Localised Collective Excitations in Doped Graphene in a Strong Magnetic Field
Andrea M.
1
1
Fischer ,
Alexander B.
2,3
Dzyubenko
and Rudolf A.
1
Römer
If I’m not by my poster,
please feel free to find me
at the conference - Andrea.
Department of Physics and Centre for Scientific Computing, University of Warwick, Coventry, CV4 7AL, UK
2
Department of Physics, California State University Bakersfield, Bakersfield, CA 93311, USA
3
General Physics Institute, Russian Academy of Sciences, Moscow 117399, Russia
S UMMARY [ AR X IV:0902.4176]
1 .4
T HEORY
1 .2
1 .2
• In graphene Kohn’s theorem doesn’t apply and interactions are very important.
Need to consider collective excitations:
1 .0
1 .0
1 .4
E n e rg y (E 0)
0 .8
ν=1
0 .0
QN N M =
1 2 z
0 .8
ν=3
†
0 .0
-0 .2
A +
D
-0 .2
n o n -d e g e n e ra te
d o u b ly d e g e n e ra te
n o n -d e g e n e ra te
d o u b ly d e g e n e ra te
(b )
(a )
-0 .4
-0 .4
-5
-4
-3
-2
-1
0
1
2
M
3
4
5
6
7
8
9
1 0
-1 0
-9
-8
-7
-6
-5
-4
-3
-2
M
z
-1
0
1
2
3
4
5
z
Figure 2: Energies of magnetoplasmons bound on an impurity versus projection of the orbital angular momentum Mz for (a)
0 .8
(b )
†
• QN N M acts on |νi, the ground state with integer filling factor ν.
1 2 z
• Hamiltonian is
N10N20
N1N2
Ĥ
m=0
∞
X
E n e rg y (E 0)
-
(b )
0 .8
0 .6
5 0
2
r a n c h , |d | x 2 0 0 0
2
u n d s t a t e s , |d | x 2 0 0 0
ra n c h
te s
0 .0
2
3
+
, σ
ν = 2
4 0
3 0
2
0 .2
U p p e r b
Q u a s ib o
L o w e r b
D a rk s ta
1 0
0
-0 .2
+
0 .0
(a )
4
-0 .2
1
2
3
4
1 .0
ν = 4
(d )
0 .8
0 .0
- 0 .2
- 0 .4
-5
-4
-3
-2
-1
M
0
z
1
2
3
4
5
-5
-4
-3
-2
-1
M
0
1
2
3
4
1
2
2
1
1
2
2
5
z
Figure 1: Energies of magnetoplasmons bound on an impurity versus projection of the orbital angular
momentum Mz for (a) a donor impurity and filling factor ν = 2, (b) a donor impurity and ν = 3, (c) an
acceptor impurity and ν = 4 and (d) a donor impurity and ν = 4. The shaded area represents the band.
(3)
Figure 3: Evolution with filling factor ν of energies and optical strengths of magnetoplasmons bound on the D+ with (a) Mz = 1
• Energies in units of E0 = (π/2)1/2e2/εlB (lB is magnetic length)
• Each LL has four sublevels due to spin and pseudospin splitting. We compute results for sublevel
filling factors ν = 1, 2, 3, 4 of the n = 0 LL. In general infinitely many excitations with the same Mz
are mixed. We focus on mixing between excitations with the same single particle cyclotron energies
which involve the n = 0 LL as an initial or final state. There are four such excitations with no spin or
pseudospin flip (see insets in Fig. 2). We present results for these excitations only, since they alone are
electrically dipole active.
• The electron-hole symmetry of graphene leads to a duality in the bound states: For sublevels ν =
1, 2, 3 of LL with number n, the eigenstates and eigenenergies of excitations are identical to those for
sublevels of LL with number −n under the following:
Mz ↔ −Mz
ν ↔ν−4
D + ↔ A− .
1 .2
1
|n01|−1 m01 |n02|−1 m02
+sn1 sn2 sn01 sn02 U|n |−1 m |n |−1 m ,
1
1
2
2
ν
- 0 .4
D +
2
|n01|−1 m01 |n02| m02
|n01| m01 |n02|−1m02
+sn1 sn01 U|n |−1 m |n | m +sn2 sn02 U|n | m |n |−1 m
1
1
2
2
1
1
2
¸ 2
0 .2
E n e rg y (E 0)
R ESULTS
(c )
2
·
|n01| m01 |n02| m02
n01m01 n02m02
Un1m1 n2m2 = an1 an2 an01 an02 U|n | m |n | m
1
1
2
2
- 0 .2
ν = 4
1
2 0
0 .0
A -
1
(2)
n01m01 n02m02
N10m01 N2m2
element, WN m N 0m0 = hΦN10m01 ΦN20m02 |Uee|ΦN1m1 ΦN2m2 i = δs1,s01 δσ1,σ10 δs2,s02 δσ2,σ20 Un1m1 n2m2
1 1 2 2
N2m2 N10m01
is direct electron-hole attraction and WN m N 0m0 is repulsive electron-hole exchange.
1 1 2 2
0 0 0 0
n1m1 n2m2
n01m01 n02m02
• Graphene matrix elements, Un1m1 n2m2 , are related to those for 2DEG, Un1m1 n2m2
active in the σ + polarization and (b) with Mz = −1 active in the σ − polarization. Optically active states are indicated by circles
2
− 2
with sizes ∼ |d±
ν | . Inset: Dipole strength |dν | vs energy for ν = 2. The spectra were convoluted with a Gaussian of width 0.03E0 .
The arrow indicates an impurity-related feature below ~e
ωc (below energy zero in the Figure).
n o n -d e g e n e ra te
d o u b ly d e g e n e ra te
trip ly d e g e n e ra te
¡
²̃N is single particle LL energy renormalised by self energy corrections due to
exchange with electrons in lower cone, VN1m = hφnm|V (r)|φnmi is impurity matrix
-0 .2
ν
1 .0
E n e rg y (E 0)
, σ
0 .8
0 .0
1
ν = 3
+
2
U p p e r b r a n c h , |d | x 1 0
2
L o w e r b r a n c h , |d | x 6 0
D a rk s ta te s
E n e rg y (E 0)
E n e rg y (E 0)
0 .6
-0 .4
(a )
¢ †
²̃N1 + VN1m cN mcN1m
1
m1,m2
m01,m02
D i p o l e s t r e n g t h , |d | ( a r b . u n i t s )
(4)
D
ν = 2
=
∞
X
¡
1 .0
where φ|n| m(r) is regular LL wavefunction for a 2DEG and χs is spin part.
p
• Corresponding energy: ²N = sign(n)~ωc |n| + ~ωssz + ~ωv σz ,
√
where ~ωc = vF 2e~B is cyclotron energy in graphene and other terms represent spin
and valley splitting.
1 .2
D
(1)
†
¢ †
−
²̃N2 + VN2m dN mdN2m
2
m=0 ³
´
X
0 0
0 0
Nm Nm
Nm Nm
†
†
WN 2m 2 N 10m01 − WN 1m 1 N 20m02 cN 0m0 dN 0m0 dN2m2 cN1m1 .
+
• Single impurity, so use axially symmetric gauge: A = 12 B × r
• Wavefunction for single electron in K valley (pseudospin ⇑):


sign(n)φ|n|−1 m(r)




1
φ
(r)
†
(δ
−1)
|n|
m

 χs ,
Φnms⇑(r) = hr|cnms⇑|0i = 2 2 n,0


0


0
D +
†
where cN m and dN m are creation operators for electron and hole, m = 0, 1, . . . is
1 1
2 2
oscillator quantum number and N = nsσ represents LL index n ∈ Z, spin s =↑, ↓ and
valley pseudospin σ =⇑, ⇓.
• Excitations labelled by Mz , the z component of orbital angular momentum, such
that expansion coefficients satisfy AN1N2Mz (m1, m2) ∼ δMz ,|n1|−m1−|n2|+m2 .
S INGLE PARTICLE IN A MAGNETIC FIELD
D +
†
AN1N2Mz (m1, m2)cN m dN m ,
1 1
2 2
m1,m2=0
a donor impurity and filling factor ν = 1, and (b) an acceptor impurity and ν = 3. The spectra exhibit the symmetry D+ ↔ A−,
Mz ↔ −Mz and ν ↔ 4 − ν. The hatched area of width 0.75E0 represents the continuum of extended magnetoplasmons.
1 .4
∞
X
†
E n e rg y (E 0)
We consider collective excitations in graphene in a strong perpendicular magnetic
field with a single Coulomb impurity. We show that localised collective modes split
off the magnetoplasmon continuum and in addition, quasibound states are formed
within the continuum. A study of the evolution of the dipole strengths and energies
of magneto-optical transitions is performed for integer filling factors ν = 1, 2, 3, 4 of
the zeroth Landau level (LL). We predict impurity absorption peaks above as well as
below the cyclotron resonance. In contrast to the 2D electron gas (2DEG) there are
strong peaks for both circular polarisations of light, whose dipole strengths increase
linearly with magnetic field. We find that the single-particle electron-hole symmetry of
graphene leads to a duality between the spectra of collective modes for the positively
charged donor impurity, D+, and the negatively charged acceptor impurity, A−. The
duality shows up as a set of complementary magneto-absorption peaks, which are
active in different circular polarizations.
O PTICS
• Interaction of electrons in graphene with light of frequency ω and left (+) and right
(−) circular polarizations is described by Hamiltonian
Ã
!
evF E σ± 0
δH± =
,
(5)
iω
0 σ∓
where E is electric field amplitude and σ± = σx ± iσy .
2
2
• Optical selection rules determined from dipole strength |d±
ν | = |hMz = ±1|δH±|νi| ,
where |Mz = ±1i is collective excitation and |νi is ground state with filling factor ν.
• Optical selection rules are no spin- or pseudospin flips, Mz = ±1 and |n1| − |n2| = ±1.
R EFERENCES
2DEG:
[1] A. Dzyubenko and Yu. E. Lozovik, Zh. Eksp. Teor. Fiz. 104, 3416 (1993).
[2] C. Kallin and B.I. Halperin, Phys. Rev. B 31, 3635 (1985).
Graphene:
[3] A. Iyengar, J. Wang, H. A. Fertig, and L. Brey, Phys. Rev. B 75, 125430 (2007).
[4] Yu. A. Bychkov and G. Martinez, Phys. Rev. B 77, 125417 (2008).