The noise spectrum of a quantum dot Eitan Rothstein Netanel Gabdank Ora Entin-Wohlman* Amnon Aharony* Physics Department, Ben Gurion University of the Negev * Also at Tel Aviv University Mesoscopic Physics Meso = Intermidiate, in the middle. Mesoscopic physics = A mesoscopic system is really like a large molecule, but it is always, at least weakly, coupled to a much larger, essentially infinite, system via phonos, many body excitation, and so on. (Y. Imry, Introduction to mesoscopic physics) A nave definition: Something very small coupled to something very large. 2/10 Mesoscopic Physics 3/10 Quantum dot There are different types of quantum dots. A large atom connecting to two ledas A metallic grain on a surface Voltage gates on 2DEG 4/10 The model The model
n Vn ,k k Left (L) electron ic reservoi r Quantum-Dot (QD) N Right (R) electroni c reservoir p -N Vg DC conductance Noise spectrum C ( ) dte I(t )I(0) i t I GV I ( I L I R ) / 2 V L R Vn , p G 2 / e 2 I I I ( IL
F / 5/10 I IR ) / 2 2C ( ) / e2 No interaction Hartree Fock / The model The Hamiltonian N E H c c Vn ,k (c dn h.c) n d dn C ( dn dn N g ) 2 2 n N n N p c p c p Vn , p (c p dn h.c) electronic charging H QD k k k k N n states on QD
momentum states in R/L reservoirs energy coupling R/L reservoir-QD d d d d Hartree Fock approximation n n m m dndn dm dm dm dm dndn dndm dm dn dm dn dndm N H effQD n dndn EC ( dndn N )dm dm EC dm dn dndm [const.] n N n m n m Hartree Fock Self-consistent dE calculation / ( L R ) / F d n d m QD 1 Im[ E H eff i]nm N nm F /
6/10 Level Occupancy and Effective Hamiltonian ~n n EC ( N mm The effective Hamiltonian N ~ H effQD ~n dndn Vmn dndm n N n m Level Occupancy N nn F 3.5 (Hartree) m ~ Vnm EC N mn Effective single-e energies ~ 3.5 n F (Hartree) EC N nn ; N nm * energy scaled with N nn N g )
EC ~ ~n ;Vnm F 3.5 (Hartree Fock) F 3.5 (Hartree Fock) Hartree terms Hartree terms Fock terms Fock terms EC EC 7/10 The Scattering Formalism Treating the problem in the Landauer Bttiker formalism Tight-Binding model Current and correlations S(E) ( r L, t ) ( r R, t ) scattering region scattering states QD 1 i 2nm [ E H eff ]nm L
S ( E ) 1 QD 1 1 inm [ E H eff ]nm R L e2 I (t ) I (t ' ) (2 )2 I (t )I (t ' ) d 4E (2 )2 e2 ( 2 )2 L R R e I (t ) 2 dEdE ' scattering states A ( , E , E ' )a a e 2 i ( E E ') t * A ( , E , E ' ) S ( E ) S ( E ' ) e dE | S ( E ) |2 ( f ( E ) f ()ELandauer )) ( 2
A ( , E , E ' )A ( , E ' ' , E ' ' ' )a ( E )a ( E ' )a ( E ' ' )a ( E ' ' ' )e 2 d E A ( , E , E ' ) A ( , E ' , E ) f ( E )(1 8/10 f ( E ' ))ei ( E E ')( t t ') i ( E E ' ) t i ( E '' E ''' ) t ' e )Bttiker( The noise spectrum Noise spectrum C ( ) dteit I(t )I(0) L and R F T0 Calculations performedk Bfor , so the 2 e Cnoise ( ) given [2 r * ( E ) r ( E ) 2t * ( E )t ( E ) r '*( E ) r ' ( E )] dEby F 4 F
2C ( ) / e 2 2C ( ) / e 2 symmetricL R EC symmetricL R F 0.5 EC [0,1] F 0.2 EC 0.6 Hartree No interaction Hartree Hartree Fock 2C ( ) / e * energy scaled with 2C ( ) / e 2 2 symmetricL R EC F 0.5 EC [0,1] asymmetric R / L 9 F 0.2
EC 0.6 Hartree Fock No interaction Hartree Hartree Fock 9/10 Summary The noise is the signal - R. Landauer The noise spectrum exhibits steps and dips o The steps quantum dot resonances o The dips energy difference between the resonances contain information on the coupling asymmetry t The interaction effect o Shifts levels to preserves neutrality o Shifts noise steps and dips Thanks Fock effect o Small quantitative effect modifies the width of the levels 10/10
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