Martes Cuánticos. Quantum Capacitors. (Quantum RC-circuits) Victor A. Gopar

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Martes Cuánticos Quantum Capacitors (Quantum RC-circuits) Victor A. Gopar -Universal resistances of the quantum resistance-capacitance circuit. Nature Physics, 6, 697, 2010. C. Mora y K.

1 Martes Cuánticos Quantum Capacitors (Quantum RC-circuits) Victor A. Gopar -Universal resistances of the quantum resistance-capacitance circuit. Nature Physics, 6, 697, C. Mora y K. Le Hur -Violation of Kirchhoff's Laws for a coherent RC Circuit. Science, 313, 499, (2006). J. Gabelli, G. Féve, et. Al -Dynamic admittance of mesoscopic conductors: Discrete potential model. PRB, 54, 8139 (1996). A. Prete, H. Thomas, M. Büttiker 1

2 Motivation Paper recommended by Markus Büttiker ( ): an experimental realization of a Gedankenexperiment [M. Büttiker, T. Petre, Phys. Letts. A, 180, 364 (1993); V.A.G, P. Mello, M. Büttiker, PRL,77,3005 (1996)] Martes Cuánticos 2

3 Motivation Nanoscopic/Mesoscopic Physics deals with simple systems that show fundamental physics. A quantum capacitor is (another) nice example of a simple circuit showing quantum phenomena Martes Cuánticos 3

4 Motivation Experimental set up with promising future for the production of electronic entanglement Nature, Octuber, 2013 Martes Cuánticos 4

5 Essentially:all this stuff is about quantum effects in small (& cold) circuits The wave nature of electrons is revealed in electronic transport experiments Martes Cuánticos 5

6 Introduction:Coherent quantum (dc) transport as a scattering problem Landauer-Büttiker approach (dc transport) T R (Conductance quantum) Electronic transport ( Notice that for T = 1, :conductance) = scattering problem ( : transmission),i.e., the resistance is finite.

7 Introduction:Coherent quantum (dc) transport as a scattering problem Landauer-Büttiker approach (dc transport) a b' b a' Martes Cuánticos

8 Introduction:Coherent quantum (dc) transport as a scattering problem Martes Cuánticos 8

9 Wave nature of electrons in transport experiments: Coherent quantum transport Experimental verification of conductance quantization! Martes Cuánticos 9

10 Wave nature of electrons in transport experiments: Coherent quantum transport Aharanov-Bohm effect in small rings ('the experiment') quantum flux: Martes Cuánticos 10

11 Introduction:Coherent quantum (dc) transport as a scattering problem At finite temperature (assuming, ) Martes Cuánticos 11

12 Introduction: dynamic (ac) quantum transport as a scattering problem But we are interested in a dynamic situation. What is the response of a coherent circuit under an ac potential? Martes Cuánticos 12

13 Introduction: dynamical (ac) quantum transport as a scattering problem The situation is much more complex, we need to describe (ac and dc) electronic transport in a self-consistent way. V:applied potential U:internal potential Martes Cuánticos 13

14 Introduction: dynamical (ac) quantum transport as a scattering problem That is, we want to know the current where, given by are the admittances, considering possible charge accumulation (internal potential variations) in the circuit as a response to the applied fields. A nontrivial problem! 14

15 Introduction: dynamical (ac) quantum transport as a scattering problem This can been done in two steps (Büttiker): Currents at the contacts are calculated in response to oscillating voltages at the contacts, keeping the internal electrostatic potential fixed. An internal response due to potential induced by the injected charges is considered External response Internal response 15

16 Introduction: dynamical (ac) quantum transport as a scattering problem For the external-response admittance, it is found: Partial density of states Charge relaxation resistance for one channel 16

17 Introduction: dynamical (ac) quantum transport as a scattering problem The internal potential U, is obtained from (and imposing current conservation) C: capacitance (Coulomb interactions) Q: charge accumulated in the conductor m in response to a internal potential U Martes Cuánticos 17

18 Introduction: dynamical (ac) quantum transport as a scattering problem It turns out that and and can be written in terms of, in a self-consistent way. The final result for the (total) admittance is: where 18 (All quantities can be expressed in terms of the scattering matrix associated to the system)

19 Introduction: dynamical (ac) quantum transport as a scattering problem An example: consider a two-plates capacitor characterized by the geometrical capacitance C Martes Cuánticos 19

20 Introduction: dynamical (ac) quantum transport as a scattering problem Macroscopically, the current response is given by : admittance Martes Cuánticos 20

21 Introduction: dynamical (ac) quantum transport as a scattering problem Let's go quantum. For a coherent/quantum two-plates capacitor half the resistance quantum 21

22 Introduction: dynamical (ac) quantum transport as a scattering problem or But actually the field penetrates the plates! : 22

seen as a (series) sum of a geometrical capacitance C and

23 Introduction: dynamical (ac) quantum transport as a scattering problem The electrochemical capacitance can be seen as a (series) sum of a geometrical capacitance C and two quantum capacitors characterized by its Thomas-Fermi screening length 23

24 Introduction: dynamical (ac) quantum transport as a scattering problem Now let's consider a simpler case: one plate is macroscopic half the resistance quantum 24

25 Introduction: dynamical (ac) quantum transport as a scattering problem i.e., half the resistance quantum A universal value! 25

26 Introduction: dynamical (ac) quantum transport as a scattering problem Can we make this system (experimentally) and check the theoretical predictions? 26

27 The experiment Here it is. A coherent RC circuit 27

28 The experiment A coherent RC circuit 28

29 The experiment A coherent RC circuit 29

30 The experiment Experimental measurements 30

31 The experiment Let's consider the impedance, Z The predicted universal value for the resistance! 31

32 The experiment For a more detailed (quantitative) description of the measurements we need to introduce A model for QPC and the small plate (the scattering matrix) A model for the transmission as a function of the gate voltage Include thermal effects 32

33 The experiment Scattering matrix of the QPC + mesoscopic plate: t Phase acquired in 33 : time spent by electrons in a roundtrip

34 The experiment The density of states can be calculated from the scattering matrix: t Model for the transmission/reflection coefficient 34

35 The experiment Including thermal effects: We now have all the ingredients to plug them into the admittance/conductance formula: 35

36 The experiment Comparison of conductances (theory & experiment) at different temperatures and frequencies 36

37 Conclusions The continuum development of experimental techniques in electronics allow us to observe fundamental and basic quantum phenonema in elementary systems, which might be considered as textbook (or Gedankenexperiments) problems in the past. This is very motivating! 37

38 Martes Cuánticos 38

39 Martes Cuánticos 39

arxiv:cond-mat/ v1 [cond-mat.mes-hall] 25 Jun 1999

arxiv:cond-mat/ v1 [cond-mat.mes-hall] 25 Jun 1999 CHARGE RELAXATION IN THE PRESENCE OF SHOT NOISE IN COULOMB COUPLED MESOSCOPIC SYSTEMS arxiv:cond-mat/9906386v1 [cond-mat.mes-hall] 25 Jun 1999 MARKUS BÜTTIKER Département de Physique Théorique, Université