Radiation induced photocurrent and quantum interference in n-p junctions.

M.V. Fistul, S.V. Syzranov, A.M. Kadigrobov, K.B. Efetov.

)(rUpH

Dirac spectrum

Effect of radiation on quasiparticles spectra: opening of dynamical gaps, proportional to the amplitude of EF

strong modification of current-voltage characteristics:

1. Photocurrent through the junction without any DC bias voltage applied (Elastic impurities weakly influence the photocurrent for experimentally relevant parameters)

2. Suppression of tunneling by strong enough radiation.

Previous study of irradiated graphene n-p junctions (PRL 98, 256803 (2007), PRB 78, 045407, (2008))

Klein tunneling

Full reflection in Schrodinger quantum mechanics

but

Perfect transmission in Dirac quantum mechanics

Now: n-p junctions in systems with a gap (bilayers, two-band semiconductors, etc.)

Result: Ramsey-like oscillations of the photocurrent!

Forbidden gap

Carbon nanotube Mono- or bilayer graphene nanotube

Floquet Theorem (analogue of Bloch theorem)

)(0 tVHH t

tihttH

)(

)()(

)()exp()( ttit T

V(t)-periodic with period T

General Solution:

T is periodic with the same period T

Scattering from energy to energy , n ...3,2,1 n

T/2

ε

Choice of coordinatesystem

Resonantmomenta

ε

Rotating reference frame in the pseudospin space

Rotating-waveapproximation closeto -static Hamiltonian

-quasiparticlespectrum

with the gap

Dynamical gap in homogeneous graphene

However: it is difficult to probe the gap, especially in the presence of disorder.

Resonantmomenta

Dynamic gaps in the presence of a p-n junction

U(r)

v

hzpres 2)(

vzUzp /))(()(

The gap exists for a certain interval of momenta and is localized in space!

At the resonant point z0 the resonant value of momentum is achieved

Landau-Zener tunneling in the momentum space:Potential close to the resonance:

-the probability of tunneling

Tunneling through the dynamic gap (rotating frame)

Cheianov, Falko (2006)

General formulae for the current

, -Energy and angle characterizing the state

The state scatters from to -leadin

,

n

outdnhnht

,

,),;,(

t -scattering amplitude

n

RLLR nhffnhPevpdpd

WI ))()()(,,(cos)2(

42

Current I through the strip of width W

2),,,(),,( nhtdnhP LRLR

Photocurrent is possible if ),,(),,( nhPnhP LRLR

Electrons in the energy interval of the width can penetrate from the right to the left leadabsorbing a photon!

- moderate intensities

Photocurrent

and are Bessel and Struve functions, W is the width

1L1I

High-intensity regime:

Effect of elastic impurities:

If , the photocurrent persists

S-intensity

Photocurrent vs. radiation intensity

Low-intensity regime:(The tunneling exponent is much smaller than 1, )1

Interference process in the presence of the gap between conduction and valence gaps.

Energy as a function of the spatial coordinate.

Energy versus momentum.

The Hamiltonian of the homogeneous system in the rotating wave approximation.

xRz

zvzczvzceff

ppppH

22

ˆR -dynamic gap

resvresc pp

The accumulated phase.

z

p

zvzcIII dpppFres

0

/2

The total quantum mechanical probability of inelastic electron transmission.

2/)1(2

III iiRL eeTTP vFT R /exp 2

F-slope

Current

n

RLRL nffnPd

geI

,2

Final expression for the photocurrent

2cos)1( TTge

I ph

Gg VEf /)/( ev

d

2

2 d-length of

the junction

deVF G /

xxxxxf /11ln1)( 222

Analogy to Ramsey fringes!

Ramsey fringes in atomic physicsNorman F. Ramsey, Phys. Rev. 78, 699 (1950)

Nobel Prize in Physics (1989)

T,, 0What is a best way to observe Ramsey fringes?Variation of

d=100nmW=1μm

close to that inJ.Huard et al., 2007B.Oezyilmaz et al. 2007

U0=0.1eV -the height of the potential barrier

S=10kW/cm2 -close to that in M.Freitag, Y. Martin, J. Misewich et al., 2003

Photocurrent is the largest, if

The photocurrent is maximized when

Exponent of tunneling through the gap

Hzff irir121052/

Hzf 1313 10510

Experimental parameters for Graphene.

Relevant experiment

Xia et al (IBM), Nanoletters. (2009)

2/0U

Hz141052

Reduced photocurrent

Maximum experimental value of the photocurrent: 25nA

Estimates for the “Ramsey fringes”

The gap: ,1.0 eVEg nmd 100

Radiation frequency: THz50

V5.2

A large number of the oscillations can be observed at VVG 3.0

Transport properties of graphene bilayers, graphene ribbons, etc,.. p-n junctions irradiated by monochromatic electromagnetic field (EF) were studied.

•The resonant interaction of propagating quasiparticles with an externalradiation opens dynamical gaps in their spectrum, resulting in a strongmodification of current-voltage characteristics.

•A photocurrent flows in all the situations considered.

• If the conduction and valence gap are separated by a gap, the photocurrent oscillates as a function of the split gate or the external frequency, which is analogous to Ramsey oscillations.