In this section, we present the Fabry-Perot resonance as the incidence energy lies on dips. For the dips enhance reflection, the interference between forward and backward waves takes place inside the region of potential barrier. To simplify the analysis, the parameters are specified such that transmissions and reflections are dominated by two sidebands.
Fig. 5.12 displays the Fabry-Perot resonance with K-valley incident and the 1st sideband energy,
1 0 0
E =E +ω =t . The transmission is dominated by central band and the oscillating period is determined from the difference between wave vectors in the central band, 2 / (π q0(2)−p0(1))=26.5ax. However, the period is 53a , double to that of estimated. x
Also we can observe the dip structure by selecting certain barrier width in Fig. 5.8. In Fig. 5.13 we give a series of dip structure by varying with barrier width. It is interesting that we can control the dip structure to be survived or not. We can reopen the Klein-tunneling; we can make zero transmission as well by choosing certain barrier width.
Fig. 5.14 demonstrates the Febry-Perot resonance for gapped graphene at low energy region with size of gap = 2∆. The incident wave is at K-valley and the next sideband energy lies on the conduction band edge (E−1 =E0−ω = ∆).
13
5
5.95 10 , 0.0084 0, 0,
s2, 5.64 10
Hz V eV
N ω
θ σ
−= × =
= ∆ = = = ×
0 20 40 60 80 100 120
0 0.2 0.4 0.6 0.8 1
L
T
Ttot
Tm=0
-1 0 1
Fig. 5.12 Non-typical Fabry-Perot resonance for the case when the incident energy stays at the dip structure. The incidence energy E0 = − is at an t0 ω ω below the band top t . The choice of 0 the time-modulated potential parameters are such that only up to first-sideband processes are important. The curve shows the dominance of the central band in the transmission. The L-period cannot be explained by the usual Fabry-Perot resonance condition: 2 / (π q0(2)−p0(1))=26.5ax,where
) 2 (
q0 , p0(1) are wavevector-pairs for the elastic channel. It is explained by a non-typical Fabry-Perot resonance that connects wavevector-pairs between the central and the first-sideband channels.
2 kax
π
E ω
( )
ax13
Fig. 5.13 T-E curves for selected L values in Fig. 5.12, which show that the dip structures can be fine-tuned by L.
( )c
13
12
5.95 10 , 0.0028
0, 0.028 , 5, 1.34 10
sHz V eV
eV N ω
θ σ
−= × =
= ∆ = = = ×
0 500 1000 1500
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
L
T
Tm=0 Ttot
-1 0 1 2
Fig. 5.14 Non-typical Fabry-Perot resonance for the case when the system stays at the dip structure, with low incident energy, and the graphene is gapped . The incident energy E0 = ∆ + is at an ω
ω
above the band bottom ∆. The observed physics in Fig. 5.12 remain valid here. The L-period is the result of a non-typical Fabry-Perot resonance condition.
E ω
2
kax
π
( )
axAppendix A
The x component of momentum we choice in calculation is quite important. By Eqn.(A1) we can get four momentum by given energy and k . Two is for forward propagating mode, the other two is for y backward propagating mode.
( )
propagating momentum, respectively.Momentums can be divided into two parts. One is real part and the other one is imaginary part. The sign of real part decide which valley the momentum belongs to. We set up τ as valley index. Here the valley index was accompany with momentum p( )τ , we define p( )1 ,q( )1 ( p( )2 , q( )2 ) as
K
( )
K' valley.The sign of imaginary part should be treated very carefully to prevent explosion from wave function.
It should be positive when the real part of momentum is positive and be negative when the real part of momentum is negative. For example, in Eqn.(A2) if the imaginary part of pis negative, then
m x
ip Ma
e η will come to be infinity while M is increasing. However, it is wrong because when the wave is propagating within potential area, it should be decay while the propagating distance is increasing.
( ) ( )
There are two cases, which have different dispersion energy profile owing to
y y 2 k a π
> and
y y 2 k a π
< according to Eqn.(A.1). Hence, the dispersion confined in Brillouin zone will be discussed separately. The different energy levels also lead to different scenario, which need to be discussed individually.
Case(a): is flat. The dashed lines divide energy levels into different scenario, which need to be discussed individually. We will get 4 momentum from Eqn(A.1). In regions E−1< <E E1 ,E>E3 ,E<E−3 we get 4 complex wave vectors. In regionsE2 < <E E3 ,E−3< <E E−2 we get 2 complex 2 real wave vectors. In regions E1 < <E E2 ,E−2 < <E E−1 we get 4 real wave vectors. We define that if k is positive corresponding to the energy region we focus on then we name it K-related wave x
vector, contrary to K related wave vector if k is negative then we name it K’-related wave vector. x And we define right-going wave vector as p left-going wave vector as q. Here the valley index was accompany with momentump( )τ , we definep( )1 ,q( )1 (p( )2 ,q( )2 ) as K
( )
K' valley. X-axis is0 1 2 3 -4
-3 -2 -1 0 1 2 3 4
ky=0.2(K)
-2 0 2 4
Dispersion Energy
FigA.2.Left: Plot momentum p( )1 in real(Blue circle) and imaginary(Red cross) part. Fix ky at 0.2K. Right: Energy dispersion, which used to compare with the left figure. The red line label the momentum we are discussing. According to energy dispersion plot with red line, the group velocity is always positive. In regionsE>E2, E<E−3,E−1< <E E1 p( )1 is given by evanescent mode.
In regions of E1≤ ≤E E2,E−3 < <E E−1, p( )1 is given by propagating mode. The imaginary part of
( )1
p become larger as the energy is further away from the band bottom, and become smaller as the energy is closer toward to the band bottom.
(
( )1)
Im p ax
(
( )1)
Re p ax
( )1
p ax p a( )1 x
E1
E2
E−1
E−3
-4 -2 0 -4
-3 -2 -1 0 1 2 3 4
ky=0.2(K)
-2 0 2 4
Dispersion Energy
Fig. A.3.Left: Plot momentum p( )2 in real(Blue circle) and imaginary(Red cross) part. Fix ky at 0.2K.
Right: Energy dispersion, which used to compare with the left figure. The red line label the momentum we are discussing. According to energy dispersion plot with red line, the group velocity is always positive. In regionsE<E−2 ,E>E3,E−1< <E E1 p( )2 is given by evanescent mode. In regions E1≤ ≤E E3,E−2 < <E E−1,p( )2 is given by propagating mode. The imaginary part of p( )2 become larger as the energy is further away from the band bottom, and become smaller as the energy is closer toward to the band bottom.
(
( )2)
Re p ax
(
( )2)
Im p ax
( )2
p ax ( )2
p ax
E1
E3
E−1
E−2
Case(b):
The dashed lines divide energy levels into different scenario, which need to be discussed individually.
We will get 4 wave vector from Eqn.(A.1). In regions E−1< <E E1 ,E>E3 ,E<E−3 we get 4 complex wave vectors. In regionsE2 < <E E3 ,E−3< <E E−2 we get 2 complex 2 real wave vectors. In regions E1< <E E2 ,E−2< <E E−1 we get 4 real wave vectors. We define that if k is x positive corresponding to the energy region we focus on then we name it K-related wave vector, contrary to K related wave vector if k is negative then we name it K’-related wave vector. And We x define right-going wave vector as p left-going wave vector as q respectively. Here the valley index was accompany with momentump( )τ , we definep( )1 ,q( )1 (p( )2 ,q( )2 ) as K
( )
K' valley. X-axis0 1 2 3 -4
-3 -2 -1 0 1 2 3 4
ky=0.8(K)
-2 0 2 4
Dispersion Energy
Fig.A.5 Left: Plot momentum p( )1 in real(Blue circle) and imaginary(Red cross) part. Fix ky at 0.8K. Right: Energy dispersion, which used to compare with the left figure. The red line label the momentum we are discussing. According to energy dispersion plot with red line, the group velocity is always positive. In regionsE>E3, E<E−2,E−1< <E E1 p( )1 is given by evanescent mode. In regions E1≤ ≤E E3,E−2 < <E E−1 p( )1 is given by propagating mode. The imaginary part of p( )1 become larger as the energy is further away from the band bottom, and become smaller as the energy is closer toward to the band bottom.
(
( )1)
Re p ax
(
( )1)
Im p ax
( )1
p ax ( )1
p ax
E1
E3
E−1
E−2
-4 -2 0 -4
-3 -2 -1 0 1 2 3 4
ky=0.8(K)
-2 0 2 4
Dispersion Energy
Fig.A.6 Left: Plot momentum p( )2 in real(Blue circle) and imaginary(Red cross) part. Fix ky at 0.8K. Right: Energy dispersion, which used to compare with the left figure. The red line label the momentum we are discussing. According to energy dispersion plot with red line, the group velocity is always positive. In regionsE>E2, E<E−3,E−1< <E E1 p( )2 is given by evanescent mode. In regions E1≤ ≤E E2,E−3 < <E E−1,p( )2 is given by propagating mode. The imaginary part of p( )2 become larger as the energy is further away from the band bottom, and become smaller as the energy is closer toward to the band bottom.
( )2
p ax p a( )2 x
(
( )2)
Re p ax
(
( )2)
Im p ax
E1
E2
E−1
E−3
Appendix B
The current operator was defined as
2 j ρv v+ ρ
= (B.1)
Where ρ and v are density operator and velocity operator, respectively.
The density operator can be written down as
2 0
1 M N A M N A, , , , M N B M N B, , , ,
ρ= a + (B.2)
a is the lattice constant, on the other word it is the length between each lattices. 0
The velocity operator can be relevant to
[
x H ,]
The Hamiltonian already introduce in Chapter 3
0
The position operator is defined as
', '
In Eqn.(B.4), RM N', ' identify the position from subblattice A and B. The distance between sublattice A and B is defined by d. M N D M N D help us ensure the difference between subblattice A and B. it works out as the following.
0
To make it more specify we can see Fig.B.1
Fig.B.1 We set up sublattice A as original point and subblattice B is at a distance d away from A site. The coorinate here is described by capitalRM N d, ,
To make the current easier to obtain, we use δi
to label the vectors we need to discuss.
Fig.B.2 We define δ δ δ1, 2, 3 as the vectors
( ) (
d , − +a2 d) (
, − +a1 d we will use later to present)
the current flow.
Now we can get commutation of
[
x H ,]
Combine (B.6) and (B.3) the velocity operator
3
Finally we get our current operator.
3
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