Transformation Electronics: Tailoring the Effective Mass of Electrons

The speed of integrated circuits is ultimately limited by the mobility of electrons or holes, which depend on the effective mass in a semiconductor. Here, building on an analogy with electromagnetic metamaterials and transformation optics, we describe a transport regime in a semiconductor superlattice characterized by extreme anisotropy of the effective mass and a low intrinsic resistance to movement—with zero effective mass—along some preferred direction of electron motion. We theoretically demonstrate that such a regime may permit an ultrafast, extremely strong electron response, and significantly high conductivity, which, notably, may be weakly dependent on the temperature at low temperatures. These ideas may pave the way for faster electronic devices and detectors and functional materials with a strong electrical response in the infrared regime. Disciplines Electrical and Computer Engineering Comments Silveirinha, M. & Engheta, M. (2012). Transformation electronics: Tailoring the effective mass of electrons. Physical Review B, 86(16), 161104. doi: 10.1103/PhysRevB.86.161104 ©2012 American Physical Society This journal article is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/623 RAPID COMMUNICATIONS PHYSICAL REVIEW B 86, 161104(R) (2012) Transformation electronics: Tailoring the effective mass of electrons Mário G. Silveirinha1,2,* and Nader Engheta1,† 1Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 2Department of Electrical Engineering, Instituto de Telecomunicações, University of Coimbra, Coimbra, Portugal (Received 20 December 2011; published 8 October 2012) The speed of integrated circuits is ultimately limited by the mobility of electrons or holes, which depend on the effective mass in a semiconductor. Here, building on an analogy with electromagnetic metamaterials and transformation optics, we describe a transport regime in a semiconductor superlattice characterized by extreme anisotropy of the effective mass and a low intrinsic resistance to movement—with zero effective mass—along some preferred direction of electron motion. We theoretically demonstrate that such a regime may permit an ultrafast, extremely strong electron response, and significantly high conductivity, which, notably, may be weakly dependent on the temperature at low temperatures. These ideas may pave the way for faster electronic devices and detectors and functional materials with a strong electrical response in the infrared regime. DOI: 10.1103/PhysRevB.86.161104 PACS number(s): 73.23.−b, 42.70.Qs, 73.21.Cd, 73.22.−f In 1969, Esaki and Tsu suggested that by either periodically doping a monocrystalline semiconductor or by varying the composition of the alloy, quantum mechanical effects should be observed in a new physical scale,1 so that the conduction and valence bands of such superlattices are structured in the form of many subbands,1,2 and in particular they predicted the possibility of a negative differential conductance.1 This pioneering work has set the stage for dispersion engineering in semiconductor superlattices. This conceptual breakthrough and other prior key proposals (e.g., the idea of quasielectric fields3) are the foundation of many interesting advances in semiconductor technology,4 and has enabled, among others, the development of the quantum cascade laser,5 and the realization of ultrahigh mobilities in semiconductor superlattices and quantum wells.6,7 Following these advancements, more recently, there has been much activity in the study of a new class of mesoscopic materials—metamaterials—whose electromagnetic properties are determined mainly by the geometry and material of its constituents, rather than from the chemical composition.8,9 Such a line of research has resulted in the development of double negative materials, which promise erasing diffraction effects and perfect lensing.8 Until now, the obvious analogy between superlattices and electromagnetic metamaterials received little attention, apart from isolated studies.10,11 Here, inspired by the paradigm offered by electromagnetic metamaterials and transformation optics,8,9 we develop the paradigm of “transformation electronics,” wherein the electron wave packets are constrained to move along desired paths, and predict a transport regime in a semiconductor superlattice based on the extreme anisotropy of the effective mass. In a semiconductor the effectivemass determines the inertia of the electron to an external stimulus. The finite value of the mobility ultimately limits the speed of integrated circuits and other devices. In most electronic circuits the electron flow is supposed to occur along a predetermined path, e.g., down the passageway connecting two transistors. However, typically only a small portion of the available free carriers responds effectively to an external electric field, i.e., those whose velocity vg = h̄−1∇kE is parallel to the impressed field. Would it however be possible to engineer the electron mass in such a way that all the available electronic states contribute to the electron flow? Moreover, would it however be possible to reverse or “cancel” the effects of the intrinsic electron resistance to movement, along the preferred direction of motion? A superlattice with the properties implicit in the first question must be anisotropic. Indeed, in order that vg = h̄−1∇kE is parallel to the desired direction of flow (let us say z), it is necessary that the energy dispersion E = E(k) depends exclusively on the wave vector component kz, and hence the effective mass tensor satisfiesmxx = myy = h̄(∂2E/∂k2 y) = ∞, i.e., the resistance to a flow in the x-y plane must be extremely large. To satisfy the requirements implicit in the second question it is necessary that mzz = h̄(∂2E/∂k2 z )−1 be near zero. Thus ideally we should havemxx = myy = ∞ andmzz = 0, and thus an effective mass tensor characterized by extreme anisotropy. Notably, heterostructures with extreme anisotropy have received some attention in recent years due to their potentials in collimating both light12 and electrons.13 However, our findings are fundamentally different from previous studies: We deal with a bulk semiconductor superlattice, and show how by combining two different semiconductors it may be possible to supercollimate the electron flow (mxx = myy = ∞) and in addition to have a weak resistance to movement (mzz = 0). A zero mass has been previously predicted to occur at contacts between semiconductors with normal and inverted band structures,14 but not an extreme anisotropy regime. To achieve this, we draw on an analogy with electromagnetic metamaterials. The intriguing tunneling phenomena observed in electromagnetic metamaterials are rooted in the fact that twomaterials such that ε1 = −ε2 andμ1 = −μ2, with ε being the permittivity and μ the permeability, “electromagnetically annihilate” one another.8,15 It is thus natural towonder if in electronics it may be possible to identify complementary materials that when paired yield m∗ ≈ 0. Since the effective mass of the carriers is expected to be determined by some averaging of the values of m∗ in the superlattice constituents, this suggests that one should look for materials wherein m∗ has different signs. Even though unusual, the carriers can have a negative effective mass, notably in semiconductors and alloys with a negative energy band gap.16 Examples of such materials 161104-1 1098-0121/2012/86(16)/161104(5) ©2012 American Physical Society RAPID COMMUNICATIONS MÁRIO G. SILVEIRINHA AND NADER ENGHETA PHYSICAL REVIEW B 86, 161104(R) (2012) are mercury telluride (HgTe) (a group II-VI degenerate semiconductor) and some alloys ofmercury cadmium telluride (HgCdTe), which have an inverted band structure,16,17 so that the 8 (P -type) valence bands lie above the conduction band 6 (S type), and the effective masses of both electrons and holes (mc,h) are negative. In Refs. 18 and 19 we develop a formal analogy between the Helmholtz equation for the electromagnetic field and a Schrödinger-type equation for the envelope wave function consistent with the standard Kane model (k·p method) for semiconductors with a zinc-blende structrure.20 Within this formalism, which is consistent with Bastard’s theory,21 the electron is described by a single component wave function ψ , which may be regarded as the spatially averaged microscopic wave function. This contrasts with the conventional k·p approach where the electron is described by a multicomponent wave function.20 For the case of Bloch waves, ψ may be identified with the zeroth order Fourier harmonic of the microscopic wave function.18,19 Related averaging procedures have been considered previously in the context of electromagnetic metamaterials.22 The wave function in the superlattice satisfies