P H Y S I C A L R E V I E W L E T T E R S week ending VOLUME 91, NUMBER 15 10 OCTOBER 2003 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 ÿ24 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. 157404-1 0031-9007=03=91(15)=157404(4)$20.00 2003 The American Physical Society 157404-1 P H Y S I C A L R E V I E W L E T T E R S week ending VOLUME 91, NUMBER 15 10 OCTOBER 2003 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 ÿ 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 mÿ1 1 X3 ? 0 0 p mÿ1 @ 0 mÿ1 A J Re F F ; (5) ? 0 ; (1) m 0 0 mÿ1 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 ÿ k2 2 k ; (2) x 2m z ÿ k2y ÿ 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? ÿ 1=mjj =3, sign 1 for the A side, and ÿ1 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 ÿkAz1 and kBz2 ÿkBz1 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. 157404-2 157404-2 P H Y S I C A L R E V I E W L E T T E R S week ending VOLUME 91, NUMBER 15 10 OCTOBER 2003 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 "jjÿ 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 ÿ 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 ÿ45 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 10ÿ4, 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 0ÿsignkxcos 0 2 "? "jj c2 ; (7) where sign 1 for the A side, and ÿ1 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 ÿ 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. 157404-3 157404-3 P H Y S I C A L R E V I E W L E T T E R S week ending VOLUME 91, NUMBER 15 10 OCTOBER 2003 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. [1] M. Born and E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1999). [2] V.V. Paranjape, Phys. Rev. B 52, 10 740 (1995). [3] G.T. Einevoll and L. J. Sham, Phys. Rev. B 49, 10 533 (1994). [4] J. Smoliner, R. Heer, and G. Strasser, Phys. Rev. B 60, R5137 (1999). [5] V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968). [6] D. R. Smith et al., Phys. Rev. Lett. 84, 4184 (2000). FIG. 4 (color online). Comparison of theoretical predictions [7] R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 with experimental data. Amphoteric refraction in a YVO4 (2001). bicrystal is divided into three regions: one negative ( B= A < [8] J. B. Pendry, Phys. Rev. Lett. 85, 3966 (2000). 0) and two positive ( B= A > 0). The data points are measured [9] N. Garcia and M. Nieto-Vesperinas, Phys. Rev. Lett. 88, with a 532 nm laser light, the curve is calculated with the 207403 (2002). refractive index of the material (given in the text). Inset: the [10] C. Luo et al., Phys. Rev. B 65, 201104 (2002). full operation range of the device. [11] P. M. Valanju, R. M. Walser, and A. P. Valanju, Phys. Rev. Lett. 88, 187401 (2002). [12] J. Pacheco et al., Phys. Rev. Lett. 89, 257401 (2002). conventional interface does not apply for the interface [13] D. R. Smith and D. Schurig, Phys. Rev. Lett. 90, 077405 considered, a full operation range of ÿ90 to 90 can be (2003). [14] J. Li et al., Phys. Rev. Lett. 90, 083901 (2003). obtained, as shown in the inset of Fig. 4. Note that the [15] I.V. Lindell et al., Microwave Opt. Technol. Lett. 31, 129 experiment performed here using the bicrystal also dem- (2001). onstrates the feasibility of obtaining similar effects for [16] R. A. Shelby et al., Appl. Phys. Lett. 78, 489 (2001). ballistic electrons. [17] N. Garcia and M. Nieto-Vesperinas, Opt. Lett. 27, 885 Many potential device applications can be derived (2002). based on the unique properties of the kind of domain [18] S. P. Ahrenkiel, in Spontaneous Ordering in Semi- boundary discussed above. They can, for example, be conductor Alloys, edited by A. Mascarenhas (Kluwer used to provide bending, angular dispersion, energy filter- Academic/Plenum Publishers, New York, 2002), p. 195. ing, and beam collimating for electrons in semiconductor [19] K. Aizu, J. Phys. Soc. Jpn. 27, 387 (1969). ballistic electron devices. The ability to steer light with- [20] Y. Zhang et al., Mater. Res. Soc. Symp. Proc. 583, 255 out reflection could be extremely valuable for high power (2000). [21] R. Wessel and M. Altarelli, Phys. Rev. B 39, 12 802 optics. Additionally, this relatively simple way to generate (1989). negative refraction may provide unique experimental op- [22] Y. Zhang, A. Mascarenhas, and L.W. Wang, Phys. Rev. B portunities for examining this unusual effect and its 63, R201312 (2001). various physical consequences. [23] http://www.casix.com 157404-4 157404-4