Marcin Swillo School of Engineering Sciences, KTH, Stockholm

Non-linear phenomena in semiconductor nanostructures

Adopt Winter School 2012

Collaboration

• Semiconductor Materials Division (KTH/ ICT):

Assoc. Prof. Anand Srinivasan

Reza Sanatinia

-SHG in semiconductor nanopillars

Outline

• Introduction • Nonlinear polarization on the surface

- Structure discontinuity - Electric quadrupole

• Surface contribution to SHG (GaP nanopillar)

• Bulk contribution to SHG (GaP nanopillar)

• Experiment

Electric displacement field:

jjj PED 0

Polarization density (applying Taylor expansion):

.3,2,1j

.3,2,1,...,, lkj

.]exp[)()( cctit

EE

If the electric field E is a superposition of monochromatic waves:

Linear

jPNonlinear

jP

Introduction

0

0

jkl

jkl

for centrosymmetric crystals

for non-centrosymmetric crystals

..., mlkjklmlkjklkjkj EEEEEEP

Nonlinear

jkjkj PED

Electric displacement field:

Wave equation:

2

2

02

2

0

2

t

P

t

EE

Nonlinear

jkjkj

Three-wave mixing: Four-wave mixing:

321 4321

lkjklj EEP )2(

mlkjklmj EEEP )3(

321 kkk 4321 kkkk

Introduction

(second order nonlinearity) (third order nonlinearity)

Introduction

Second harmonic generation (SHG):

pump 21

P. Franken, A. Hill, C. Peters, G. Weinreich, "Generation of Optical Harmonics". Physical Review Letters 7 (4), 118 (1961).

R. W. Terhune, P. D. Maker, and C. M. Savage, “Optical Harmonic Generation in Calcite”, Physical Review Letters 8 (10), 404 (1962)

The first observation of SHG (crystalline quartz –> non-centrosymmetric crystal):

The first observation of SHG in calcite (centrosymmetric crystal):

● Third order nonlinearity:

● Second order nonlinearity: 211111111

)2(

1 )(EdEEP

- Using external electric field: - Electric quadrupole

ml

DC

kjklmj EEEP )3(

● Second order nonlinearity: - Structure discontinuity at the surface

E

Au

- Local field enhancement - Structure discontinuities - Electric quadrupole contribution

Nonlinear polarization on the surface

A. Bouhelier, M. Beversluis, A. Hartschuh, L. Novotny Physical Review Letters 90, 013903 (2003)

Second harmonic generation on the glass surface

Electrostatic approximation:

Structure discontinuities

x

y

z

lkjklj EEP )2(

x

y

z

x’

y’

z’

x’ = x y’ = -y z’ = -z

)2()2(

'''1515

)2( 22 xxxzxzx PPEEdEEdP

E E

015 d

Using symmetry operations the only nonvanishing elements are:

yx

xz

yz

z

y

x

z

y

x

EE

EE

EE

E

E

E

d

d

ddd

P

P

P

2

2

2

2

2

2

35

26

131211

yx

xz

yz

z

y

x

z

y

x

EE

EE

EE

E

E

E

d

d

ddd

P

P

P

2

2

2

2

2

2

35

26

131211

x

y

z

1312 dd

3526 dd

Using symmetry operations some nonvanishing elements are identical:

11d

Electric dipole contribution

lkjklj EEP )2(

Structure discontinuities

Electric quadrupole contribution to SHG

]exp[})]([)0({),( tixxEEtxE x

]2exp[))2((2

)(

)(2

)(]exp[

)()(

22

0

222

0

2

0

3

22

0

2

0

2

0

3

22

0

0

2

tim

xEENe

m

xEENeti

m

ENetP xx

Electron oscillator for electric quadrupole approximation:

LP )2(PStatic polarization

e

E(x, t)

x

● Third order nonlinearity (2,0,,): 02

x

y

z

'

35

'

26 dd

'

11d

Electric quadrupole contribution

0

x

Ex

Electric quadrupole contribution to SHG

D. Epperlein, B. Dick, and G. Marowsky, „Second-Harmonic Generation in Centro-Symmetric Media”, Applied Physics B 44, 5-10 (1987)

1

2

2'

11

i

sd

Preferred a high refractive index Semiconductors

yx

xz

yz

z

y

x

z

y

x

EE

EE

EE

E

E

E

d

d

ddd

P

P

P

2

2

2

2

2

2

'

35

'

26

'

13

'

12

'

11

1312 dd

3526 dd

11d

Structure discontinuities (electric dipole)

'

35

'

26 dd

'

11d

Electric quadrupole contribution

Amorphous solid

x

y )2(

z

Crystal Class: -43m

GaP crystal

Bulk nonlinearity:

362514 ddd

1312 dd

11d'

11d

3625 dd '

36

'

25 dd

14d

Structure discontinuities (electric dipole)

Electric quadrupole contribution

[001] x

y

z

Nonlinear susceptibility on the surface

Picture from: Benjah-bmm27

yx

xz

yz

z

y

x

z

y

x

EE

EE

EE

E

E

E

d

d

d

P

P

P

2

2

2

2

2

2

36

25

14

3526 dd '

35

'

26 dd

Array of GaP nanopillar waveguides

x

y

Electric field Ex of guided fundamental mode

Wavelength: 840 nm Diameter: 150 nm

kx

kpump kSH kpump

kx

]2

[Ak

ASinc x

A

2

A

4

A

6

A

2

A

4 0

]2

[]2

[],[ 11)2( LkSinc

AkALSincdkk zxzx

kz

]2

[Lk

LSinc z

A

11

)2( d0)2(

kpump

Nanopillar Air

L

surf

ace

kpump Ex pump

z

x

𝑃𝑥2

= 𝜒(2)𝐸𝑥𝐸𝑥

Surface contribution to SHG

kz

Ex

x [m]

y [

m]

00,10,20,30,40,50,60,70,80,9

1

50 80 110 140 170 200 230 260 290 320 350

Diameter [nm]

SH

Inte

nsi

ty [

a.u

]

𝑃⊥2

= 𝜒(2)[𝐸𝑥∙ 𝒆⊥]2

Nanopillar

Air

bpump

Ex

e E

z

x

y

Surface contribution to SHG

25

)2( d

bpump

bSH

Nanopillar Air Air

L

Ex

]2

[2][ 25)2( LkLSincdk zz

bpump bSH

bpump

kz

kz

]2

[Lk

LSinc z Ez

kpump Ex pump

Ex

Ez

z

x

𝑃𝑦2

= 𝜒(2)𝐸𝑥𝐸𝑧

Bulk contribution to SHG

Ey

00,10,20,30,40,50,60,70,80,9

1

50 80 110 140 170 200 230 260 290 320 350

Diameter [nm]

SH

Inte

nsi

ty [

a.u]

𝑃𝑦2

= 2𝑑25𝐸𝑥𝐸𝑧 Ex

Ez

x [m]

x [m]

y [

m]

y [

m]

Bulk contribution to SHG

(d) (c) (b)

RIE CAIBE

(a) Deposition of colloidal silica nanospheres (500nm)

(b) Size reduction by RIE (c) Etching by Ar/Cl2 CAIBE (d) Nanopillars ca. 1m high, diameter

100-250 nm

(a)

Semiconductor Materials Division (KTH)

Nanopillar fabrication steps

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

50 80 110 140 170 200 230 260 290 320 350

Measurements

Surface simulation

Bulk simulation

Fitting

Diameter [nm]

SH

In

ten

sity

[a.

u]

[0 0 1] direction

SHG- 420 nm

Pump 840 nm

50X, NA= 0.5

150 nm 250 nm

SHG measurement

R. Sanatinia, M. Swillo, S. Anand, Nano Letter., 2012, 12 (2), p. 820

SHG efficiency (detected): 2∙10-7 % Coherence lenght for SHG: ~ 100 nm

0,99

1,09

1,19

400 420 440

[0 0 1]

direction

SHG- 420 nm

50X, NA= 0.5

Pump 840 nm

5X, NA= 0.15

Polarizer

Epump

0

15 °

30 °

45 °

60 °75 °90 °105 °

120 °

135 °

150 °

165 °

180 °

195 °

210 °

225 °

240 °255 ° 270 ° 285 °

300 °

315 °

330 °

345 °

Epump

Polarization of SH light

SHG measurement

Nanopillar diameter: 150 nm, 250 nm