Search for time reversal symmetry effects in disordered conductors and insulators

beyond weak localization.

40 years of Mesoscopics Physics: Colloquium in memory of Jean-Louis Pichard June 25-26, 2018

Marc Sanquer CEA/DRF/INAC & UGA

G. Bergmann 1984

JLPichard PhD (Orsay, 1984) Contribution à une théorie quantique des phénomènes de transport par études numériques de systèmes désordonnés: localisation d’Anderson

(Image by Yang-Zhi Chou and Matthew Foster/Rice University)

JLP obtains the electronic localization length in 2D ribbons or 3D bars ( beyond the strict 1D case)

Lt (under special conditions, Lt= cross section of the ribbon)

at a time where computers are not very efficient…

JLP and G. Sarma, Journal of Physics C-solid state physics, L127-132, (1981)

First report of reproducible mesoscopic conductance fluctuations (« fingerprint »)

Conductance in Restricted-Dimensionality Accumulation Layers A. B. Fowler, A. Hartstein, and R. A. Webb Phys. Rev. Lett. 48, 196 (1982)

Long ( few microns) and narrow channel

Random Matrix Theory and mesoscopic conductance fluctuations

a eigenvalues of the X matrix:

g measures the numbers of a between 0 and 1 . If the {a } are randomly distributed, then var(g) ≈ , But var(g) ≈ 1 (UCF)… Therefore there are correlations between the {a }

M

Time reversal symmetry effects =1 =2

=1 Porter-Thomas distribution

Muttalib, Pichard,Stone PRL 59 (1987)

ensemble of Hermitian matrices with random matrix elements

Spectral Rigidity Level repulsion p(s=0)= 0 p(s

Eigenvalues repulsion in various physical problems: nuclear physics , chaos-logy (from Pier Mello, Les Houches)

=1, Porter-Thomas distribution

TESTING the DEPENDENCE OF THE UCF ON TIME REVERSAL SYMMETRY

D. Mailly & M. Sanquer, J. Phys 1, 357 (1992)

=1 =2

L=10m, L=2.8 m, W=90nm GaAs:Si (10 18 cm-3) without a gate

P.A. Mello, PRL60, 1089 (‘88)

D. Mailly & M. Sanquer, J. Phys 1, 357 (1992)

=1 =2: 1/f noise reduction

L=10m, L=2.8 m, W=90nm GaAs:Si ( 2 10 17 cm-3) with an Al-gate

Prediction by Feng, Lee, Stone PRL 56, 1960 (1986)

N. O. Birge B. Golding, W. H. Haemmerle PRL62, 195 (1989)

Bi-films (thickness 11-90nm, size few m 2 up to 10 m × 100 m )

JLP, MS et al. PRL 65, 1812 (1990)

Sensitivity of the localization length to time reversal symmetry

Macroscopic films VRH regime

≈ Nl

W/O SOC

With SOC

GaAs:Si

a:YxSi1-x

(quasi 1D bar)

JLP, MS et al. PRL 65, 1812 (1990)

= (N + 2- )l

Sensitivity of the localization length to time reversal symmetry

GaAs:Si, w/o SOC

Khavin, Gershenson, Bogdanov PRB 58 (‘98)

=12 N ≈7

GaAs:Si, w/o SOC

Agreement w/o SOC but Controversy with SOC: Is there an increase of the localization length under application Of a magnetic field (resminiscent os weak antilocalization) or not ?

PRL 66, 1517 (‘92)

diffusive loops / forward directed path analysis Problem of ergodicity

Bouchaud J.-P., 1991, J. Phys. France, 1, 985 (w/o SOC) Lerner Imry Europhys. Lett., 29 (l), pp. 49-54 (1995) Medina Kardar PRB46 9984 (92)

With SOC

Forward directed path analysis (Nguyen Spivak Shklovskii JETP Lett. 41,42 (85))

Medina Kardar PRB46 9984 (92)

Bouchaud J.-P., 1991, J. Phys. France, 1, 985 (w/o SOC)

J-P Bouchaud, D. Sornette Europhys. Lett. 17, 721 (92) (with SOC)

J-P Bouchaud, D. Sornette Europhys. Lett. 17, 721 (92) (with SOC)

W. Poirier, MS, D. Mailly, Phys. Rev. B59, 10856 (1999)

Magneto-conductance of small MESFETs ( L=70×70×500nm)

W. Poirier, MS, D. Mailly, Phys. Rev. B59, 10856 (1999)

Magneto-conductance of small MESFETs ( L=70×70×500nm)

N ≈1 N ≈8

[Fit parameters L=160nm, l =20nm]

Nguyen,Spivak, Shklovskii , JETP Lett. 41,42 (85)

W. Poirier, MS, D. Mailly, Phys. Rev. B59, 10856 (1999)

Magneto-conductance of small MESFETs ( L=70×70×500nm)

Silicon nanodevices ? Smaller, larger carrier density, smaller mean-free path [Compared to GaAs ] Design of doped silicon bars of various lengths and cross-sections

L=200nm comparable to L l = 4-8nm kF l ≈1 ≈ N l ≈ L

Issue: Coulomb Blockade ?

1989: first report on periodic conductance oscillations in semi-conductors

“Conductance Oscillations Periodic in the Density of a One-Dimensional Electron Gas” Scott-Thomas, J.H.F., S.B. Field, M.A. Kastner, H.I. Smith, and D.A. Antonadis, 1989, Phys. Rev. Lett. 62, 583. Abstract: By use of x-ray lithography Si inversion layers have been fabricated with width ∼25 nm and mobility ∼15 000 cm2/V s. These display oscillations in their conductance that are periodic in the number of electrons per unit length, even in zero magnetic field. The oscillations reflect an oscillatory activation energy of the conductance and are accompanied by unusual nonlinearities suggestive of pinned charge-density waves.

2006: Silicon quantum dot based on the FDSOI technology M. Hofheinz, X. Jehl , M. Sanquer , G. Molas, M. Vinet and S. Deleonibus , “A simple and controlled single electron transistor based on doping modulation in silicon nanowires”, Applied Physics Letters vol.89, 143504 (2006).

journée nanosciences 27Nov. 2012 M. Sanquer

Simulation INAC+LETI

The MOS-SET

Underlapped NW

One gate

MOS-SET @ CEA Ec=30meV, 20*20*10nm3, C=6aF R1 MW (Prati et al. Nanotechnology 2011)

R= 100kW

MOS-SET @300K Ec=85meV, C=2aF R= 5 MW (Deshpande et al. IEDM2012)

Ec= e2/C =12 meV

Tsi=10nm, spacers 15nm, L*W=40*20nm

Size down to 3,5nm×10nm

Lavieville et al Nano Lett. 15 2958 (2015)

Quantum dots; statistics of the conductance for CB peaks (and valleys) ( Jalabert, Stone, Alhassid et PRL 68, 3468 , 1992) =1 =2: First, breaking TR symmetry reduces amplitude of fluctuations… Second, breaking TR symmetry increases the mean amplitude =4 =2 (Ahmadian, Aleiner PRB 73, 073312 (2006) the average peak height is reduced in the case [=2]. With SOC, the application of the magnetic field causes the average peak conductance to drop by a factor 1.37, similar to antilocalization for bulk systems.