Bilayer Quantum Hall Systems

Bilayer quantum Hall systems

Cristiane MORAIS SMITH

Institute for Theoretical Physics, Utrecht University, The Netherlands

Capri Spring School – p.1/25 Outline

Low-D systems: observation of quantum effects Here: 2D

2D electron gas and the quantum Hall effect (QHE): electron-liquid and electron-solid phases Single-layer: effect of spin Bilayer 2D electron gas: BEC of excitons

Capri Spring School – p.2/25 Single layer: Quantum Phases

M=7

Wigner crystal Bubble crystal Stripe phase

Electron-solid: Wigner crystal, Bubbles, Stripes Electron-liquid: Laughlin liquid, Moore-Read, Read-Rezayi states First-order quantum phase transitions between them

Capri Spring School – p.3/25 IQHE: single particle picture

one electron in ¢¡ : ¥ ©

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¨ 4 3 heB/m degenerate Landau 2 levels (LLs) Landau Levels 1 Density of states per n = 0

¨ ¨ LL: m ﬁlling factor : " ¨ !

Capri Spring School – p.4/25 Quantum Hall Ferromagnet

What about SPIN? Each Landau Level splits into two levels (Zeeman energy) Quantum Hall Ferromagnet at ¨ : ..... g n=0 ..... m=0 m=1 m=2 m=Nφ −1

Capri Spring School – p.5/25 Quantum Hall Ferromagnet

What about SPIN? Each Landau Level splits into two levels (Zeeman energy)

Magneto-excitons bosons ..... g n=0 ..... m=0 m=1 m=2 m=Nφ −1

Capri Spring School – p.5/25 Interacting 2DEG at

non-interacting bosons RPA interaction term Skyrmion/anti-Skyrmion pair ¢

Bosonization theory: 2DES at ¡

Doretto, Caldeira, Girvin, PRB 71, 45339 (2005)

Capri Spring School – p.6/25 non-interacting bosons RPA interaction term Skyrmion/anti-Skyrmion pair ¢

Bosonization theory: 2DES at ¡

Doretto, Caldeira, Girvin, PRB 71, 45339 (2005) Interacting 2DEG at ¨

Capri Spring School – p.6/25 non-interacting bosons RPA interaction term Skyrmion/anti-Skyrmion pair ¢

Bosonization theory: 2DES at ¡

Doretto, Caldeira, Girvin, PRB 71, 45339 (2005) Interacting 2DEG at ¨ ¦

¥ ¥ ¢ ¨ ¡ ¤ £ ¤ £ ¤ £

Capri Spring School – p.6/25 non-interacting bosons RPA interaction term Skyrmion/anti-Skyrmion pair ¢

Bosonization theory: 2DES at ¡

Doretto, Caldeira, Girvin, PRB 71, 45339 (2005) Interacting 2DEG at ¨ ¦ ¥ ¥ ¢¡

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¦ § £ § § ¦ ¦ © ¥ ¥ ¥ ¥

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Capri Spring School – p.6/25 ¢

Bosonization theory: 2DES at ¡

Doretto, Caldeira, Girvin, PRB 71, 45339 (2005) Interacting 2DEG at ¨ ¦ ¥ ¥ ¢¡

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non-interacting bosons RPA

interaction term Skyrmion/anti-Skyrmion pair

Capri Spring School – p.6/25 Spin excitations

Two kinds of excitations: 1.25

1 l)

ε 0.75 / 2 (e

q 0.5 w

0.25

0 0 2 4 6 8 10 Neutral: spin-waves |ql|

Capri Spring School – p.7/25 Spin excitations

Two kinds of excitations:

Charged: skyrmions

Capri Spring School – p.7/25 What about PSEUDOSPIN? Bosonization theory for QH bilayers ¢

Bosonization theory: 2DES at ¡

Doretto et al. PRB 72, 35341 (2005) ¡ ¨ ? - include SPIN in Hamiltonian theory ¢ - use Bosonization theory for ¨ Spin-excitations of the QH FM of composite fermions

Capri Spring School – p.8/25 ¢

Bosonization theory: 2DES at ¡

Doretto et al. PRB 72, 35341 (2005) ¡ ¨ ? - include SPIN in Hamiltonian theory ¢ - use Bosonization theory for ¨ Spin-excitations of the QH FM of composite fermions

What about PSEUDOSPIN? Bosonization theory for QH bilayers

Capri Spring School – p.8/25 Bilayer 2D electron gas

BEC of excitons Eisenstein and MacDonald, Nature 2004

Capri Spring School – p.9/25 Bilayer QH system

d nm nm

Bi−layer QHE ¡ ¡

¢ ¥ Ratio xxxxxxxxxxxxxxx ¨ © ¨§ £¥¤ ¡ ¦ ¤ : electron interlayer tunnelling/Coulomb energy

Capri Spring School – p.10/25 Bilayer QH system - experiments Magnetotransport data - different ¡ ¡

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∆V

I ¡ ¥¤ Incompressible-compressible phase transition Kellogg et al., PRL 88, 126804 (2002)

Capri Spring School – p.11/25 Bilayer QH system - experiments Magnetotransport data - different ¡ ¢

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∆V

I ¡ ¥¤ Incompressible-compressible phase transition Kellogg et al., PRL 88, 126804 (2002)

Capri Spring School – p.11/25 + v F = 0 B − v

Evidence excitonic superﬂuid

Bilayer QH system - experiments Magnetotransport data - counterﬂow ¡

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Kellogg et al., PRL 93, 036801 (2004)

Capri Spring School – p.12/25 Bilayer QH system - experiments Magnetotransport data - counterﬂow ¡

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¦ ¤ ¤

+ v F = 0 B − v

Evidence excitonic superﬂuid Kellogg et al., PRL 93, 036801 (2004)

Capri Spring School – p.12/25 Bilayer QH system - experiments ¡ ¡

Tunnelling conductance + B xxxxxxxxx ¤

2.5

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) -7 -1

-1 Ω 1.5 10

Ω 0.2 -6 (10 -200 0 200 B|| = 0 1.0 V (µV) dI/dV 0.1 (meV) eV* 0.5

0 0 10 20 30 0.0 B|| = 0.6T 6 -1 q (10 m ) -200 0 200 V (µV) Spielman et al., PRL 87, 036803 (2001)

Capri Spring School – p.13/25 Bilayer QH system - theory

......

d m 0 1 2 3 N φ −1

Spin-polarized electrons

Lowest Landau level

¨ ¨ ©

Capri Spring School – p.14/25 PROBLEM: no roton minimum is seen experimentally!

Bilayer QH system - theory

Fertig, PRB 40, 1087 (1989); MacDonald et al., PRL 65, 775 (1990) ¦ ¦ ©¥¤ £ £ © ¡¢

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¦ ¦ : BEC of excitons

Diagrammatic calc.

Linear mode ¡

¨ ¤

Capri Spring School – p.15/25 PROBLEM: no roton minimum is seen experimentally!

Bilayer QH system - theory

Fertig, PRB 40, 1087 (1989); MacDonald et al., PRL 65, 775 (1990) ¦ ¦ ©¥¤ £ £ © ¡¢

¥ ¥ ¥ £ ¨

¦ ¦ : BEC of excitons

Diagrammatic calc.

Linear mode ¡ ¥¤ ¨

Capri Spring School – p.15/25 PROBLEM: no roton minimum is seen experimentally!

Bilayer QH system - theory

Fertig, PRB 40, 1087 (1989); MacDonald et al., PRL 65, 775 (1990) ¦ ¦ ©¥¤ £ £ © ¡¢

¥ ¥ ¥ £ ¨

¦ ¦ : BEC of excitons

Diagrammatic calc.

Linear mode ¡ ¥¤ ¨

Capri Spring School – p.15/25 Bilayer QH system - theory

Fertig, PRB 40, 1087 (1989); MacDonald et al., PRL 65, 775 (1990) ¦ ¦ ©¥¤ £ £ © ¡¢

¥ ¥ ¥ £ ¨

¦ ¦ : BEC of excitons

Diagrammatic calc.

Linear mode ¡ ¥¤ ¨

PROBLEM: no roton minimum is seen experimentally!

Capri Spring School – p.15/25 N = N

Instability BEC of (interlayer) excitons

Bilayer QH system - boson model

......

d m 0 1 2 3 N φ −1

Capri Spring School – p.16/25 N = N

Instability BEC of (interlayer) excitons

Bilayer QH system - boson model

......

d m 0 1 2 3 N φ −1

Capri Spring School – p.16/25 Instability BEC of (interlayer) excitons

Bilayer QH system - boson model

......

d m 0 1 2 3 N φ −1

¨ ¨ !

Capri Spring School – p.16/25 Bilayer QH system - boson model

......

d m 0 1 2 3 N φ −1

¨ ¨ !

Instability BEC of (interlayer) excitons

Capri Spring School – p.16/25 Bilayer QH system - theory

Fermion model ¦ £¥¤

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£ © © ¡ £ £ £ £ § © £¨§ £ § £ ¦ Electron interlayer tunnelling: ¤ ¤

Coulomb interaction:

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Capri Spring School – p.17/25 Bilayer QH system - boson model

Fermion model ¦ £¥¤

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Coulomb interaction:

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Capri Spring School – p.18/25 Boson model ( )

where

Bilayer QH system - boson model £

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Capri Spring School – p.19/25 Bilayer QH system - boson model £

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where ¡

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Capri Spring School – p.19/25 Bilayer QH system - boson model

Doretto, Caldeira and C.M.S. PRL 97, 186401 (2006) Boson model: Bogoliubov approx. ¦ ¥ ¥ ¡

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Diag. canonical transformation

Capri Spring School – p.20/25 0.5 d/l

l) 0.2

ε 0.4 /

2 0.5

(e 0.8 q 0.3 Ω 1.0 1.5 0.2 2.0

0.1 0 1 2 3 4 |ql|

Bilayer QH system - boson model

Boson model: Bogoliubov approx. ¨§ £ ¡

¨ ¦ Dispersion relation quasi-particles: ¤ ¤

0.5

0.4 d/l 0.2 l)

ε 0.3 / 0.5 2

(e 0.8 q 0.2

Ω 1.0 1.5 0.1 2.0

0 0 1 2 3 4 |ql|

Capri Spring School – p.21/25 Bilayer QH system - boson model

Boson model: Bogoliubov approx. ¨§ £ ¡ ¨ ¦ ¤ ¤ Dispersion relation quasi-particles: ¤

0.5 d/l

l) 0.2

ε 0.4 /

2 0.5

(e 0.8 q 0.3 Ω 1.0 1.5 0.2 2.0

0.1 0 1 2 3 4 |ql|

Capri Spring School – p.21/25 3 No QHE

2 Novel phase l d/

Exciton 0.4 condensate 0.1 0 0.02 0.04 0.06 0.08 0.1 ∆ 2 ε SAS/(e / l) Murphy et al., PRL 72, 728 (1994)

Phase Diagram

Boson model: Popov approx.

2 l d/

Exciton 0.4 condensate 0.1 0 0.02 0.04 0.06 0.08 0.1 ∆ 2 ε SAS/(e / l)

Capri Spring School – p.22/25 Phase Diagram

Boson model: Popov approx.

3 No QHE

2 Novel phase l d/

Exciton 0.4 condensate 0.1 0 0.02 0.04 0.06 0.08 0.1 ∆ 2 ε SAS/(e / l) Murphy et al., PRL 72, 728 (1994)

Capri Spring School – p.22/25 Bilayer QH system - boson model

Boson model: Bogoliubov approx.

Condensate fraction:

0.8 ∆ SAS

bosons 0.6 0 >/N

0 0.4 0.10

0 0 0.5 1 1.5 2 2.5

d/l Capri Spring School – p.23/25 Conclusions

Single-layer 2DEG: quantum phases - electron liquid: Laughlin, CF-mixture - electron solid: WC, bubbles, stripes Quantum phase transitions: -solid/liquid, solid/solid Bilayer 2DEG: - BEC of excitons - phase transition not yet understood

Capri Spring School – p.24/25 Collaborators

Single-layer QHE: Mark Goerbig and Pascal Lederer Bilayer BEC of excitons: Ricardo Doretto and Amir Caldeira

Capri Spring School – p.25/25