PHYSICAL REVIEW B 88 , 155432 (2013)
Bolin Liao,
1
Mona Zebarjadi,
2
Keivan Esfarjani,
2 and Gang Chen
1 ,
*
1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2 Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, New Jersey 08854, USA
(Received 4 April 2013; revised manuscript received 5 October 2013; published 22 October 2013)
We propose and investigate a design for “electron cloaks” comprised of two electrodes, one top gate and one back gate, on either side of a graphene sheet arranged in a concentric disk configuration. Dirac electrons with specific energies can flow through these electron cloaks with negligible scattering, while electrons with different energies experience significant scattering. The scattering widths of the electron cloaks are analyzed using the partial wave formalism applied to the Dirac equation, and the contributions of the first two partial waves to the scattering widths are set to zero simultaneously via a proper combination of the potentials on the two electrodes.
We show that this strategy is sufficient for reducing the total scattering widths to below 0.01% of the physical widths of the cloaks. This new design differs from the well-known Klein tunneling phenomenon in that, in our case, the transparency is isotropic and energy selective. These characteristics, in tandem with tunable Fermi levels and/or the gate voltages on the electrodes, enable the electron cloaks to serve as core units in the designs of new sensors, switches, or transistors.
DOI: 10.1103/PhysRevB.88.155432
PACS number(s): 72 .
80 .
Vp
I. INTRODUCTION
Since its first discovery in 2004,
1
graphene, with its extraordinary electrical and mechanical properties,
2 , 3
has attracted a significant amount of interest in the research community as a promising candidate for next-generation electronics. An essential characteristic of graphene which makes it qualitatively distinct from traditional electronic materials is that its electrons obey relativistic Dirac-like dynamics
2 – 8
instead of the nonrelativistic effective-mass Schr¨odinger dynamics.
From a full gamut of exotic phenomena which result from
Dirac dynamics, Klein tunneling,
9
i.e., the perfect transmission of Dirac electrons through a tall and wide potential barrier, were predicted to also occur in graphene
10
and is one reason for its ultrahigh carrier mobility.
11 , 12
Klein tunneling is a direct consequence of the charge-conjugation symmetry of the Dirac equation and the conservation of pseudospin.
10
In the case of Klein tunneling, the transparency of the barrier is angle selective in that normal-incidence electrons pass through freely, regardless of their energies, while there is, in general, finite reflection for electrons with a non-normal angle of
incidence.
10
In this paper, we propose and study a design of artificial scattering centers on graphene, which appear transparent to electrons with a specific energy regardless of their incident direction, and present a sharp transition of the scattering rate within a small energy range. Recently, we reported properly
tuned core-shell nanoparticles that act as electron filters 13
and we demonstrated that if the filtering energy is tuned inside the
Fermi window, such invisible dopants could be used to enhance the carrier mobility and thus the thermoelectric power factor of semiconductor materials.
14
In a recent publication, Fleury et al.
gave a systematic account of the mechanism and the condition
of quantum cloaking in such a structure.
15
As we pointed out in our earlier report,
13
such electron filters are easier to implement in a two-dimensional (2D) structure since the artificial barriers can be realized simply using gate electrodes. Our previous work and that of Ref.
15
were based on the effective-mass
Schr¨odinger equation. We present our results on graphene in
1098-0121/2013/88(15)/155432(6) this paper, following a similar strategy now applied to Dirac dynamics. It is worth mentioning that “invisible cloaks” for relativistic fermions in both 3D and 2D have previously been designed
16 , 17
using the technique of transformation optics. The resulting structures, however, are too difficult to implement in practical situations. The technique described herein, based on the partial wave formalism, is analogous to the Mie-theorybased approach adopted by Al`u et al.
in designing optical cloaks.
18 , 19
This formalism leads to a design for an electron cloak that is potentially feasible.
II. METHODOLOGY
Consider the geometry shown in Fig.
1 , with two round
electrodes, arranged in a concentric disk configuration, deposited on either side of the graphene sheet. In a previous paper,
20
Cserti et al.
studied a circular p n junction structure on graphene for the purpose of focusing electrons. In this paper, we are concerned with the scattering property of electrons off this artificial structure, and the geometry is chosen so that the scattering events are isotropic and it is possible to obtain an analytical solution. With a circularly symmetric external electrostatic potential V ( r ), the Dirac Hamiltonian
for electrons near the cones reads 3
H
K
= v
F
σ
· k
+
V ( r ) (1) where v
F
= (3 / 2) | t | a
0 is the Fermi velocity, with | t | ≈ 2 .
8 eV being the nearest-neighbor coupling energy and a
0
=
1 .
42 ˚ the interatomic distance between nearest neighbors, σ
=
( σ x
,σ y
) are the Pauli matrices, and k is the wave number operator. One canonical way of calculating the elastic scattering cross section of electrons off a circularly symmetric potential is the partial wave method,
21
which has also been discussed in detail in the context of Dirac electrons in graphene.
22 – 25
Here we only cite the crucial results that will be used in later discussions. For Dirac electrons in graphene, their orbital momenta and pseudospins are coupled via the vectorial form of the wave function, and thus the consequence of the circular symmetry of the scattering potential is the conservation of the
155432-1 ©2013 American Physical Society

LIAO, ZEBARJADI, ESFARJANI, AND CHEN PHYSICAL REVIEW B 88 , 155432 (2013)
Accordingly, the total scattering width [defined as the ratio of the total number of scattered electrons per unit time to the incident electron flux, analogous to the three-dimensional (3D)
cross section] can be calculated from the phase shifts via 22
w
=
8 k j
=
1 / 2 , 3 / 2 ,...
sin
2
δ j
.
(3)
FIG. 1. (Color online) Schematic of the arrangement of the top and the back electrodes on a graphene sheet. The potential profile generated is shown in the bottom figure, and the radii of the electrodes are labeled. In practice a thin insulating separation layer is needed between the gates and the sample, which is not shown in this drawing for clarity.
z component of the so-called “isospin-orbital” momentum
26
j z
= l z
+ (1 / 2) σ z
, i.e., the Dirac Hamiltonian commutes with j z
. As a result, the partial waves are labeled by the quantum number j z
= ±
1 / 2 ,
±
3 / 2 . . . , and the wave function of a general scattering state can be expanded in terms of the partial waves (eigenstates of j z
) as
ψ
E
( r ) = c j
φ
Ej
( r ) j
= j c j
F
Ej
( r ) e i ( j
−
1 / 2) θ iG
Ej
( r ) e i ( j
+ 1 / 2) θ
, (2) where c j
’s are expansion coefficients, and F
Ej
( r ) and G
Ej
( r ) are arbitrary radial functions to be determined by a given scattering potential. In the absence of any scattering potential, the radial functions turn out to be Bessel functions of the first kind
( k
= |
E
|
/v
F
F
Ej
( r )
=
J j − 1 / 2
( kr ), G
Ej
( r )
= sgn( E ) J j + 1 / 2
( kr which approach a 2D circular wave at r → +∞ . That the G
Ej
) is the wave number corresponding to the energy), component depends on the sign of the energy (positive for electron states, and negative for hole states) reflects the fact that the electron states and the hole states possess opposite
chirality, 27
defined as the projection of the pseudospin onto the direction of propagation. If a finite-range scattering potential, or one which decays rapidly enough,
21
is turned on, the asymptotic behavior of the radial functions at r
→ +∞
, when the Bessel functions approach radial waves, will not be altered due to the short range of the potential. The only impact of the potential is to “pull in” or “push out” the partial waves by a certain amount, giving rise to a “phase shift” δ j for each partial wave. This phase shift can subsequently be calculated by matching the wave functions ( F
Ej
/G
Ej in this case
22
and the logarithmic derivative ψ /ψ in the case of Schr¨odinger equation) at the cut-off radius of the potential. The phase shifts quantify the distortions of each partial wave caused by the scattering potential and thus represent the weights of the contributions of the partial waves to the total scattering.
Although the summation includes infinite terms, in general only a few terms of the lowest orders will contribute nonnegligibly. This is due to the fact that the first maximum of the Bessel function of order n scales roughly as O ( n ), and thus if ka n ( a is the cut-off radius of the potential), the amplitude of the n th partial wave inside the potential region is so small that it feels a negligible impact from the scattering
potential.
21 As discussed in our previous work, 13
a choice for the size of the potential region which is both simple and practical is for ka 1. Under this condition, partial waves with j 5 / 2, involving Bessel function of order 2 and above, will not contribute much to the summation in Eq.
(3) . This
condition translates to a
3
| t
|
2 E a
0
=
4 .
2 eV
E
×
1 .
42 ˚ .
(4)
Thus, if we want to make the potential transparent to electrons with energy E
=
100 meV, the radius of the potential region needs be smaller than
∼
6 nm, which can be relaxed to
∼
60 nm if we choose an electronic energy of E
=
10 meV.
Note that in the case of nonrelativistic electrons with effective mass (i.e., spherical nanoparticles inside bulk matrices), the maximum cloaking radius discussed above is significantly smaller. For example, for an electron in SiGe with the same energy of 10 meV but with an effective mass of 1, the maximum radius of the potential region is
∼
2 nm. Even for electrons in GaAs, with much smaller effective masses of 0.063, the maximum radius is only
∼
8 nm, which makes the 3D realization more difficult. Here we set the “cloaking” energy at E
=
100 meV because it corresponds to a moderate carrier concentration of n
=
3 .
7
×
10 11 cm
−
2 , which can be obtained practically in experiments.
1
Provided the condition ka 1 is met, the remaining task is to make the first two terms in Eq.
(3)
vanish. We will show later that it is indeed the case that higher-order terms only contribute negligibly to the total scattering width. It can be seen from Eq.
(3)
that the way to make a certain term in the summation vanish is to set the phase shift of the corresponding partial wave to a multiple of
π , in which case the scattered partial wave outside the potential region differs from its unscattered counterpart by, at most, a global sign change.
Given the formalism above, we first calculate the phase shifts caused by the core-shell potential. Assume the radius of the core is a c and that of the shell is a , the corresponding electrostatic potentials are V
1 and V
2
, respectively, and the electronic energy is E > 0, without loss of generality.
Denote the wave numbers in the core and the shell regions as α
= |
E
−
V
1
|
/v
F and β
= |
E
−
V
2
|
/v
F
. Then the radial wave functions in each region can be written as
22
( J i and
Y i are i th Bessel functions of the first and second kind,
155432-2

ISOTROPIC AND ENERGY-SELECTIVE ELECTRON . . .
PHYSICAL REVIEW B 88 , 155432 (2013) respectively)
F j,r<a c
= A [ J j
−
1 / 2
( αr )] ,
G j,r<a c
= sgn( E
−
V
1
) A [ J j
+
1 / 2
( αr )] ,
F j,a c
<r<a
=
B [ J j − 1 / 2
( βr )
+
B Y j − 1 / 2
( βr )] ,
(5)
(6)
(7)
G j,a c
<r<a
= sgn( E − V
2
) B [ J j
+
1 / 2
( βr ) + B Y j
+
1 / 2
( βr )] ,
F j,r>a
=
C [ J j − 1 / 2
( kr ) cos δ j
−
Y j − 1 / 2
( kr ) sin δ j
] ,
G j,r>a
= sgn( E ) C [ J j
+
1 / 2
( kr ) cos δ j
−
Y j
+
1 / 2
( kr ) sin δ j
] ,
(8)
(9)
(10) where A,B,B , and C are constants to be fixed by matching boundary conditions. At the boundaries ( r
= a c components of the wave function must be continuous, or in other words, F /G must be continuous.
22 and
Matching r
=
F /G a ), both at each boundary, the phase shifts are
δ j
= arctan
J j + 1 / 2
( βa ) J j
J j + 1 / 2
( βa ) Y j
− 1 / 2
− 1 / 2
( ka )
+
B J
( ka )
+
B Y j − 1 / 2 j − 1 / 2
( ka ) Y j
( ka ) Y j
+ 1 / 2
( βa )
−
ζ
2
+ 1 / 2
( βa )
−
ζ
2
J j − 1 / 2
( βa ) J j
J j − 1 / 2
( βa ) Y j
+ 1 / 2
+ 1 / 2
( ka )
−
ζ
2
( ka )
−
ζ
2
B J j + 1 / 2
B Y j + 1 / 2
( ka ) Y j
( ka ) Y j
− 1 / 2
( βa )
− 1 / 2
( βa )
,
(11) where the parameter B is given by
B
=
ζ
J
1
ζ
2
J j
−
1 / 2 j
+
1 / 2
( αa c
( αa
) Y c
) J j
+
1 / 2 j
−
1 / 2
( βa c
( βa c
) − ζ
)
−
J j
+
1 / 2
1
ζ
2
J j
−
1 / 2
( αa c
( αa c
) J j
−
1 / 2
) Y j
+
1 / 2
( βa c
)
( βa c
)
, (12) where ζ
1
= sgn( E
−
V
1
) and ζ
2
= sgn( E
−
V
2
). We note that the continuity of F and G does not guarantee the continuity of their first derivatives. The discontinuities of the first derivatives can be quantified by putting Eq.
(2)
back into the Dirac equation, Eq.
(1) , and it turns out
F and G are coupled in the following form: dG dF
Ej dr
Ej dr
( r
( r )
)
−
+
( j
+ 1 / 2) r
( j
−
1 / 2)
F r
G
Ej
Ej
( r
(
) r )
=
= − k k
(
(
V
V
)
)
F
G
Ej
Ej
( r ) ,
( r ) ,
(13)
(14) where k ( V ) is the local wave number that depends on the local potential. Assume k
= k out outside of the boundary and k
= k in inside the boundary, then the discontinuity of the first derivatives can be calculated as dG
Ej
( r ) dr out
− dG
Ej
( r ) dr in
=
( k out
− k in
) F
Ej boundary
,
(15) dF
Ej
( r ) dr out
− dF
Ej
( r ) dr in
= − ( k out
− k in
) G
Ej boundary
.
(16)
These discontinuities, however, do not render the previous formalism unphysical, since the probability flux in this case
21
J
= v
F
ψ
+
σ ψ is only a function of the wave-function components, rather than their derivatives.
To proceed further, we first examine the effects on the phase shifts of partial waves of different orders exerted by the potential barriers and wells. In Fig.
2
the phase shifts of partial waves with orders 1 / 2 and 3 / 2 are plotted against varied
155432-3 electrostatic potentials of a single scatterer. Two characteristics can be seen from this figure: A potential well “pulls in” the partial wave and results in a positive phase shift, while a potential barrier does the opposite; both potential barrier and well have a stronger effect on the lower-order partial wave than the higher-order one. These characteristics were also observed in a 3D semiconductor core-shell nanostructure, delineated in our previous work.
13
We follow the same procedure as in that work to find the proper values of the electrostatic potentials.
Readers are referred to our previous paper
13
for details and only the resulting solution curves of in Fig.
3 . Values of
V
1 and V
2
V
1 and V
2 are given here on the curves in Fig.
3
represent solutions which make the phase shifts of the first two partial waves equal to 0 or π , and the intersection of these solutions
1
0.5
0
−0.5
j=0.5
j=1.5
−1
−0.2
0 0.2
Electrostatic Potential (V)
FIG. 2. (Color online) Effects of a potential well and a potential barrier on the phase shifts of the two lowest-order partial waves. For this demonstration, the electron energy is chosen to be 100 meV, and the radius of the potential region is set to be 9 nm.

LIAO, ZEBARJADI, ESFARJANI, AND CHEN PHYSICAL REVIEW B 88 , 155432 (2013)
70
60
50
40
30
20
10
0
−1.5
j=0.5, PS=
π j=1.5, PS=
π j=1.5, PS=0 j=0.5, PS=0
−1
V
2
(eV)
−0.5
0
FIG. 3. (Color online) The combinations of V
1 and V
2 that make the phase shifts of the two lowest-order partial waves 0 or π . The intersection (marked out by the dashed circle in the figure) indicates the solution for making both phase shifts π at the same time, and thus the contributions from the two lowest-order partial waves to the total scattering width vanish. Here the parameters we use are a
=
4 nm, a is c
=
2 nm, and E
=
100 meV. The solution given at the intersection
V
1
=
128 meV and V
2
= −
0 .
969 eV.
is where the contributions to the total scattering width from these two partial waves vanish simultaneously. It is seen that the intersection only happens when both phase shifts are set to π , as a result of the mismatch of the effects of the potential barrier and the well on the two partial waves.
III. RESULTS AND DISCUSSION
In Fig.
4
the total scattering width and the separate contributions from the first few partial waves are plotted against the electron energy. As expected, the contributions to the scattering width from the two lowest-order partial waves with j
=
1 / 2 and 3 / 2 vanish at the designed “cloaking energy” of E
=
100 meV, and a sharp transition of the scattering rate happens around this cloaking energy. A residue scattering width smaller than 0.01% of the physical width (the diameter of the cloak, i.e., 2 a in this case) comes from the contributions the radial distribution functions [
√ rF ( r ) and
√ rG ( r )] with
1
0
−1
0 j=0.5, F
10 20 30
1
0
−1 j=1.5, F
0 10 20 30
1
0
−1 j=0.5, G
0 10 r (nm)
20 30
1
0
−1 j=1.5, G
0 10 r (nm)
20 30
FIG. 5. (Color online) Comparison of the radial distribution functions of both components of the two lowest-order partial waves with (green lines) and without (orange lines) the scattering potential.
Outside of the potential region ( r > 4 nm), a phase shift of π is observed for all four wave-function components, making the wave functions with and without the scattering potential essentially indistinguishable from each other.
and without the scattering potential in Fig.
5 , where the
π phase shifts outside of the potential region are observed. The wave functions with and without the scattering potential are essentially indistinguishable from each other outside of the potential region.
In practical applications, one way of controlling the switching behavior of the device is via changing the electrostatic potential on one of the electrodes (as opposed to shifting the Fermi level). To demonstrate this operation, we calculate the scattering width of the electrons with fixed energy ( E
=
100 meV), while sweeping the potential on the core electrode
V
1
, and the result is shown in Fig.
6 . Again a sharp transition
is observed around the “correct” cloaking potential V
1
=
128 meV within an energy range of about 20 meV. These results suggest that the core-shell gated graphene structures can serve as basic electronic units for switches, transistors, and sensors.
It is important to note that, in practice, the simple sharpedged potential is not realistic due to electrostatic smearing, and also intervalley scattering may be induced which is not included in our discussion. To evaluate the effect of the electrostatic smearing of the potential edges and to test
10
2
10
2
Total
10
0 j=0.5
10
0
Total j=1.5
10
−2
10
−2 j=2.5
10
−4
0 50 100 150
Energy (meV)
200 250 300
FIG. 4. (Color online) The total scattering width (normalized to the physical width) and contributions from different partial waves. A sharp dip is observed at E
=
100 meV as designed, and the residue scattering width comes from higher-order partial waves.
10
−4
0 50
V
1
100
(meV)
150 200
FIG. 6. (Color online) The total scattering width at fixed energy
( E
=
100 meV) versus the electrostatic potential on the core electrode
V
2
V
1
. The other parameters used are a
=
4 nm, a
= −
0 .
969 eV.
c
=
2 nm, and
155432-4

ISOTROPIC AND ENERGY-SELECTIVE ELECTRON . . .
PHYSICAL REVIEW B 88 , 155432 (2013)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
(a)
1
10
1
(b)
2
γ
=0
γ =0.03nm
γ
=0.1nm
γ
=0.3nm
3 r (nm)
4 5 6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
γ
=0
γ
=0.1nm
γ
=0.2nm
1 2 3 4 k (1/nm)
5 6 7 8
FIG. 8. (Color online) The normalized Hankel transform of the potential profile for different smearing parameters.
10
0
10
−1
10
−2
10
−3
10
−4
50
10
2
10
1
10
0
10
−1
10
−2
(c)
100
Energy (meV)
γ
=0
γ
=0.03nm
γ
=0.1nm
γ
=0.3nm
150
10
−3
0 20 40 60
Energy (meV)
80 100
FIG. 7. (Color online) Robustness of the cloaking behavior against the electrostatic smearing. (a) The smeared-out potential profile with different smearing parameters γ ; (b) the corresponding total scattering width with different smearing parameters; and (c) the total scattering width versus energy with γ
=
0 .
3 nm while the cloaking size is increased to a
=
12 nm , a c
=
6 nm .
the robustness of the “cloaking behavior,” we developed a numerical program to calculate the scattering width for an arbitrary concentric scattering potential. We chose a
Fermi-Dirac function-typed f ( x )
= { exp [( x
−
μ ) /γ ]
+
1
} −
1 smearing of the potential, for it is simple and the extent of smearing can be easily quantified by picking different values of the parameter γ . The resulting potential profiles are plotted in Fig.
7(a)
below. The calculated scattering widths
155432-5 versus energy with different smearing parameters are plotted in Fig.
7(b) . As expected, for small smearing parameters
( γ 0 .
1 nm), the sharp dip in the scattering width is still well defined, and only the dip becomes shallower and the cloaking energy is slightly shifted. Whereas for a larger smearing parameter ( γ
=
0 .
3 nm in this case), the sharp dip disappears and the cloaking behavior is lost. However, we can always increase the size of the cloak and decrease the cloaking energy to recover the cloaking behavior. To demonstrate this point, we set the smearing parameter γ
=
0 .
3 nm, which is “large” in the previous case, but increase the size of our cloak to a
=
12 nm and a c
=
6 nm, and correspondingly reduce the cloaking energy to E
=
40 meV. Following the same strategy of design, we found the appropriate potentials are V
1
=
35 meV and V
2
= −
317 meV. The scattering width versus energy is plotted in Fig.
7(c)
below. The cloaking behavior is resumed in this case, with only the cloaking energy being slightly shifted up from the designed cloaking energy
(40 meV) due to the electrostatic smearing.
Perfectly sharp potentials considered in this paper can provide large crystal momentum, leading to intervalley scatterings. For smeared-out smooth and continuous potential profiles, on the other hand, we expect the intervalley scattering should be largely suppressed. To verify this, we carry out
2D spatial Fourier transforms of the potential profiles and check how fast the Fourier component decays with increasing magnitude of the k vector. For a 2D circularly symmetric function, its Fourier transform is also circularly symmetric and can be calculated equivalently as its Hankel transform of order zero (also named Fourier-Bessel transform):
F
0
( k )
=
0
+∞ f ( r ) J
0
( kr ) r dr.
(17)
The results (for the cloak of size a = 4 nm) are plotted in Fig.
8 . The threshold for intervalley scattering to start
playing a role is approximately k
=
1 /a
0
≈
7 .
04 nm
− 1
. As we can see from the plot, actually even in the case of perfectly sharp potential, the intervalley scattering only plays a minor role, which is further suppressed when the potential edges are smeared out. If we further increase the size of the cloaking zone, the potential profile tends to be more spread out and

LIAO, ZEBARJADI, ESFARJANI, AND CHEN smooth and the intervalley scattering can always be safely neglected.
Another practical concern has to do with the potential range within which the linear dispersion and Dirac dynamics remain valid. In Ref.
3 , the cut-off energy was suggested as
E cut-off
0 .
4 t
≈ 1 .
1 eV, and thus the designed electrostatic potentials have to lie in this range. In general this would not be a significant issue since one can always adjust the other parameters (sizes and the cloaking energy) to accommodate lower electrostatic potentials in the design. The design shown in
Fig.
7(c)
serves as an example: a = 12 nm , a
35 meV , and V
2 c
= 6 nm ,V
1
=
= −
317 meV with a cloaking energy at E
=
40 meV.
The presence of disorders is another important factor in considering the experimental implementation of our design. For the above formalism to be valid, the electrons should transport through the cloaking region coherently, which requires their mean free paths due to other scattering mechanisms (electronphonon, electron-disorder, etc.) be significantly longer than the characteristic size of the cloaking region. This criterion could be met in high-quality graphene samples with boron nitride substrate at sufficiently low temperatures.
3
When the disorder density gets so high that the disorders start interacting with the cloaking electrodes, the above formalism is no longer valid and its impact on the cloaking behavior needs to be reexamined. Given the multitude of different types of disorders in graphene (impurities, structural defects, adsorptions, etc.), and considering the fact that the effect of disorders cannot be incorporated into our analytic framework in any straightforward way, a quantitative estimation of the
PHYSICAL REVIEW B 88 , 155432 (2013) robustness of the cloaking behavior against disorders is lacking at present and will be included in our future pursuits.
IV. SUMMARY
In summary, a design of electron “cloaks,” or filters, is demonstrated theoretically on a graphene sheet. The scattering width of these artificial scattering centers is minimized to negligible values with regard to electrons with certain energy.
The physical origin of the transparency, namely, the π phase shifts for the two lowest-order partial waves, is discussed. One concern for practical applications is that the overall scattering level is relatively low for the current design, and a high on-off ratio may not be achievable even with the sharp change of the scattering rate. With the great flexibility in terms of the parameter choices, however, we believe there is enough room for optimizing the switching behavior of the cloaking devices, and even larger room to explore for novel ways of controlling electron transport by artificial scattering centers at nanometer scale.
ACKNOWLEDGMENTS
This paper is based upon work supported partially by
S 3 TEC, an Energy Frontier Research Center funded by the
U.S. Department of Energy, Office of Basic Energy Sciences under Award Number DE-FG02-09ER46577 (for potential applications of the cloaking idea on power generation), and partially by AFOSR MURI via Ohio State University (for potential applications in cryogenic cooling).
* Author to whom correspondence should be addressed: gchen2@mit.edu
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