8 at Ω = ± 3Γ . An additional enhancement of the rotation angle can be achieved by using few QWs with the Bragg spacing instead of one [3.89].

Fig. 3.17. Images of the magnetic flux pattern at the surface of the 10 µm Indium layer obtained at 1.8 K and H = 28 Oe (left), H = 77 Oe (right). Normal and superconducting domains appear in black and gray, respectively. The size of each image is 264×191 µm2. The analyser was uncrossed by 15◦ with respect to the polarizer. [3.89]

The above magneto-optical layers allowed an observation of the intermediate state pattern in a superconductor layer with spatial and magnetic resolutions of 1 µm and 10 mT (Fig. 3.17). As it can be seen from the figure, the intermediate state is formed by normal flux tubes (dark spots) and long corrugated lamellae with typical width and spacing of 10 µm. These normal state regions are separated by superconducting state domains (in gray). Figure 3.17 demonstrates wide possibilities to use the low-dimensional physics for studies of hybrid systems containing semiconductor and superconductor layers.