Mechanisms of Terahertz Radiation
Generation in Graphene Structures
Institute for Nuclear Problems, Belarus State University, Belarus
The XII-th International School-Seminar
The Actual Problems of Microworld Physics
Gomel, Belarus, July 22 - August 2, 2013
1. Motivation;
2. Some facts about some graphene structures;
3. Demand for new Terahertz sources;
4. Brief review of ideas and achievements in the field of
graphene based terahertz sources;
5. Our ideas and proposals;
6. Conclusion
Graphene
Graphene, a 2-dimensional flat monolayer of carbon atoms arranged in a honeycomb
lattice , is a promising candidate to be the basic building material for nanoscale
electronic applications. Due to the hexagonal lattice structure of graphene, an interesting
and elegant electronic structure arises, namely that of a gapless semi-metal with a linear
dispersion relation in the vicinity of the Fermi level at the K-points in the Brillouin zone.
Graphene was discovered experimentally in 2004 [K.S. Novoselov et al., Science 306,
666 (2004)].
Some interesting facts about graphene
CERN on one’s desk:
Dirac electrons
Klein tunnelling:
Experimental confirmation of the Klein tunnelling:
Stander, Huard &Goldhaber-Gordon, 2009; Young & Kim, 2009
Quantum Hall and other quantum effects
Novoselov et. al. Nature, 2005
Very high room-temperature electron mobility 2.5 x 105 cm2V-1 s-1 and
possibility to increase mobility to the value 2·106 cm²·V−1·s−1
Ability to sustain extremely high densities of electric current (a million times
higher than copper)
Moser, J., Barreiro, A. & Bachtold, A. Appl. Phys. Lett. 91, 163513 (2007).
Optical absorption of exactly πα≈2.3%
Nair, R. R. et al. Fine structure constant defines visual transparency of graphene. Science
320, 1308 (2008).
Graphene bilayer
Unrolled circumferential vectors c for a (4,4) armchair nanotube (black
arrow), a (4,0) zigzag nanotube (blue arrow) and a chiral (4,2) nanotube (red arrow) are
shown on a graphene plane. a1 and a2 are the unit cell vectors of graphene. The chiral
angle and the translational periodicity vector ` of the (4,2) nanotube (green arrow)
are also shown. Dashed lines indicate the area spanned by c and l which corresponds
to the unrolled unit cell of the (4,2) nanotube.
CARBON NANOTUBE –
quasi-one-dimensional carbon macromolecula
Graphene crystalline lattice
(m,0) for zigzag CNT
(m,m) for armchair CNT
SWCNT (m,n)
Rc=ma1+na2
The typical CNT radius ~ 1-20 nm, CNT length ~ 10 nm- few cm
(p,0) zigzag nanoribbons;
(p,1) armchair nanoribbons.
FIG. a) A typical structure of nanoribbons. A solid circle stands for a carbon atom
with one electron, while an open circle for a different atom such as a hydrogen. A
closed area represents a unit cell. It is possible to regard the lattice made of solid
circles as a part of a honeycomb lattice. b) A nanoribbon is constructed from
a chain of m connected carbon hexagons, as depicted in dark gray, and by
translating this chain by the translational vector T=qa+b, q<m. A nanoribbon
is indexed by a set of two integers (p,q) with p=m−q.
Motohiko Ezawa PHYSICAL REVIEW B 73, 045432 2006.
THz: Unexplored Territory
Applications?
•
•
•
•
•
•
THz spectroscopy/imaging
Radar
Biology, medicine
Communications
Signal processing
High speed computers …?
• Plasmon amplification
A.Bostwick, T. Ohta, T. Seyller, Karsten Horn, E. Rotenberg, Nature, 3,
36, (2007);
Rana F. Graphene terahertz plasmon oscillators. IEEE Trans.
NanoTechnol. 7, 91–99 (2008)
• Cherenkov-type emissions in nanotubes and
graphene
• Harmonic generation, Rabi oscillations and
Rabi-waves.
O. V. Kibis, M. Rosenau da
Costa, M. E. Portnoi,
Generation of Terahertz
Radiation by Hot Electrons in
Carbon Nanotubes
NanoLetters, 7 (11), 3414,
(2006).
Rana F. Graphene terahertz plasmon oscillators. IEEE Trans. NanoTechnol. 7, 91–99
(2008)
Experimental observation Taiichi Otsuji1, Hiromi Karasawa1, Tsuneyoshi
Komori1, Takayuki Watanabe and Victor Ryzhii 2010.
Electron-hole injection
Ubitron-type generator
If a fast electron moves in a periodic potential
and velocity oscillates in time with the frequency
, its momentum
The frequency of radiation
can be tuned in these devices by varying the accelerating voltage which determines
the electron.
S. A. Mikhailov, Graphene-based voltage-tunable coherent terahertz emitter. Phys.
Rev. B 87, 115405 (2013).
Dispersion law in graphene
Graphene stripes
CHERENKOV LASING IN CNT
K. G. Batrakov, P. P. Kuzhir, S. A. Maksimenko, Proc.SPIE 6328, 63280Z (2006).
K.G. Batrakov, O.V. Kibis, Polina P. Kuzhir, M. R. Costa, and M. E. Portnoi, Terahertz
processes in carbon nanotubes. Journal of Nanophotonics, Vol. 4, 041665 (2010).
K.G. Batrakov, P. P. Kuzhir, S. A. Maksimenko, Physica B: Condensed Matter, 405 3050
(2010).
K. G. Batrakov, P. P. Kuzhir, S. A. Maksimenko and C. Tomsen, Phys. Rev. B 79, 125408
(2009).
K. G. Batrakov, P. P. Kuzhir, and S. A. Maksimenko, Physica E 40, 1065 2008).
K. G. Batrakov, P. P. Kuzhir, and S. A. Maksimenko, Physica E 40, 2370 (2008).
The basis of generation in nanotubes and graphene
1) Very large current density (up to 10 10 A/cm2 ) [M. Radosavljevi´c,
J. Lefebvre, and A. T. Johnson, “High-field electrical transport and breakdown in
bundles of single-wall carbon nanotubes”, Phys. Rev. B 64, 241 307® (2001);
S.-B. Lee, K. B. K. Teo, L. A. W. Robinson, A. S. Teh, M. Chhowalla,
et al., J. Vac. Sci. Technol. B 20, 2773 (2002)];
2) Ballistic electron transport (up to 10 μm ) [C. Berger, P. Poncharal, Y. Yi,
W. A. de Heer,Ballistic Conduction in Multiwalled Carbon Nanotubes,
J. Nanosci. Nanotechn., 3, 171 (2003)];
3) The possibility of strong electromagnetic wave slowing down
[G. Ya. Slepyan, S. A. Maksimenko, A. Lakhtakia, O. Yevtushenko,
A. V. Gusakov, Phys. Rev. B 60, 17136 (1999)].
Self-consistent equations
Interaction between electron beam and produced
electromagnetic wave leads to electron beam modulation. This
process can be described by self-consistent system:
for electromagnetic field:
and for electrons:
Dispersion equation
Absorption
term
Emission term
electron group velocity in nanotube
If width of emission line exceeds the magnitude of
quantum recoil. then traditional form of second-order
Cherenkov resonance is realized:
Otherwise, quantum recoil contributes to resonance
condition and dispersion equation has the form
Threshold current and instability increment of
generation
Gain is extremely large as
compared with the gain
per unit length for macrodevices
Boundary conditions on nanotube tips
and dispersion equations give threshold
condition and instability increment
K. G. Batrakov, P. P. Kuzhir, S. A. Maksimenko,
Proc.SPIE 6328, 63280Z (2006).
K. G. Batrakov, P. P. Kuzhir, S. A. Maksimenko
and C. Tomsen, Phys. Rev. B 79, 125408 (2009).
K.G. Batrakov, O.V. Kibis, Polina P. Kuzhir, M. R.
Costa, and M. E. Portnoi, Terahertz processes in
carbon nanotubes. Journal of Nanophotonics,
Vol. 4, 041665 (2010).
K.G. Batrakov, P. P. Kuzhir, S. A. Maksimenko,
Physica B: Condensed Matter, 405 3050 (2010).
Generation
is already possible at the current stage of nanotechnolog
development.
Method for the instability control
The points of maximum group velocity
and
respectively low excitation energy
can be
advantageous for lasing.
In the point of group velocity extremum the
negative influence of the beam energy spread is
smaller, and therefore more electrons interact with
the wave: the radiation effectiveness can be
increased.
It is also possible to intensify the effect of
radiation instability in nanotube
due to the
generation in the region of small effective mass of
quasiparticle.
Compensation of electron beam spread
Dispersion equation
Extremum of group velocity
Then, expansion near this point gives
So, negative influence of beam spread can be reduced.
Advantage of using two-layer or multilayer nanotube or graphene for emission
Estimated wave retardation in single walled nanotube reach 50 times (G. Slepyan,
A. Lakhtakia, S. Maksimenko…)
For long wavelength   d (d is the distance between nanotube layers), frequencies
may be approximately presented as:
  ~  1 ( R1 )   2 ( R 2 )
  ~|  1 ( R1 )   2 ( R 2 ) |
Phase velocity corresponding to frequency   , v ph     / k can be significantly less,
than phase velocities corresponding to single walled nanotubes with radii R i !!!
Estimated wave retardation in two-wall carbon nanotube reach ~ 150 times.
K. G. Batrakov, P. P. Kuzhir, and S. A. Maksimenko, Electrical and Computer
Engineering Series archive,Proceedings of the 1st WSEAS international
conference on Nanotechnology table of contents,Cambridge, UK,2009, pp. 96–100.
However  1 ( R1 )   2 ( R 2 ) , therefore decreasing of phase velocity in two-wall
nanotube is limited.
Bilayer graphene has no such drawback.
Graphene bilayer
The basic equations
  e 0

r , t  r , t  0

r, t  ks b ks
bks 
tiE ks b ks 
tie  
ks|
r, t
|k 1 s 1 b k 1 s1
b
tiE ks b ks 
tie
km 
k 1 s1
k 1 s 1 |
r, t
|ksb 

k 1s1
k 1 s1
Resulting self-consistent equation for potential has the form:
 4
e 
2
1,2
k 2 s2 k 1 s1
k 1 s 1 |
r, 

|k 2 s 2 
0

0

0

0
0 b
b
b
b
k 2 s2 k 2 s2
k 1 s1 k 1 s1 0
E k 2 s2 E k 1 s1
are the wave functions of graphene bilayer
B. Partoens and F. M. Peeters, PHYSICAL REVIEW B 74, 075404 2006
irreducible
polarizability
The root with minus “-” corresponds to antiphase (acoustic) oscillation.
If kd<<1, then the phase velocity of antiphase oscillation is more less of usual one!!!
spatially separated double-layer graphene
It can be can be fabricated by folding an single layer graphene over substrate.
H. Schmidt, T. Ludtke, P. Barthold, E. McCann, V. I. Fal’ko, and R. J. Haug, Appl.
Phys. Lett. 93, 172108 (2008).
In spatially separated double-layer graphene electron tunneling between graphene
planes can be eliminated. However, the plasmon modification still exists if kd<<1.
Konstantin G. Batrakov ; Vasily A. Saroka ; Sergey A. Maksimenko ; Christian
Thomsen, J. Plasmon polariton deceleration in graphene structures. Nanophoton.
6(1), 061719 (Dec 05, 2012). doi:10.1117/1.JNP.6.061719
The electromagnetic wave frequency (1/s)
versus wave vector k (1/cm).
Harmonic generation in terahertz range
Odd harmonics generation
One-photon and
multi-photon Rabi
oscillation
Rabi oscillation and harmonic generation in AB-stacked bilayer
graphene
The effect of trigonal warping on the electronic structure of bilayer
graphene
General view of the
dispersion surface
McCann & Falko (2006)
and McCann, Abergel & Falko (2007)
Cross-section of the dispersion surface
In the presence of electric field
the Hamiltonian has the form:
After second quantization :
In the Heisenberg representation dynamics is described by equation:
H.K.Avettissian, G.F.Mkrtchian,
K.G.Batrakov, S.A.Maksimenko, Nonlinear
Interactions of Coherent Radiation with
Bilayer Graphene and High Harmonic
Generation, Physics, Chemistry and
application of Nanostructures, p.195-198,
2013.
Dynamics of density matrix:
The method of characteristic is used for transition from partial to ordinary
differential equations .
The characteristic coincides to the classical electron dynamics in the electron field:
Parameter which characterizes electromagnetic field-graphene electron
interaction
1)
-one photon excitation, the multiphoton effects are
suppressed;
2)
- multiphoton excitation.
Condition of n-photon process
, where
Particle distribution function after the interaction during the 32 wave oscillation
periods,χ=0.1,( a) and (b) present Dirac sea excitation of two different graphene
valleys, ω=0.9 meV, (c) and (d) is for the wave frequency ω=5 meV.
Particle distribution function after 32 wave periods for different parameters χ:
(a) χ=0.2, (b) χ=0.3,(c) χ=0.4,(d) χ=0.5, ω=10 meV , T/ω=0.01.
Temperature dependence: ω=8 meV , χ=0.5, (a) T/ω=3, (b) T/ω=1, (c) T/ω=0.1,
(d) T/ω=0.01
Harmonics generation
The emission rate of the n-th harmonic is
proportional to the n-th Fourier
components of the field-induced current:
Harmonic emission rate at the multiphoton excitation in bilayer graphene for
different parameters χ:(a) χ=0.2, (b) χ=0.3,(c) χ=0.4,(d) χ=0.5, ω=1 meV.
Required Intensities
The wave intensity can be expressed through the χ parameter as:
Therefore for THz photons (wavelengths from 30 μm to 3 mm), multiphoton
interaction regime can be achieved at the intensities 10 – 105 W cm-2 . So,
with radiation fields of moderate intensities ∼ 3 kW cm−2 in the THz
domain one can observe multiphoton excitation of Fermi-Dirac sea and
efficient generation of moderately high harmonics in the bilayer graphene.
Conclusion
• Graphene structure is very perspective material for
terahertz nanoelectronics.
Acknowledgments
БРФФИ № Ф06Р-101,
БРФФИ № Ф08Р-009 ,
БРФФИ № Ф11АРМ-006,
EU FP7 TerACan project FP7-230778,
BMBF(Germany) project BLR 08/001,
EU FP7 CACOMEL project FP7-247007,
EU FP7 BY-NanoERA project FP7-266529, Call ID FP7-INCO-2010-6, 20102013..
Thank you for attention