Download PLMCN10-orals-12-Monday-Mo-33

Bose-Einstein Condensation of
Trapped Polaritons in a Microcavity
Oleg L. Berman1, Roman Ya. Kezerashvili1,
Yurii E. Lozovik2, David W. Snoke3, R. Balili3,
B. Nelsen3, L. Pfeiffer4, and K. West4
1Physics
Department, New York City College of Technology
of City University of New York (CUNY), Brooklyn NY, USA
2Institute of Spectroscopy, Russian Academy of Sciences,
Troitsk, Moscow Region, Russia
3Department of Physics and Astronomy,
Univeristy of Pittsburgh, Pittsburgh PA, USA
3Bell Labs, Lucent Technologies, Murray Hill NJ, USA
OUTLINE
• 2D EXCITONS AND POLARITONS IN QUANTUM WELLS EMBEDDED IN
A MICOCAVITY
• EXPERIMENTS DEVOTED TO TRAPPED POLARITONS IN A
MICROCAVITY
• BEC AND SUPERFLUIDITY of 2D POLARITONS IN A HARMONIC
POTENTIAL IN A MICOCAVITY
• GRAPHENE
• BEC OF TRAPPED QUANTUM WELL AND GRAPHENE POLARITONS IN
A MICROCAVITY IN HIGH MAGNETIC FIELD
• CONCLUSIONS
Semiconductor microcavity structure
Bragg refractors:
Trapping Cavity Polaritons
cavity photon:
E  c kz2  k||2  c ( / L)2  k||2
quantum well exciton:
E  Egap   bind
2 2
h2 N 2
k||


2
2mr (2L)
2m
Tune Eex(0) to equal Ephot(0):
cavitypolariton
photon
upper
exciton
lower polariton
Mixing leads to “upper polariton” (UP) and “lower polariton” (LP)
LP effective mass ~ 10-4 me
Exciton life time ~ 100 ps
Polariton life time ~ 10 ps
Spatially traped polaritons (using applied stress)
R. B. Balili, D. W. Snoke, L. Pfeiffer and K. West,
Appl. Phys. Lett. 88, 031110 (2006)
Spatially trapped polaritons
(using applied stress)
1. R. B. Balili, D. W. Snoke, L. Pfeiffer and K. West,
Appl. Phys. Lett. 88, 031110 (2006)
2. R. B. Balili, V. Hartwell, D. W. Snoke, L. Pfeiffer and K. West,
Science 316, 1007 (2007).
Theory: O. L. Berman, Yu. E. Lozovik, and D. W. Snoke, Phys Rev B 77, 155317 (2008).
• Starting with the quantum well exciton energy higher than
the cavity photon mode, stress was used to reduce the
exciton energy and bring it into resonance with the photon
mode.
• At the point of zero detuning, line narrowing and strong
increase of the photoluminescence are seen.
• An in-plane harmonic potential was created for the
polaritons, which allows trapping, potentially making
possible Bose-Einstein condensation of polaritons
analogous to trapped atoms.
• Drift of the polaritons into this trap was demonstrated.
Spatial profiles of polariton luminescence
Spatial profiles of polariton luminescence- creation at side of trap
Angle-resolved luminescence spectra
50 mW
400 mW
600 mW
800 mW
x p 
Hamiltonian of trapped polaritons
Total Hamiltonian:
Htot= Hexc+ Hph+ Hex-ph
Exciton Hamiltonian:
Trapping potential:
V(r)=1/2 γr2
Exciton spectrum: εex(P) = P2/2M
Photon Hamiltonian:
Photon spectrum:
λ is a spacing of the Bragg
refractors (size of the cavity)
Hamiltonian of excitonphoton interaction:
15 meV
Rabi splitting
Hamiltonian of non-interacting polaritons
(after unitary Bogoliubov transformations)
After the diagonalization of the total Hamiltonian
applying the unitary transformations we get
Effective Hamiltonian of lower polaritons Heff
(after unitary Bogoliubov transformations)
small parameters α and β
(very low temperature, small momentum and cloud size)
Effective mass of a polariton Meff:
Effective external trapping potential: Veff(r)=1/4 γr2
BEC in Popov’s approximation
Anomalous averages
are neglected
at temperatures
O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008).
γ=960 eV/cm2
1
γ=860 eV/cm2
0.9
γ=10 eV/cm2
γ=760 eV/cm2
0.8
Condensate profiile n0(r) in the trap
γ=100 eV/cm2
n(r=0) = 109 cm-2
Condensate fraction N0/N as a
function of temperature T
Superfluidity
O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008).
NS=N-Nn
Superfluid
component
Normal
component
Linear Response on rotation
Superfluid fraction
O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008).
γ=100 eV/cm2
γ=50 eV/cm2
γ=10 eV/cm2
γ→0
Superfluid fraction Ns/N as a function of temperature T
Normal density was calculated as a linear response of the total
angular momentum on rotation with an external velocity
Conclusions:
O.L. Berman, Yu. E. Lozovik, and D. W. Snoke, Physical Review B 77, 155317 (2008).
• The condensate fraction and
• the superfluid component are
• decreasing functions of temperature,
• and increasing functions of the curvature of the parabolic
potential.
Graphene
Graphene was obtained and studied experimentally for the first time in 2004 by K. S. Novoselov,
S. V. Morozov, A. K. Geim,et.al. from the University of Manchester (UK).
2D atomic honeycomb crystal lattice of carbon (graphite)
Perfect Graphene crystal and resultant Band Structure.
The effective masses of electrons and holes and the energy gap in graphene equals 0
Landau Levels in 2DEG:
Ens   C (ns  )
1
2
Ens 
Vf
rB
Landau Levels
in Graphene:
  ns  V f  e  B  ns
Landau levels in Gallium Arsenide and in Graphene:
Quantum Hall Effect in Graphene:
ωC is cyclotron
frequency, nS is the
number of an energy
level and rB is magnetic
length
eB
C 
m
  
rB  

eB
1
2
Magnetoexcitons in graphene
Hamiltonian (A. Iyengar, J. Wang, H. A. Fertig, and L. Brey, Phys. Rev. B 75, 125430 (2007))
Conserving quantity magnetic momentum
(instead of momentum without magnetic field)
Wave function of e-h pair
Generalization of the approach
used by
Lerner and Lozovik, JETP (1980)
Semiconductor microcavity structure
or graphene layers
Graphene in an optical microcavity in high
magnetic field in the potential trap
Magnetoexcitons:
Magnetoexcitons:
1.
Electron: Landau level 1;
Hole: Landau level 0
•
All LLs below 1 are
filled. All other LLs
are empty.
2.
Electron: Landau level 0;
Hole: Landau level -1
•
All LLs below 0 are
filled. All other LLs
are empty.
graphene
O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009).
Graphene in an optical microcavity in high
magnetic field in the potential trap
O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009).
O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Nanotechnology 21, 134019 (2010).
• The potential trap can be produced by
applying an external inhomogeneous
electric field.
• The trap is caused by the inhomogeneous
shape of the cavity.
Effective Hamiltonian of trapped
magnetoexcitons and cavity photons
O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009).
Htot= Hmex+ Hph+ Hmex-ph
εex(P) = P2/2mB
V(r)=1/2 γr2
potential of a trap
index of refraction of cavity
length of cavity
Rabi splitting
Effective Hamiltonian of lower polaritons in a trap
(after unitary Bogoliubov transformations)
O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009).
Effective mass of a magnetopolariton:
Critical temperature of a magnetopolariton BEC:
Rabi splitting corresponding to the criation of
magnetoexciton in graphene:
matrix term of the Hamiltonian of
the electron-photon interaction
corresponding to magnetoexciton
generation transition
volume of microcavity
Rabi splitting
The ratio of the BEC critical temperature to the square root of the total number of
magnetopolaritons as
function of the magnetic field B and different spring constants γ.
O.L. Berman, R.Ya. Kezerashvili and Yu. E. Lozovik, Physical Review B 80, 115302 (2009).
The ratio of the BEC critical temperature to the square root of the total number of
magnetopolaritons as
function of the magnetic field B and the spring constant γ.
O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Physical Review B 80, 115302 (2009).
O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Nanotechnology 21, 134019 (2010).
Conclusions
O.L. Berman, R.Ya. Kezerashvili and Yu. E. Lozovik, Physical Review B 80, 115302 (2009).
O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Nanotechnology 21, 134019 (2010).
• The BEC critical temperature for graphene
and quantum well polaritons in a
microcavity Tc(0) decreases as B-1/4 and
increases with the spring constant as γ1/2.
• We have obtained the Rabi splitting related
to the creation of a magnetoexciton in a
high magnetic field in graphene which can
be controlled by the external magnetic field
B.