Aharonov-Bohm effect and persistent currents in quantum rings A. A. Lopes Department of Physics University of Aveiro September 2008 Lopes (UA - DP) Quantum rings September 2008 1 / 31
Outline Introduction 1 Free electron gas ring 2 Tight-binding ring 3 Interacting spinless fermions 4 Summary 5 Lopes (UA - DP) Quantum rings September 2008 2 / 31
Introduction Outline Introduction 1 Free electron gas ring 2 Tight-binding ring 3 Interacting spinless fermions 4 Summary 5 Lopes (UA - DP) Quantum rings September 2008 3 / 31
Introduction Aharonov-Bohm effect Figure: Aharonov-Bohm effect general geometry. Aharonov-Bohm effect phase shift ∆ ϕ = q � l = 2 π φ A . d � � = φ ′ (1) � φ o C Lopes (UA - DP) Quantum rings September 2008 4 / 31
Introduction Aharonov-Bohm effect It is a quantum mechanical topological effect Produces a phase shift in the electron’s wavefunction The electron’s properties are affected even if they don’t “feel“ a force Leads to persistent currents Leads to periodic thermodynamical properties Lopes (UA - DP) Quantum rings September 2008 5 / 31
Introduction Persistent currents Currents that exist in quantum rings even with no potential difference Are due to Aharonov-Bohm’s effect Have been measured experimentally, using SQUIDs, agreeing with theory One level current I n = − ∂ E n ∂φ = bM n (2) Lopes (UA - DP) Quantum rings September 2008 6 / 31
Free electron gas ring Outline Introduction 1 Free electron gas ring 2 Tight-binding ring 3 Interacting spinless fermions 4 Summary 5 Lopes (UA - DP) Quantum rings September 2008 7 / 31
Free electron gas ring Description Description and energy Independent electrons Electron’s don’t feel a local potential from the ions Free electron gas Hamiltonian � 2 � p − q � � A H = (3) 2 m Energy � 2 � n + φ E n = E 0 , n = 0 , ± 1 , ± 2 , ... (4) φ o Lopes (UA - DP) Quantum rings September 2008 8 / 31
Free electron gas ring Energy and magnetic moment Energy Energy � E o Energy � E 0 14 14 12 12 10 10 8 8 6 6 4 4 2 2 3 Φ � Φ o 3 Φ � Φ o � 3 � 2 � 1 1 2 � 3 � 2 � 1 1 2 (a) No potential (b) Weak potential Figure: Plot of the first five energy levels for a quantum ring using the free electron model (a) and using the free electron model with a weak potential (b). Lopes (UA - DP) Quantum rings September 2008 9 / 31
Free electron gas ring Energy and magnetic moment Magnetic moment M � M 0 M � M o 4 4 2 2 3 Φ � Φ o 3 Φ � Φ o � 3 � 2 � 1 1 2 � 3 � 2 � 1 1 2 � 2 � 2 � 4 � 4 (a) No potential (b) Weak potential Figure: Plot of the first five magnetic moment levels for a quantum ring using the free electron model (a) and using the free electron model with a weak potential (b). Lopes (UA - DP) Quantum rings September 2008 10 / 31
Free electron gas ring Persistent currents Persistent currents I � I 0 I � I 0 1.0 1.0 0.5 0.5 0.5 Φ � Φ o 0.5 Φ � Φ o � 0.5 � 0.5 � 0.5 � 0.5 � 1.0 � 1.0 (a) N e odd (b) N e even I � I 0 0.5 � 0.5 0.5 Φ � Φ o � 0.5 (c) Averaged over N e Figure: For N e even and N e odd the p.c. are different but both have the same periodicity. Averaging the persistent currents over N e result in a halving of the period and of the maximum current. Lopes (UA - DP) Quantum rings September 2008 11 / 31
Tight-binding ring Outline Introduction 1 Free electron gas ring 2 Tight-binding ring 3 Interacting spinless fermions 4 Summary 5 Lopes (UA - DP) Quantum rings September 2008 12 / 31
Tight-binding ring Description Description Independent electrons Electrons are strongly bound to ions Spinless fermions Lopes (UA - DP) Quantum rings September 2008 13 / 31
Tight-binding ring Description Hamiltonian Tight-binding Hamiltonian with flux N � � e i ( φ ′ / N ) c † � H = − t j c j + 1 + h . c . (5) j = 1 Lopes (UA - DP) Quantum rings September 2008 14 / 31
Tight-binding ring Energy and magnetic moment Energy and magnetic moment Energy k − φ ′ � � E = − 2 t cos (6) N with k lying in the first Brillouin zone. Magnetic moment k − φ ′ � � M = − M o sin (7) N where M o = 2 t π R 2 . N φ o Lopes (UA - DP) Quantum rings September 2008 15 / 31
Tight-binding ring Persistent currents Persistent currents I � I 0 I � I 0 1.0 1.0 0.5 0.5 0.5 Φ � Φ o 0.5 Φ � Φ o � 0.5 � 0.5 � 0.5 � 0.5 � 1.0 � 1.0 (a) N = 2 , N e odd (b) N = 2 , N e even I � I 0 I � I 0 1.0 1.0 0.5 0.5 0.5 Φ � Φ o 0.5 Φ � Φ o � 0.5 � 0.5 � 0.5 � 0.5 � 1.0 � 1.0 (c) N = 7 , N e odd (d) N = 7 , N e even Figure: Persistent currents for an odd and an even number of electrons. Lopes (UA - DP) Quantum rings September 2008 16 / 31
Tight-binding ring One impurity ring One impurity ring - Description Figure: A ring enclosing an external flux with an impurity at site N. Lopes (UA - DP) Quantum rings September 2008 17 / 31
Tight-binding ring One impurity ring Hamiltonian Hamiltonian N N − 2 � � � c † � e i ( φ ′ / N ) c † H = ε 0 j c j − t j c j + 1 + h . c . j = 1 j = 1 (8) N � � � e i ( φ ′ / N ) c † + ε N c † N c N − t N j ′ c j ′ + 1 + h . c . j ′ = N − 1 ε N - impurity on-site energy ε 0 - non impurity on-site energy t N - hopping factor from and to the impurity t - the hopping factor between non-impurity sites Lopes (UA - DP) Quantum rings September 2008 18 / 31
Tight-binding ring One impurity ring Bound state Energy � t Energy � t 3 3 2 2 1 1 6 2 ΠΦ � Φ o 6 2 ΠΦ � Φ o � 6 � 4 � 2 2 4 � 6 � 4 � 2 2 4 � 1 � 1 � 2 � 2 (a) Perfect ring (b) Ring with impurity Figure: Energy vs magnetic flux plot for a perfect ring and for a ring with an impurity with ε 0 = 3 ( N = 20). Lopes (UA - DP) Quantum rings September 2008 19 / 31
Tight-binding ring Disordered Ring Disordered ring - Description In a disordered ring we can consider that t is different between different sites We can also consider that the on-site energy is different between different sites Lopes (UA - DP) Quantum rings September 2008 20 / 31
Tight-binding ring Disordered Ring Energy Energy � t � Energy � t 2 2 1 1 6 2 ΠΦ � Φ o 6 2 ΠΦ � Φ o � 6 � 4 � 2 2 4 � 6 � 4 � 2 2 4 � 1 � 1 � 2 � 2 (a) Perfect ring (b) Disordered ring Figure: Energy vs magnetic flux plot for a perfect ring and a disordered ring ( N = 4 and t j chosen randomly). Lopes (UA - DP) Quantum rings September 2008 21 / 31
Tight-binding ring Disordered Ring Ensemble persistent current Figure: Persistent current curves for an ensemble of disordered rings (averaged over 100 disorder configurations) with t j chosen randomly, N = 4 and ε F = − 0 . 8 Lopes (UA - DP) Quantum rings September 2008 22 / 31
Interacting spinless fermions Outline Introduction 1 Free electron gas ring 2 Tight-binding ring 3 Interacting spinless fermions 4 Summary 5 Lopes (UA - DP) Quantum rings September 2008 23 / 31
Interacting spinless fermions Description Description t-V Hamiltonian with flux N N � � � c † � � � H = − t j c j + 1 + h . c . + V n j n j + 1 . (9) j = 1 j = 1 where V is the Coulomb energy between two nearest-neighbors electrons and n j = c † j c j is the occupation number at site j . Figure: The t-V model is a spinless tb model with Coulomb interaction between nearest-neighbors. Lopes (UA - DP) Quantum rings September 2008 24 / 31
Interacting spinless fermions Strong coupling limit Energy in the strong coupling limit (V/t » 10) (a) Global view (b) Zoom in region V=1 Figure: Energy-flux plots for for a system with N = 6 and N e = 3 in the strong coupling limit. Lopes (UA - DP) Quantum rings September 2008 25 / 31
Interacting spinless fermions Strong coupling limit Energy when not in the strong coupling limit (a) Global view (b) Zoom in the top energy sub- space Figure: Energy plot for a ring with N = 5 and N e = 3. Lopes (UA - DP) Quantum rings September 2008 26 / 31
Interacting spinless fermions Strong coupling limit Persistent currents (a) t = 1, V = 0 . 1, (b) t = 1, V = 1, (c) t = 1, V = 10, period = 2 π period = 2 π period = π Figure: P .c. for the fundamental level of the top subspace for a system with N = 5 and N e = 3. Lopes (UA - DP) Quantum rings September 2008 27 / 31
Interacting spinless fermions Strong coupling limit One impurity t-V ring (a) Without impurity (b) With impurity Figure: Comparison of the energy vs flux plots for a t-V ring with N = 5 and N e = 3 with and without an impurity. Lopes (UA - DP) Quantum rings September 2008 28 / 31
Summary Outline Introduction 1 Free electron gas ring 2 Tight-binding ring 3 Interacting spinless fermions 4 Summary 5 Lopes (UA - DP) Quantum rings September 2008 29 / 31
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