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Total Negative Refraction in Real Crystals for Ballistic Electrons and Light
Yong Zhang, B. Fluegel, and A. Mascarenhas
National Renewable Energy Laboratory, 1617 Cole Boulevard, Golden, Colorado 80401, USA
(Received 17 April 2003; published 9 October 2003)
It is found that there exists a category of material interfaces, readily available, that not only can
provide total refraction (i.e., zero reflection) but can also give rise to amphoteric refraction (i.e., both
positive and negative refraction) for electromagnetic waves in any frequency domain as well as for
ballistic electron waves. These two unusual phenomena are demonstrated experimentally for the
propagation of light through such an interface.
DOI: 10.1103/PhysRevLett.91.157404 PACS numbers: 78.20.Ci, 42.25.Gy, 73.40.c, 73.50.h
The phenomenon of refraction of light at the interface refraction can be either positive or negative, depending
of two transparent media A and B is the underlying on the incident angle, despite all components of " and
mechanism for steering light in many optical devices being positive. These two findings, in principle, apply to
[1]. However, the necessity of a refractive index mismatch the full spectrum of electromagnetic waves as well as
(nA Þ nB) for achieving this effect inevitably results in a ballistic electron waves, and thus should simplify the
finite reflection loss. In the propagation of an electron study of negative refraction.
wave through discontinuous media, one encounters a The unique interface proposed here can be viewed as a
situation quite analogous to that for light. Perhaps the homojunction that belongs to a special category of twin-
closest analogy to the refraction of light would be that ning structures in uniaxial crystals. For such a twin
of a ballistic electron beam propagating through a heter- structure, the interface is a reflection symmetry plane
ojunction of semiconductors A and B which differ only in for the two twin components: their symmetry axes are
their effective masses (mA Þ mB). Here again, refraction coplanar with the normal to the interface and oriented
inevitably is associated with a finite reflection, because of symmetrically with respect to the interface. Figure 1
the effective mass mismatch [2]. Furthermore, for most of shows a real twin structure of this type frequently ob-
the commonly encountered situations, there will always served in spontaneously ordered III-V semiconductor al-
be an energy discontinuity between A and B [24], caus- loys [18]. The ordering direction or the symmetry axis
ing an additional intensity loss for the transmission across switches from the crystallographic direction [ 1111] (A
such an interface. The first intriguing finding to be pre- side) to 1 111 (B side) across the twin plane whose normal
sented in this Letter is a unique type of interface that is in the [ 1110] direction. This type of domain twin struc-
enables refraction without any reflection, i.e., total re- ture can be found in many naturally or synthetically
fraction, for either an electron or a light beam. formed crystals that are classified as ferroelastic mate-
Recently, the phenomenon of negative refraction [5] rials [19]. With the advances in semiconductor growth
has attracted a great deal of attention, because of its techniques, they can now be obtained during epitaxial
implications for realizing a ``superlense'' with a resolu-
tion smaller than the wavelength of light, as well as for
observing a reversal of the Doppler shift and Vavilov-
Cerenkov radiation [614]. It was first suggested by A B
Veselago [5] that negative refraction can occur at the
interface of a normal medium, with both permittivity "
and permeability being positive, and an abnormal
medium, with both " and being negative. It has been
pointed out lately that if the abnormal side is a uniaxial
medium, negative refraction can arise with just one of the
four components of " and being negative [15]. There
has so far been only one experimental demonstration of
negative refraction, which occurs in a small window of
microwave frequencies with a low transmission typically FIG. 1. Electron microscopy of a domain twin. A typical
below ĸ24 dB [7,16], and the validity of the interpreta- high-resolution cross-sectional TEM picture of domain twin
tion is still under debate [11,17]. The second interesting structures frequently observed in CuPt ordered III-V semi-
finding presented in this Letter is that the same type of conductor alloys. The ordering directions are [ 1111] (left) and
interface that can yield total refraction for electrons and [1 111] (right). The vertical dashed line indicates the twin
light can in fact yield amphoteric refraction, i.e., the boundary.
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growth in a controllable manner with domain sizes rang- term the reflected wave, and the ``c'' term the transmitted
ing from nm to m [20]. wave. The boundary conditions for the continuity of the
We first consider the transmission of a ballistic electron wave function and the current at the interface [21] lead to
beam at a semiconductor twin boundary, using the struc- two equations: a b c and a ĸ b c. The solutions
ture shown in Fig. 1 as a prototype system. For such a for these boundary conditions are simply c a and
homojunction, there is obviously no band offset between b 0. This surprising result implies that the twin bound-
the two twin components. Thus, the two regions can ary is indeed reflectionless and transparent to electron
simultaneously be transparent for electrons (with energy propagation.
above the conduction band edge) and light (with energy This prompts the question as to whether the electron
below the fundamental band gap). However, the effective beam will still be refracted at all. It is easy to see that
mass and refractive index of A and B are not matched for there is indeed a refraction for the wave front defined by
general directions, except for the direction of the twin the direction of k, since with kAz1 Þ kBz1 the incident angle
plane normal. Therefore, intuitively, a finite reflection A Arctan kx=kAz1 differs from B Arctan kx=kBz1 .
would be expected at the interface for any non-normal However, in an anisotropic crystal, it is more meaningful
incidence of electron or light. The results given below are to examine the group velocity, defined as v rkE k = h,
in fact counterintuitive. that coincides with the direction of the electron flow,
In a principal coordinate system, the effective mass described by the probability current density J. In an
tensor of the uniaxial semiconductor takes the form anisotropic semiconductor, J is given as [21]
0 mĸ1 1 X3
? 0 0 p
mĸ1 @ 0 mĸ1 A J Re F F ; (5)
? 0 ; (1) m
0 0 mĸ1 1
jj where m
where m is the component of the effective mass
? and mjj are effective masses (in units of the free tensor. In general, we find that across the interface, the
electron mass m0) for the wave vector k perpendicular current perpendicular to the interface, J
and parallel to the uniaxis. In a coordinate system with z z, is continuous,
but the current parallel to the interface, J
along the twin plane normal [ 1110], x along [001], and y x, is not, which
results in a pure deflection or bending of the incident
along [110], the electron dispersion electron beam, since the reflection is identically zero.
h2 k2
p Negative refraction is said to occur when the sign of J
E k ĸ k2 2 k ; (2) x
2m z ĸ k2y ĸ sign2 xkz changes across the boundary. In Fig. 2, the interrelation of
0 m
m
where m
m is the average effective mass defined as 1= m
m
2=m? 1=mjj =3, is the anisotropy parameter defined
as 1=m? ĸ 1=mjj =3, sign 1 for the A side, and
ĸ1 for the B side. Since the structure is uniform along the
y direction, one can choose the x-z plane as the incidence
plane (i.e., ky 0) without loss of generality. For an
incident electron with a given energy E and wave vector
kx (these quantities are required to be conserved across
the interface), there are two allowed solutions for the
wave vector kz from the dispersion Eq. (2): kAz1and kAz2
for the A side, kBz1 and kBz2 for the B side, respectively.
Note that because of the anisotropy, the simple relations
kAz2 ĸkAz1 and kBz2 ĸkBz1 are not valid any more.
However, it can be shown that for each side only one
solution can give rise to a positive z component of the
group velocity (chosen to be kAz1 and kBz1); while the other FIG. 2 (color). Refraction of a ballistic electron beam at the
yields a negative z component of the group velocity ( kAz2 interface of the semiconductor twinning structure. The vertical
and kBz2). gray line indicates the interface. Arrows A and B indicate the
The wave functions for the two sides can be written as orientations of the uniaxis on each side with 0 35:3 . The
beams in the two regions (left and right) can be either on the
FA x; z a exp i kxx kAz1z b exp i kxx kAz2z ; same side (corresponding to negative refraction) or on different
(3) sides (corresponding to positive refraction) of the interface
normal, depending on the value of the wave vector parallel
FB x; z c exp i k to the interface k
xx kB x (in unit of k0 2 =a, a is the lattice
z1z ; (4) constant of the semiconductor). The energy of the incident
where the ``a'' term describes the incident wave, the ``b'' electron is 0.2 eV.
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the incident and refracted current J, described by the wave is identically zero, and that of the transmitted wave
incident angle A Arctan JAx=Jz and the refraction always equals that of the incident wave. Similar to the
angle B Arctan JBx=Jz , is illustrated by numerical situation for the electron beam, the negative refraction
results with typical parameters achievable in III-V alloys does occur over a range of incidence angles. The largest
( m
m 0:114 and 2:117 for a fully ordered GaInP bending or the strongest negative refraction in fact
[22]). It is of particular interest to notice that it is possible happens when kx 0, where the propagation direc-
to vary the incident angle from positive to negative, while tion of the light wave, defined by the Poynting vector
keeping the refraction angle positive, i.e., the refraction S E H, is given as sin A0 sin2 0 "jjĸ
q
can be amphoteric. The angle between the symmetry axis
p " "2 , and sin
and the twin plane normal is ? = 2 ?sin2 0 "2jjcos2 0 B0 ĸ sin A0.
0 Arccos 2=3 35:3
for this specific example, but the qualitative conclusions To experimentally illustrate the effect, we use a YVO4
are in fact valid for any arbitrary value of bicrystal with 0 ĸ45 to emulate the proposed twin
0, with the
effect maximized at structure.YVO4 is a uniaxial positive crystal with n0
0 45 .
We next discuss the transmission of light at a twin 2:017 68 and ne 2:250 81 at 532 nm [23]. The device is
boundary similar to that of Fig. 1, but with an arbitrary formed by bonding two nominally identical crystals in
angle optical contact. The accuracy for the optical axis orienta-
0. Analogous to Eq. (1), the dielectric tensor of an
anisotropic crystal has the following form in the principal tion is 0:5 for each crystal. The input and output planes
coordinate system: are antireflection coated at 532 nm. Figure 3 shows the
0 1 refraction of a 532 nm laser beam at the interface of the
"? 0 0 bicrystal at two typical incident angles, which yields both
" @ 0 " A
? 0 : (6) positive and negative refraction. The power loss of the
0 0 " transmitted beam, which ideally should be zero, is mea-
jj sured to be in the order of 10ĸ4, due to the imperfection of
It can be shown that for an electromagnetic wave whose the device (e.g., the relative orientation of the optical axes
electric field is polarized along the y direction (i.e., and the quality of the optical contact). In fact, on the
orthogonal to both the uniaxis of the A and B side and images shown in Fig. 3, no reflection is visible to the
normally referred to as an ordinary wave), the twin naked eye at the bicrystal interface. Figure 4 shows the
boundary like the one shown in Fig. 1 has no effect at comparison between the measured and calculated light
all on the incident wave (i.e., A B with a 100% trans- propagation directions, which yields a perfect agreement.
mission). However, for a wave whose electric field is Since the limitation of total internal reflection for a
polarized in the incidence plane x-z (normally referred
to as an extraordinary wave), the effects of the twin
boundary are in fact very similar to the results obtained
above for the electron beam. The dispersion relation can
be obtained by solving Maxwell's equation for plane
waves propagating within the x-z plane:
kzcos 0 signkxsin 0 2 k !2
zsin 0ĸsignkxcos 0 2
"? "jj c2 ;
(7)
where sign 1 for the A side, and ĸ1 for the B side.We
have assumed the medium is nonmagnetic (i.e., the rela-
tive permeability 1). Again, for each side there is one
solution for kz that can have a positive z component of the
group velocity. The electric (E) and magnetic (H) waves
in regions A and B can be written for both sides in a
similar manner as for Eqs. (3) and (4), and on applying
the boundary conditions, we arrive at the two same
equations as those obtained for the electron waves: a
b c due to the continuity of the tangential component
of the magnetic field in the y direction, and a ĸ b c
due to the continuity of the tangential component of the FIG. 3 (color). Images of light propagation in a YVO4 bi-
crystal. The upper panel shows an example of normal (positive)
electric field in the x direction, where a, b, and c are, refraction, the lower panel shows an example of abnormal
respectively, the amplitude of the x component of the E (negative) refraction. Note that no reflection is visible at the
field for the incident, reflected, and transmitted waves. bicrystal interface to the naked eye. The interface is illumi-
Thus, we again find that the amplitude of the reflected nated by inadvertently scattered light.
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We thank S. P. Ahrenkiel for the electron microscopy
image and M. Hanna for sample growth. The work was
supported by the U.S. Department of Energy, Office of
Sciences, Basic Energy Sciences under Contract No. DE-
AC36-99GO10337.
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