Spatially separated excitons in 2D and 1D David Abergel March 10th, 2015 D.S.L. Abergel 3/10/15 1 / 24
Outline Introduction 1 Spatially separated excitons in 2D – The role of disorder 2 Spatially separated excitons in 1D 3 D.S.L. Abergel 3/10/15 2 / 24
Introduction D.S.L. Abergel 3/10/15 3 / 24
The fundamental idea Key ingredients: Independent contacts to each layer High degree of nesting of Fermi surfaces Low SP tunneling rate between layers Picture credit: Kharitonov et al. , Phys. Rev. B 78 Phase coherence between the two layers Transport of excitons can be measured: Picture credit: Su et al. , Nat. Phys. 4 . Apply current in lower layer, measure voltage drop in upper layer (drag measurement). D.S.L. Abergel 3/10/15 4 / 24
A long-standing prediction Prediction was formation of ‘superconductivity’ with gap of the order of room temperature. D.S.L. Abergel 3/10/15 5 / 24
The impact of disorder in 2D with Enrico Rossi, Rajdeep Sensarma, and Martin Rodriguez-Vega, and Sankar Das Sarma. D.S.L. Abergel 3/10/15 6 / 24
Double layer graphene – Excitonic superfluidity The condensate has yet to be observed despite several experimental attempts. Question is: Why? D.S.L. Abergel 3/10/15 7 / 24
Double layer graphene – Excitonic superfluidity The condensate has yet to be observed despite several experimental attempts. Question is: Why? Possibility 1: Excitonic gap is too small. The form of the inter-layer screening used in the calculation of the gap is crucial: e 2 For SiO 2 or BN substrates, α = κ � v F ≈ 0 . 5 . For vacuum (suspended graphene), α = 2 . 2 . Unscreened interaction ⇒ room temperature condensate!!! Static screening ⇒ vanishing gap. Dynamic screening ⇒ ??? Sodemann et al. , Phys. Rev. B 85 , 195136 (2012). D.S.L. Abergel 3/10/15 7 / 24
Double layer graphene – Excitonic superfluidity The condensate has yet to be observed despite several experimental attempts. Question is: Why? Possibility 1: Excitonic gap is too small. The form of the inter-layer screening used in the calculation of the gap is crucial: e 2 For SiO 2 or BN substrates, α = κ � v F ≈ 0 . 5 . For vacuum (suspended graphene), α = 2 . 2 . Unscreened interaction ⇒ room temperature condensate!!! Static screening ⇒ vanishing gap. Dynamic screening ⇒ ??? Sodemann et al. , Phys. Rev. B 85 , 195136 (2012). Possibility 2: Disorder D.S.L. Abergel 3/10/15 7 / 24
Disorder in graphene systems STM can reveal atomic-scale structure of crystal. Also resolve the Dirac point, Which can be used to extract the local charge density. Rutter et al. , Nat. Phys. 7 , 649 (2009). Monolayer: Bilayer: Deshpande et al. , Phys. Rev. B 79 , 205411 (2009). Rutter et al. , Nat. Phys. 7 , 649 (2011). Scale bar is 20nm. Scale bar is 8nm. D.S.L. Abergel 3/10/15 8 / 24
Causes of inhomogeneity Scalar potential acts as a local shift in the chemical potential: Charged impurities: Ripples, corrugations, and strain: Zhang et al. , Nat. Phys. 5 , 722 (2009). Gibertini et al. Phys. Rev. B 85 , 201405(R) (2012). D.S.L. Abergel 3/10/15 9 / 24
Possibility 2 – Disorder Main question: Does charge inhomogeneity affect the formation of the condensate? + Upper layer Lower layer − D.S.L. Abergel 3/10/15 10 / 24
Possibility 2 – Disorder Main question: Does charge inhomogeneity affect the formation of the condensate? Upper layer µ u > 0 E + k x d µ u Upper layer − µ l < 0 Lower layer µ l Lower layer − µ = µ u + µ l δµ µ ¯ ¯ µ l This is similar to magnetic disorder in 2 superconductivity. µ u δµ = µ u − µ l D.S.L. Abergel 3/10/15 10 / 24
Our calculation There are three stages to the calculation: Theory for homogeneous unbalanced system. 1 ◮ Temporarily ignore inhomogeneity, calculate effect of imperfectly nested Fermi surfaces. Analysis of realistic inhomogeneity. 2 ◮ Calculate statistics for δµ ( r ) in situations corresponding to contemporary experiments. Combine these two results to assess impact of inhomogeneity on 3 condensate formation. D.S.L. Abergel 3/10/15 11 / 24
Step 1: T c in clean system – unscreened interaction (a) Unscreened, d =1nm (b) Unscreened, d =5nm Unscreened interaction: 200 200 V ( q ) = 2 πe 2 150 150 ǫq _ (meV) _ (meV) 100 100 µ µ ∆( δµ ) unchanged for 50 50 δµ < 2∆(0) . Equivalent to 0 0 0 10 20 30 0 10 20 30 Clogston–Chandrasekhar limit. δµ (meV) δµ (meV) No evidence of FFLO state. 0 50 100 150 T c (K) (c) 200 (d) 200 _=50meV δµ =0 µ _=100meV δµ =10meV µ _=150meV 150 δµ =20meV 150 µ E + ∆ T c (K) T c (K) δµ = 0 100 100 E - ∆ 50 50 E - _ 0 0 µ 0 50 100 150 200 0 10 20 30 _ v F k δµ (meV) µ (meV) D.S.L. Abergel 3/10/15 12 / 24
Step 1: T c in clean system – unscreened interaction (a) Unscreened, d =1nm (b) Unscreened, d =5nm Unscreened interaction: 200 200 V ( q ) = 2 πe 2 150 150 ǫq _ (meV) _ (meV) 100 100 µ µ ∆( δµ ) unchanged for 50 50 δµ < 2∆(0) . Equivalent to 0 0 0 10 20 30 0 10 20 30 Clogston–Chandrasekhar limit. δµ (meV) δµ (meV) No evidence of FFLO state. 0 50 100 150 T c (K) (c) 200 (d) 200 _=50meV δµ =0 µ _=100meV δµ =10meV µ _=150meV E + 150 δµ =20meV 150 µ ∆ T c (K) T c (K) δµ = ∆ 100 100 E - ∆ 50 50 E - _ 0 0 µ 0 50 100 150 200 0 10 20 30 _ v F k δµ (meV) µ (meV) D.S.L. Abergel 3/10/15 12 / 24
Step 1: T c in clean system – unscreened interaction (a) Unscreened, d =1nm (b) Unscreened, d =5nm Unscreened interaction: 200 200 V ( q ) = 2 πe 2 150 150 ǫq _ (meV) _ (meV) 100 100 µ µ ∆( δµ ) unchanged for 50 50 δµ < 2∆(0) . Equivalent to 0 0 0 10 20 30 0 10 20 30 Clogston–Chandrasekhar limit. δµ (meV) δµ (meV) No evidence of FFLO state. 0 50 100 150 T c (K) (c) 200 (d) 200 _=50meV δµ =0 µ _=100meV δµ =10meV E + µ _=150meV 150 δµ =20meV 150 µ ∆ T c (K) T c (K) δµ = 2 ∆ 100 100 E - ∆ 50 50 E - _ 0 0 µ 0 50 100 150 200 0 10 20 30 _ v F k δµ (meV) µ (meV) D.S.L. Abergel 3/10/15 12 / 24
Step 2: Analysis of inhomogeneity Broken translational symmetry makes it impossible to analytically calculate exact density distribution for random disorder. We employ a numerical method: Thomas-Fermi theory. Functional method (` a la DFT). The kinetic energy operator is also replaced by a functional of the density. This restricts the applicability to the regime where |∇ n/n | < k F , which is satisfied for double layer graphene. D.S.L. Abergel 3/10/15 13 / 24
Step 2: Analysis of inhomogeneity Energy functional is n u ( r ) n l ( r ′ ) E [ n u , n l ] = E u [ n u ( r )] + E l [ n l ( r )] + e 2 �� d 2 r d 2 r ′ | r − r ′ | 2 + d 2 2 κ � Layer energy functional includes contributions from disorder potential, and electron–electron interactions: E [ n ] = E K [ n ( r )] + e 2 d r n ( r ) n ( r ′ ) � � d r ′ 2 κ | r − r ′ | + e 2 � � d r V D ( r ) n ( r ) − µ d r n ( r ) . κ Ground state density landscape is found by numerically minimizing the energy functional with respect to the density distribution. Density distribution gives local chemical potential for each layer, and hence the local δµ . D.S.L. Abergel 3/10/15 14 / 24
Step 2: Analysis of inhomogeneity Using TFT, we calculate the spatial profile of δµ for a given manifestation of charged impurity disorder: d B = 1nm SiO 2 c.f. Austin d B = 20nm hBN SiO 2 c.f. Manchester D.S.L. Abergel 3/10/15 15 / 24
Step 3: Links back to BCS theory We can perform this calculation for many ( ≈ 600 ) disorder realizations and collect statistics for the distribution of δµ . This distribution characterized by it’s root-mean-square (rms) value. (a) d B = 1nm, d = 1nm (c) d B = 20nm, d = 1nm 40 δµ rms (meV) δµ rms (meV) 10 30 20 5 10 0 0 Predictions for ∆ from BCS theory: 0 100 200 0 100 200 _ (meV) _ (meV) µ µ Unscreened: ∆ ∼ 30 meV, (b) d B = 1nm, d = 5nm (d) d B = 20nm, d = 5nm Static screening: ∆ ∼ 0 . 01 meV, n imp = 2 × 10 11 40 n imp = 2 × 10 10 Dynamic screening: ∆ ∼ 1 meV. n imp = 2 × 10 9 δµ rms (meV) δµ rms (meV) 10 30 20 5 10 0 0 0 100 200 0 100 200 _ (meV) _ (meV) µ µ D.S.L. Abergel 3/10/15 16 / 24
Summary Excitonic superfluidity is severely impacted by charge inhomogeneity in the two layers. The very cleanest contemporary samples may be on the cusp of allowing the condensate. ◮ If estimates of the gap size using dynamical screening are to be believed. D.S.L. Abergel 3/10/15 17 / 24
Generalization to 1D D.S.L. Abergel 3/10/15 18 / 24
Excitons in core-shell nanowires B. Ganjipour et al. , Appl. Phys. Lett. 101 , 103501 (2012). D.S.L. Abergel 3/10/15 19 / 24
Excitons in core-shell nanowires B. Ganjipour et al. , Appl. Phys. Lett. 101 , 103501 (2012). d Case 2 allows for pairing. Ground state populations. Alternate geometries also possible. d D.S.L. Abergel 3/10/15 19 / 24
Assumptions and disclaimers No true long-range order in 1D. Particle correlations have power law decay ⇒ quasi-order. D.S.L. Abergel 3/10/15 20 / 24
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