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 ¢¡ : ¥ © £¥¤§¦ ¨ 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 .6/25 p – School ing Spr i Capr ¢ ¡ (2005) ¡ at © © £ 45339 © £ 2DES 71, £ ¤ © ¥ £ PRB ¤ ¦ ¥ ¨ vin, ¤ theory: Gir at © £ £ £¨§ a, ¢ ¡ 2DEG ¨ Caldeir , acting Doretto Inter Bosonization pair mion A RP mion/anti-Skyr Skyr bosons m ter acting action non-inter inter .6/25 p – School ing Spr i Capr ¢ ¡ (2005) at § ¥ £ © ¥ 45339 ¨ ¦ § ¦ ¥ £ 2DES 71, © £ ! ¥ © ¦ ¨ £ ¥ ¦ ¥ ¦ £ PRB ¤ © ¨ £ vin, £ § § theory: © § Gir at ¦ ¢¡ a, ¢ ¡ 2DEG ¨ Caldeir , acting Doretto Inter Bosonization pair mion A RP mion/anti-Skyr Skyr bosons m ter acting action non-inter inter .6/25 p – School ing Spr i Capr ¢ ¡ pair mion (2005) at § ¥ £ © ¥ 45339 ¨ ¦ A § ¦ ¥ £ 2DES 71, © RP £ ! ¥ © ¦ ¨ £ ¥ mion/anti-Skyr ¦ ¥ ¦ £ PRB ¤ © ¨ Skyr £ vin, £ § § bosons theory: © § Gir at m ¦ ¢¡ a, ter ¢ acting ¡ 2DEG ¨ Caldeir action , acting non-inter inter Doretto Inter Bosonization 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 .10/25 p – School gy ing Spr i ener Capr nm nm ¥ ¢ tunnelling/Coulomb ¨ er y ¡ d ¨ interla on electr system : ¡ ¨ ¨§ xxxxxxxxxxxxxxx ¡ © Bi−layer QHE QH ¦ ¤ er £¥¤ ¨ Ratio © Bilay Bilayer QH system - experiments Magnetotransport data - different ¡ ¡ ¢£ ∆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 ¡ ¢ ¢£ ∆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 ¡ ¤ £¥¤ ¢ ¡ ¦ ¤ ¤ Kellogg et al., PRL 93, 036801 (2004) Capri Spring School – p.12/25 Bilayer QH system - experiments Magnetotransport data - counterflow ¡ ¤ £¥¤ ¢ ¡ ¦ ¤ ¤ + 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 ¢ ¢ ¨ ¥ ¤ 2.0 £ ¨ ¤ m/s ) -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 W ...... ...... 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) ¦ ¦ ©¥¤ £ £ © ¡¢ ¥ £ ¥ ¥ ¨ ¦ ¦ : 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 W ...... ...... 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 W ...... ...... 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 W ...... ...... d m 0 1 2 3 N φ −1 ¨ ¨ ! © ¥§¦ ¡ £ ¢ ¤ N ¡¢ ¡ £ ¢ = N Capri Spring School – p.16/25 Bilayer QH system - boson model W ...... ...... d m 0 1 2 3 N φ −1 ¨ ¨ ! © ¥§¦ ¡ £ ¢ ¤ N ¡¢ ¡ £ ¢ = N Instability BEC of (interlayer) excitons Capri Spring School – p.16/25 .17/25 p – School ing ¢ Spr i ¨ Capr © ! © ¨ ¡ ¤ ¤ ¡ © © £ § ¢ ¤ ¨ ¥ © £ ¤ ¢ ¦ © ¥ £ ¤ © ¦ ¤ © theory £ ¤ £ § £ - £ ¦ ¤ © £ £¥¤ £¨§ § © ! ¡ © ¨ ¨ system tunnelling: § er y ¨ QH interaction: © interla model er £ on Fermion Electr Coulomb Bilay .18/25 p – ¢ School ¨ £ ing ¡ Spr i ¤ © © Capr ¢ © ¡ ¢ © © ! © © ¢ ¨ ¢ ¢ ¨ © § £¡ ¨ © ¡ ¤ ¢ ¢ ¤ © £ ¦ ¤ ¡ ¥ © ¢ model ¤ ¦ ¥ © © ¢ ¤ boson © - ¦ ¤ £ ¦ £¥¤ © ¡ ¨ system interaction: ¨ model QH © ¢ er Fermion Coulomb Bilay .19/25 p – School ing Spr ¡ i © Capr ¢ ¢ © £ ¢ ¥ £ £ ¥ ¢ £ © ¦ ¥ ¦ ¦ ¥ £ ¦ © © ¡£ ¡ © £ ¢ model £ ) ¡ © ¨ ¢ ¢ ¡ ¡ boson ¦ ¤ © - ¡ ¦ § ¡ £ ¤ £ ¦ ( ¦ § ¨ ¨ system ¨ © model representation: © QH ¢ ¢ ¢ er Fermion bosonic Bilay ) ( model Boson where .19/25 p – School ing Spr i Capr § ¥ ¢ ¡ £ ¥ © ¨ ¦ ¦ £ © ¥¤ ¦ ¥ © § £ ¡£ ! ¢ ¦ ¨ ¥ ¦ ¢ ¨ © © £ © ¦ ¡ £ ¡£ § § ¦ model § © © ¡ ¦ ¡ ) £ £ ¦ £ ¨ ¥ £ ¨ boson ¦ ¥ - ¦ ¦ ¤ ¤ ¤ £ ¡ £ § ( ¨ © system ¨ ¦ £ ¤ ¦ model QH er Boson where Bilay .20/25 p – ¨ School £ ing ¥ Spr £ i ¨ ¡ ¦ Capr ¥ £ ¨ (2006) ¦ ¥ £ ¦ ¥ ¥ © ¦ ¦ § ¥ model 186401 £ ¨ ¦ 97, ¥ £ ¨ ¥ boson PRL (1,1,1) x. - o £ xxxxxxxxxxx ¥ £ ¦ mation ¥ appr .M.S © v C ¦ or ¦ § xcitons e-function e £¥¤ and ansf ¦ v ¡ system £ £ tr a a ¡ ¨ w ¡ Bogoliubo s in’ QH ¢ ¦ of Caldeir , er model: ¢ ¨ canonical BEC Halper ¢ £ Doretto Boson Diag. Bilay 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/ 1 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. 3 2 l d/ 1 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/ 1 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: 1 0.8 ∆ SAS bosons 0.6 0 >/N 0 0.4 0.10 <N 0.2 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.