Microscopic model: k-linear terms in the Hamiltonian

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Microscopic model:

k-linear terms in the Hamiltonian

2 m h ² k ²

ε = _{± β} k _x

ε

hh1 (+3/2)

e1 (-1/2) e1

(+1/2)

ε

0 kx

( 3/2) hh1 +- ( 1/2)

e1 +-

0 kx

k-linear terms

( 1/2) lh1 +-

spin-orbit interaction

hh1

(-3/2) BIA (Dresselhauser) IIA

SIA (Rashba)

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Microscopic model: direct transitions

kx

ε

+ kx 0

σ+

jx

e2 (-1/2) e2

(+1/2)

e1 (-1/2) e1

(+1/2)

PRL 86, 4358 (2001)

j ∝ W( ε ) ν ( kx - ) ( τ

_{pi -}

τ

_pf

)

h ω > h ω LO τ

pi >>

τ

_pf

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Microscopic model: direct transitions

kx

ε

+ kx 0

σ+

jx

e2 (-1/2) e2

(+1/2)

e1 (-1/2) e1

(+1/2)

kx

ε

σ−

- 0 kx

jx

e2 (-1/2) e2

(+1/2)

e1 (-1/2) e1

(+1/2)

PRL 86, 4358 (2001)

j ∝ W( ε ) ν ( kx - ) ( τ

_{pi -}

τ

_pf

)

h ω > h ω LO τ

pi >>

τ

_pf

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j / P

x

( 10

-9

A / W

)

115 125 135

-2 -1 0 1 2

h

^ω ^(meV)

0.0 0.4 0.8

A b sor p tion (a.u.)

n-GaAs QWs

σ+

σ−

Experiment: e1-e2 direct transitions

PRB (2003), cond-mat/0303054

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Resonant inversion of the CPGE

kx

ε

+ kx 0

σ+ ^, ^D ^ω1

jx

e2 (-1/2) e2

(+1/2)

e2 (-1/2) e2

(+1/2)

PRB (2003), cond-mat/0303054

kx

ε

- 0 kx

jx

e2 (-1/2) e2

(+1/2)

e2 (-1/2) e2

(+1/2)

σ+ ^, ^D ^ω2

j

_x

(CPGE) ∝ ( τ

_p1

- τ

_p2

) d η

₂₁

( h ω ) d h ω

IP

_c

h ω

D ω1 > D ω2

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Microscopic model: direct transitions

kx

j x ^ε

σ+

hh1 (+3/2) (-3/2)

kx kx

- +

e1 (+1/2)

hh1

0

e1 (-1/2)

inter band

j ∝ W( ε ) ν (k) τ _p ( ε )

PRL 86, 4358 (2001)

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Direct interband transitions

Regensburg group (S.Ganichev) &

Hannover group(M.Oesstreich), solid state comm. (2003)

M. Sakaki, Y.Ohno, H.Ohno, ICPC-2004

1.4 1.5 1.6 1.7

0.0 0.6 1.2

Excitation energy, Dω ( eV ) j x

/ P ( nA / W )

0.03 0.04

0.05 Degree of polarization

x [110]

e

[332]

y

(113)A- grown p - GaAs MQWs T = 293K

1.60 1.65 1.70 1.75 0

50 100 150 200

τ s (ps)

Dω (eV)

0 500 1000 1500

0.00 0.04 0.08

T=293K, P =15mW Dω = 1.74 eV Dω = 1.58 eV

Degree of polarization

Time ( p s )

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Phenomenological description of PGE I

=0

j 0 σ _{i j} ^E _j ⁰

α _{i jkl} Ê _j ⁰ Ê _k ω Ê _l ω

λ _{i jk} ^E _j ω ^E _k ω

E _j 0

Fourier amplitudes: j _i ( ω ), E _j ( ω )

j ⁰ λ _{i jk} ^E _j ω ^E _k ω

Photogalvanic effect (PGE) Photoconductivity

Ohm's law

E.L. Ivchenko, G.E. Pikus, Superlattices and Other Heterostructures. Symmetry and Optical Phenomena, (second edition, Springer, Berlin 1999)

B.I. Sturman, V.M. Fridkin, The Photovoltaic and

Photorefractive Effects in Non-Centrosymmetric Materials, Gordon and Breach Science Publishers, New York, 1992.

see for review:

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Phenomenological description of PGE II

Photogalvanic effect (PGE):

j _i ( PGE )= α _{i j l} e _j e * _l = χ _{i j l} [ e _j e _l * + e _l e * _j ] / 2 + γ _{i j} i ( e e e × e e e * ) _j

χ _{i j l} γ _{i j}

- third rank piezoelectric tensor

- second rank gyration pseudo-tensor γ _{i j} i ( e e e × e e e * ) _j

j _i ( CPGE ) = ∝ E ₀ ² P _circ = E ₀ ² sin 2 ϕ

Linear PGE Circular PGE

(symmetric part) (anti-symmetric part)

at first theoretically considered by: Ivchenko&Pikus

and, independently, by Belinicher&Sturman

observed in a bulk tellurium by V.M. Asnin, A.A. Bakun, A.M. Danishevskii, and A.A. Rogachev,

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inversion asymmetric, but non-gyrotropic no CPGE ( linear photogalvanic effect only )

Symmetry:

· Bulk GaAs, InAs etc.: point group T

_d

· (001) - oriented QWs: point group D

₂_d

-- only at oblique incidence CPGE

or C

2v

· (113) - oriented QWs: point group C

_s

CPGE at normal and at oblique incidence

b) c)

E2 E1

a)

D

2_d

C

2v

. . .

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angular momentum directed motion of circ. pol. photons of carriers

Mechanical analogues of the CPGE

transversal -wheel

longitudinal - propeller

mirror

reflection plane

jx

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Helicity dependent current

Cs symmetry

(113)A or miscut - grown QWs

0 45 90 135 180

-4 -2 0 2 4

σ_- σ₊

0 0 0

0

(113)A- grown p-

T = 300 K, λ = 77 µm GaAs/AlGaAs MQWs

ϕ ϕ

ez

jx

j x /P ( 10-9 A /

W

)

CPGE current occurs normal to the mirror reflection plane!

jx = ( γ xy ey + γ xz ez )E

0

²

sin 2 ϕ

Physica E 14, 166 (2002)

> >

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D2d or C2v symmetry

in (001) - grown QWs, ey II [110]

Helicity dependent current

Cs symmetry

(113)A or miscut - grown QWs

0 45⁰ 90⁰ 135⁰ 180⁰

ϕ ϕ

-20 -10 0 10 20

(001)- grown n- InAs/AlGaSb QW T = 300 K, λ = 77 µm

ey

jx σ_-

σ₊

j x /P ( 10-9 A /

W

)

0 45 90 135 180

-4 -2 0 2 4

σ_- σ₊

0 0 0

0

(113)A- grown p-

ϕ ϕ

ez

jx

j x /P ( 10-9 A /

W

)

CPGE current occurs under oblique excitation only!

CPGE current occurs normal to the mirror reflection plane!

jx = ( γ xy ey + γ xz ez )E

0

²

sin 2 ϕ

Physica E 14, 166 (2002)

> >

jx = γ xy ey E

0

²

sin 2 ϕ

>

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C

2v symmetry

Normal and Oblique Incidence

-50 -25 0 25 50

0 2 4 6

(113)A- grown p-

jx

θ₀

σ₊

θ

0

j x /P ( 10-9 A /

W

)

-50 -25 0 25 50

-40 -20 0 20 40

(001)- grown n-

T = 300 K, λ = 77 µm type InAs/AlGaSb QW

jx

θ₀ σ₊

θ

0

j x /P ( 10-9 A /

W

)

Cs symmetry

Θ 0 - angle of incidence

Physica E 14, 166 (2002)

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Si:Ge quantum wells

symmetric Si:Ge QWs do not show CPGE.

PRB, (2002)

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Si:Ge quantum wells

0 45 90 135 180

-15 -10 -5 0 5 10 15

(001) p- Si Ge

j x /P ( 10-11 A /

W

)

ϕ ey

jx

a) b)

E2 E1

symmetric Si:Ge QWs

do not show CPGE. asymmetric Si:Ge QWs !

PRB, (2002)

stepped QW T = 300 K λ = 10.6 µm

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Spin-galvanic effect

j _α = Σ Q _αβ S _β

current averaged spin β

e.g. for C2v-symmetry and x [110]: Qxy, Qyx are non-zero:

jx = Qxy Sy

Nature 417, 153 (2002)

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Spin-galvanic effect

j _α = Σ Q _αβ S _β

current averaged spin β

e.g. for C2v-symmetry and x [110]: Qxy, Qyx are non-zero:

jx = Qxy Sy

Nature 417, 153 (2002)

e

S0z

Sy

jx

2DEG

S0

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Spin-galvanic effect

ez

S0z Bx

Sy

jx ω

Larmor

2DEG

j _α = Σ Q _αβ S _β

current averaged spin β

e.g. for C2v-symmetry and x [110]: Qxy, Qyx are non-zero:

jx = Qxy Sy

Nature 417, 153 (2002)

e

S0z

Sy

jx

2DEG

S0

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