1. Quantum transport and its classical limit Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Lecture 1 Capri spring school on Transport in Nanostructures, March 25-31, 2007

2. Quantum Transport About the manifestations of quantum mechanics on the electrical transport properties of conductors These lectures: signatures of quantum interference Quantum effects not covered here: Interaction effects Shot noise Mesoscopic superconductivity sample

3. Quantum Transport These lectures: signatures of quantum interference What to expect? Magnetofingerprint Nonlocality R 1+2 =R 1 +R 2 B (mT) G 1+2 (e 2 /h) G 1 =G 2 =2e 2 /h B GG B (10 -4 T) G (e 2 /h) Figures adapted from: Mailly and Sanquer (1991) Webb, Washburn, Umbach, and Laibowitz (1985) Marcus (2005)

4. Landauer-Buettiker formalism sample x y W N: number of propagating transverse modes or “channels” a n : electrons moving towards sample b n : electrons moving away from sample Note: |a n | 2 and |b n | 2 determine flux in each channel, not density N depends on energy , width W Ideal leads

5. Scattering Matrix: Definition sample |S mj;nk | 2 describes what fraction of the flux of electrons entering in lead k, channel n, leaves sample through lead j, channel m. Probability that an electron entering in lead k, channel n, leaves sample through lead j, channel m is |S mj;nk | 2 v nk /v mj. More than one lead: N j is number of channels in lead j Use amplitudes a nj, b nj for incoming, outgoing electrons, n = 1, …, N j. Linear relationship between a nj, b nj : S: “scattering matrix”

6. Scattering matrix: Properties sample Linear relationship between a nj, b nj : S: “scattering matrix” Current conservation: S is unitary Time-reversal symmetry: If  is a solution of the Schroedinger equation at magnetic field B, then  * is a solution at magnetic field –B.

7. Landauer-Buettiker formalism Reservoirs sample Each lead j is connected to an electron reservoir at temperature T and chemical potential  j.  j, T Distribution function for electrons originating from reservoir j is f(  -  j ).

8. Landauer-Buettiker formalism Current in leads sample  j, T I j,in I j,out In one dimension: = ( nk h) -1 Buettiker (1985)

9. Landauer-Buettiker formalism Linear response sample  j, T I j,in I j,out  j =  – eV j Expand to first order in V j : Zero temperature

10. Conductance coefficients sample  j =  -eV j IjIj Current conservation and gauge invariance Time-reversal Note: only if B=0 or if there are only two leads. Otherwiseand in general.

11. Multiterminal measurements In four-terminal measurement, one measures a combination of the 16 coefficients G jk. Different ways to perform the measurement correspond to different combinations of the G jk, so they give different results! I V V V I I Benoit, Washburn, Umbach, Laibowitz, Webb (1986)

12. Landauer formula: spin Without spin-dependent scattering: Factor two for spin degeneracy With spin-dependent scattering: Use separate sets of channels for each spin direction. Dimension of scattering matrix is doubled. Conductance measured in units of 2e 2 /h: “Dimensionless conductance”.

13. Two-terminal geometry r, r’: “reflection matrices” t, t’: “transmission matrices” t’r in out   f()f() f()f()   |t’| 2 |r|2|r|2 t r’ |t|2|t|2 |r’| 2 eV  (meV) f()f() Anthore, Pierre, Pothier, Devoret (2003)

14. Quantum transport Landauer formula t’r sample t r’ What is the “sample”? Point contact Quantum dot Disordered metal wire Metal ring Molecule Graphene sheet

15. Example: adiabatic point contact N(x)N(x) N min x g 10 6 2 4 8 0 V gate (V) -2.0-1.8-1.6-1.4-1.2 Van Wees et al. (1988)

16. Quantum interference In general:  g small, random sign t nm, , t nm,  : amplitude for transmission along paths ,   

17. Quantum interference Three prototypical examples: Disordered wire Disordered quantum dot Ballistic quantum dot

18. Scattering matrix and Green function Recall: retarded Green function is solution of In one dimension:  k =  and v = h -1 d  k /dk Green function in channel basis: r in lead j; r’ in lead k Substitute 1d form of Green function If j = k:

19. Quantum transport and its classical limit Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Lecture 2 Capri spring school on Transport in Nanostructures, March 25-31, 2007

20. Characteristic time scales h/Fh/F   erg DD HH F l L Ballistic quantum dot:  ~  erg ~ L/v F, l ~ L Diffusive conductor:  erg ~ L 2 /D Inverse level spacing: relevant for closed samples Elastic mean free time

21. Characteristic conductances Conductances of the contacts: g 1, g 2 Conductance of sample without contacts: g sample if g >> 1 ‘Bulk measurement’: g 1,2 >> g sample Quantum dot: g 1,2 << g sample g dominated by sample g dominated by contacts general relationships:

22. Assumptions and restrictions Always: F << l. Well-defined momentum between scattering events Diagrammatic perturbation theory: g >> 1. This implies  D <<  H Only ‘nonperturbative’ methods can describe the regime g ~ 1 or, equivalently, times up to  H. Examples are certain field theories, random matrix theory.

23. Quantum interference corrections Weak localization Small negative correction to the ensemble- averaged conductance at zero magnetic field Conductance fluctuations Reproducible fluctuations of the sample- specific conductance as a function of magnetic field or Fermi energy G G B B Anderson, Abrahams, Ramakrishnan (1979) Gorkov, Larkin, Khmelnitskii (1979) Altshuler (1985) Lee and Stone (1985)

24. Weak localization (1) Nonzero (negative) ensemble average   g at zero magnetic field   g B =+ ‘Hikami box’ + permutations  g g ‘Cooperon’ Interfering trajectories propagating in opposite directions

25. Weak localization (2) Nonzero (negative) ensemble average   g at zero magnetic field g B  g g   Sign of effect follows directly from quantum correction to reflection. Trajectories propagating at the same angle in the leads contribute to the same element of the reflection matrix r. Such trajectories can interfere.

26. Weak localization (3) Disordered wire: Disordered quantum dot: N 1 channels N 2 channels B (10 -4 T) G (e 2 /h) Mailly and Sanquer (1991) (no derivation here) (derivation later)   

27. Weak localization (4)   B (10 -4 T) G (e 2 /h)  Magnetic field suppresses WL.  g g Chentsov (1948) 0 -10 -3  R/R 10 -3 H(kOe )

28. Weak localization (5) Typical dwell time for transmitted electrons:  erg Typical area enclosed in that time: sample area A. WL suppressed at flux  ~ hc/e through sample. Typical area enclosed in time  erg : sample area A. Typical area enclosed in time  D : A(  D /  erg ) 1/2. WL suppressed at  ~ (hc/e)(  erg /  D ) 1/2 << hc/e.   

29. Weak localization (6) In a ring, all trajectories enclose multiples of the same area. If  is a multiple of hc/2e, all phase differences are multiples of 2  :   g oscillates with period hc/2e. ‘hc/2e Aharonov-Bohm effect’    Note: phases picked up by individual trajectories are multiples of , not 2  ! Altshuler, Aronov, Spivak (1981) Sharvin and Sharvin (1981)

30. Conductance fluctuations (1) Fluctuations of   g with applied magnetic field  g g “diffuson” interfering trajectories in the same direction “cooperon” interfering trajectories in the opposite direction   ’’ ’’   ’’ ’’   ’’ ’’   ’’ ’’ Umbach, Washburn, Laibowitz, Webb (1984)

31. Conductance fluctuations (2) Fluctuations of   g with applied magnetic field  g g Disordered wire: Disordered quantum dot: N 1 channels N 2 channels B (mT) G (e2/h)G (e2/h) Jalabert, Pichard, Beenakker (1994) Baranger and Mello (1994) Marcus (2005)

32. Conductance fluctuations (3)  g g    In a ring: sample-specific conductance g is periodic funtion of  with period hc/e. ‘hc/e Aharonov-Bohm effect’ Webb, Washburn, Umbach, and Laibowitz (1985)

33. Random Matrix Theory Quantum dot N 1 channels N 2 channels Ideal contacts: every electron that reaches the contact is transmitted. For ideal contacts: all elements of S have random phase. Ansatz: S is as random as possible, with constraints of unitarity and time-reversal symmetry, Dimension of S is N 1 +N 2. Assign channels m=1, …, N 1 to lead 1, channels m=N 1 +1, …, N 1 +N 2 to lead 2 “Dyson’s circular ensemble” Bluemel and Smilansky (1988)

34. RMT: Without time-reversal symmetry Quantum dot N 1 channels N 2 channels Ansatz: S is as random as possible, with constraint of unitarity Probability to find certain S does not change if We permute rows or columns We multiply a row or column by e i  Average conductance: No interference correction to average conductance

35. RMT: with time-reversal symmetry Quantum dot N 1 channels N 2 channels Additional constraint: Probability to find certain S does not change if We permute rows and columns, We multiply a row and columns by e i , while keeping S symmetric Interference correction to average conductance Average conductance:

36. RMT: with time-reversal symmetry Quantum dot N 1 channels N 2 channels Weak localization correction is difference with classical conductance For N 1, N 2 >> 1: Same as diagrammatic perturbation theory Jalabert, Pichard, Beenakker (1994) Baranger and Mello (1994)

37. RMT: conductance fluctuations Quantum dot N 1 channels N 2 channels Without time-reversal symmetry: With time-reversal symmetry: Same as diagrammatic perturbation theory There exist extensions of RMT to deal with contacts that contain tunnel barriers, magnetic-field dependence, etc. Jalabert, Pichard, Beenakker (1994) Baranger and Mello (1994)

38. Quantum transport and its classical limit Piet Brouwer Laboratory of Atomic and Solid State Physics Cornell University Lecture 3 Capri spring school on Transport in Nanostructures, March 25-31, 2007

39. Ballistic quantum dots Past lectures: Qualitative microscopic picture of interference corrections in disordered conductors; Quantitative calculations can be done using diagrammatic perturbation theory Quantitative non-microscopic theory of interference corrections in quantum dots (RMT). This lecture: Microscopic theory of interference corrections in ballistic quantum dots Assumptions and restrictions: F > 1 Method: semiclassics, quantum properties are obtained from the classical dynamics

40. Semiclassical Green function Relation between transmission matrix and Green function Semiclassical Green function (two dimensions)  : classical trajectory connection r’ and r S: classical action of    : Maslov index A  : stability amplitude  r’ r r ’’ ’

41. Comparison to exact Green function Semiclassical Green function (two dimensions) Exact Green function (two dimensions) Asymptotic behavior for k|r-r’| >> 1 equals semiclassical Green function

42. Semiclassical scattering matrix Insert semiclassical Green function and Fourier transform to y, y’. This replaces y, y’ by the conjugate momenta p y, p y ’ and fixes these to Result: Legendre transformed action  y  Jalabert, Baranger, Stone (1990)

43. Semiclassical scattering matrix Legendre transformed action Stability amplitude transverse momenta of  fixed at  y  Transmission matrix Reflection matrix

44. Diagonal approximation Reflection probability Dominant contribution from terms  = . probability to return to contact 1  

45. Enhanced diagonal reflection Reflection probability If m=n: also contribution if  time- reversed of  :   Without magnetic field:  and  have equal actions, hence Factor-two enhancement of diagonal reflection Doron, Smilansky, Frenkel (1991) Lewenkopf, Weidenmueller (1991)

46. diagonal approximation: limitations   One expects a corresponding reduction of the transmission. Where is it? Note: Time-reversed of transmitting trajectories contribute to t’, not t. No interference! Compare to RMT: captured by diagonal approximation missed by diagonal approximation We found The diagonal approximation gives

47. Lesson from disordered metals   =+ ‘Hikami box’ + permutations Weak localization correction to transmission: Need Hikami box.   Weak localization correction to reflection: Do not need Hikami box.

48. Ballistic Hikami box? In a quantum dot with smooth boundaries: Wavepackets follow classical trajectories.

49. Ballistic Hikami box? Marcus group But… quantum interference corrections  g and var g exist in ballistic quantum dots!

50. Ballistic Hikami box? Initial uncertainty is magnified by chaotic boundary scattering. : Lyapunov exponent Aleiner and Larkin (1996) Richter and Sieber (2002) Time until initial uncertainty ~ F has reached dot size ~L: L= F exp(  t) t = “Ehrenfest time” Interference corrections in ballistic quantum dot same as in disordered quantum dot if t E <<  D

51. Ballistic weak localization Probability to remain in dot: tEtE   loop also for disordered quantum dot: included in RMT special for ballistic dot Aleiner and Larkin (1996) Adagideli (2003) Rahav and Brouwer (2005)

52. Semiclassical theory Landauer formula S , S  : classical action angles of ,  consistent with transverse momentum in lead, A , A  : stability amplitudes Jalabert, Baranger, Stone (1990)

53. Semiclassical theory Landauer formula (0,0)(s,u)(s,u) s, u: distances along stable, unstable phase space directions Action difference S  -S  = su Richter and Sieber (2002) Spehner (2003) Turek and Richter (2003) Müller et al. (2004) Heusler et al. (2006) t enc c: classical cut-off scale s  e - t u  e t encounter region: |s|,|u| < c

54. Semiclassical theory (0,0)(s,u)(s,u) ’’ t enc c: classical cut-off scale P 1, P 2 : probabilities to enter, exit through contacts 1,2 Landauer formula s, u: distances along stable, unstable phase space directions Aleiner and Larkin (1996) Adagideli (2003) Rahav and Brouwer (2005)

55. Classical Limit L but… var g remains finite! Brouwer and Rahav (2006) Take limit F /L 0 without changing the classical dynamics of the dot, including its contacts diverges in this limit! 0 THE END Aleiner and Larkin (1996)