Electron dynamics in graphene with gate-defined quantum dots Rafael Leslie Heinisch, Andreas Pieper, and Holger Fehske ¨ Greifswald Institut fur ¨ Physik, Universitat Abstract
DFG Graphene Schwerpunktprogramm 1459
Model and computational method • tight-binding Hamiltonian X † X † H= Vi ci ci − t (ci cj + cj†ci )
We study transport through circular graphene quantum dots. We use numerically exact Chebyshev expansion and kernel polynomial methods in the framework of a tight-binding honeycomb lattice model [1]. Thereby, we confirm the scattering resonances and bound states for small dots found in the Dirac approximation [2]. Our focus lies on the regime where individual modes of the electrostatically defined dot dominate the charge carrier dynamics. In particular, we discuss the scattering of an injected Dirac electron wave packet for a single quantum dot, electron confinement in the dot, the optical excitation of dot-bound modes, and the propagation of an electronic excitation along a linear array of dots.
E
hij i
i
on-site potential Vi = V inside the dot
�i
• wave packet propagation |ψ(τ )i = U (τ, τ0)|ψ(τ0)i • expansion in Chebyshev polynomials of first kind Tl (x ) U (∆τ ) = e
−ib∆τ ~
" c0
d∆
~
# X d ∆τ ˜) cl +2 Tl (H ~ l =1
Q
cl (d ∆τ /~) = (−i ) Jl (d ∆τ /~), Jl Bessel function,
˜ = (H − b)/d rescaled Hamiltonian H
• particle density ni (τ ) = |hi |U (τ, τ0)|ψ(τ0)i|2 • current density ~ji (τ ) = |hi |(−~J /e)U (τ, τ0)|ψ(τ0)i|2 P † † current operator ~J = −(ite/~) (~rj − ~ri )(ci cj − cj ci )
Schematic representation of the scattering geometry. An
Dirac electron scattering at a single dot.
electron wave packet of energy E, momentum k and width
For E < V , the incident (ψi ) and reflected
∆x propagates along the zigzag direction in a broad graphene
(ψr ) electron waves reside in the conduc-
sheet. It impinges on a circular, electrostatically defined dot with
tion band, while the transmitted (ψt ) wave
radius R and applied potential V . (To guarantee a smooth po-
inside the dot corresponds to a state in the
tential Vi is interpolated linearly within a small range R ± 0.1R.)
valence band.
Dot-bound normal modes
10
a2
E/t = 0.01 R = 10 nm
a0
a0
a1
(1)
5
n(r)
a1
(4) a0
(3)
a1
-5
a2
-6
10
10
-6
10
-7
10
(2)
0 -0.1
�r
R
l
20 15
�t(qb)
M
Scattering by a single dot a2
V
-8
-0.05
0
0.05
0.1
10
0.15
-7
0
20
(2) V/t = 0.0863
60
80
0
20
r [nm]
V/t (1) V/t = 0.0674
40
(3) V/t = 0.0965
40
10 80
60
r [nm]
(4) V/t = 0.1218
160 101
80
a1
a2
E/t=0.0039
E/t=-0.0215
10nm
y [nm]
(h) 0
0 100
10000
20000
30000
6
-80 4
10-1 80 x [nm]
160
240
τ1 γ = -92.2 nm
-80
(3b)
0
80 x [nm]
160
240
τ2 γ = 92.2 nm
-80
0
80 x [nm]
160
240
-80
0
80 x [nm]
160
240
-2
τ3 γ = 368.9 nm
(3c)
102
-6
12
101
10 8 6
100
4 2
-50
2
-25
0 25 x [nm]
50
-50
-25
0 25 x [nm]
50
-50
-25
0 25 x [nm]
50
-10
-5
0 x [nm]
5
-2
0 x
2
4
tom: density profiles obtained by exact diagonalisation for a dot with
6
R = 10 nm and V /t = 0.081615. Magnifications are shown above. (i)
Each tile gives the electron density on one lattice site. Top: ra-
1.5
dial density profile for different grading widths R ± 0.01R (black),
1
R ± 0.3R (red) and R ± 0.5R (blue). The mode a1 is a bound state,
0.5
0 -50
-4
2.5
ref
0 -25
Dot-localised states associated with the normal modes am . Bot-
-6
14
jr (ϕ) [arb. units]
y [nm]
-4
16
50 25
0
0 -1
10
-0.5
0
0.5
while the mode a2 is a long-living decaying state.
1
ϕ/π
• scattering resonances correspond to (decaying)
Scattering and particle confinement by a gate-defined circular quantum dot. The scattering efficiency Q for plane-wave
Plane wave scattering of a Dirac
scattering in the Dirac approximation (top) shows resonances when specific modes am are excited. Below we show density
electron. Density (top), current den-
snapshots obtained numerically for a tight-binding Hamiltonian for the scattering of a wave packet with ∆x = 148 nm after
sity (middle), and far-field radial com-
passing through the dot. The lower panels show three time steps during the scattering process at the a1 resonance (marker (3)).
ponent of the reflected current (bottom)
Here we also show the current density. The dot is indicated by the dashed circle.
for the a1 resonance.
• series of scattering resonances am for particular
Dirac Hamiltonian
mode characteristics
• ring shaped density (am>0) • 2(2m + 1) vortices in
values of R, V , and E • mode a0: strong scattering, weak trapping • modes am>0: strong scattering and strong trapping • between modes: wave packet passes through dot undisturbed
current field • 2m + 1 preferred scattering directions
H = −i γ~∇σ + V Θ(R − r ) scattering resonances for |V − E |R = γ~ jm,s , jm,s sth zero of Bessel function Jm
γ group velocity at Dirac point
Optical excitation of dot-bound normal modes (a) µ/t = -0.01
1.4 3.5
(a) µ/t = -0.01
1.2 1
a2
(b) µ/t = -0.03
E2/t
-0.02
-0.04
-0.06
0.6
a3 a4
-0.08 -0.08
0.8
a4
a3
-0.06
-0.04
a2
a1
-0.02
0
0.04
2.5
100
1.5
0.5
103 102 101 0 10
0 0
0.05
E1/t
0.1
ω/t
0
0.05
0.1
ω/t
Current matrix element jy (E1, E2) (left) and optical conductivity (right) for a chain of ten dots in an armchair graphene nano-ribbon. The dot radii R = 10 nm, their separation D = 40 nm, and the potential V /t = 0.081615. In (a) the peak at ω/t = 0.02 is due to transitions from the a2 to the a1 mode. In (b) this peak corresponds to the a3 -to-a2 transition. Varying the chemical potential - by tuning the voltage at the back gate - one can bring different transitions am → am±1 into observational range. (Grey curve: pristine nano-ribbon, Blue curve: slightly different dot radii.)
reflected by optical conductivity π~ X σα = |hi |Jα|j i|2[f (Ei ) − f (Ej )]δ(ω + Ei − Ej ) ωΩ i ,j Z ∞ π~ = dEjα(E , E + ω)[f (E ) − f (E + ω)] ω −∞ • current matrix element jα (E1, E2) =
1
Ω
Tr[Jαδ(E1 − H )Jαδ(E2 − H )]
(1a) τ = 500−h/t
(2) τ = 2000−h/t
(1b) τ = 500−h/t
(3) τ = 6000−h/t
40 20 0 -20 -40 40 20 0 -20 -40
2
0.2
• optical excitation of dot bound modes
10
101
1
0.06
(b) µ/t = -0.03
2
3
0.4
0 0.02
σy / (e2/8−h)
a1
jy / (e2/−h2)
0
Transport through a linear array of dots
4
0.02
dot-bound states in equilibrium E = 0 true bound-states for particular R, V E 6= 0 hybridisation with extended states → quasi-bound states • bound states fourfold degenerate total angular momentum j = −m − 1/2, j = m + 1/2 (orbital angular momentum m and pseudospin) valley K and K 0 → 4 dot localised states obtained by exact diagonalisation • sublattice pattern reflects vortex structure
y [nm]
(3a)
0
y [nm]
-80
2
y
-160
• selective increases in jα(E1, E2) if E1 and E2 coincide with energies of dot-bound modes reflect selection rule hm1|Jα|m2i ∝ δm1,m2±1
• sharp peak in optical conductivity due to am → am+1 transition, ωn,m ≈ ~γ(jm,1 − jn,1)/R
DPG Fr¨ uhjahrstagung Dresden 2014
-80
-40
0 40 x [nm]
80
120
-240 -200 -160 -120 -80
-40
0 40 x [nm]
80
120 160 200 240
Propagation of an electronic excitation along a linear chain of dots. The eleven dots are aligned on a graphene sheet with Nz × Na = 12000 × 4000 sites corresponding to (1476 × 852) nm2 . The parameters are R = 10 nm, D = 40 nm and V /t = 0.081615. At time τ = 0 the a1 mode (energy E = 0) is excited at the central dot. The left panels show the current field (top) and the density profile (bottom) after τ = 500~/t (right panels: later times).
• wave functions of bound states overlap → electron transfer between dots
• different timescales between lattice sites (faster) and between dots (slower)
• preferred scattering directions of modes a1 → propagation to the right faster than to the left
[1] A. Pieper, R. L. Heinisch and H. Fehske, Europhys. Lett. 104, 47010 (2013) [2] R. L. Heinisch, F. X. Bronold, and H. Fehske, Phy. Rev. B 87, 155409 (2013)
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