Electron mobility in graphene without invoking the Dirac equation Chaitanya K. Ullal, Jian Shi, and Ravishankar Sundararamana) Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180 (Received 1 July 2018; accepted 10 February 2019) The Dirac point and linear band structure in graphene bestow it with remarkable electronic and optical properties, a subject of intense ongoing research. Explanations of high electronic mobility in graphene often invoke the masslessness of electrons based on the effective relativistic Dirac- equation behavior, which are inaccessible to most undergraduate students and are not intuitive for non-physics researchers unfamiliar with relativity. Here, we show how to use only basic concepts from semiconductor theory and the linear band structure of graphene to explain its unusual effective mass and mobility, and compare them with conventional metals and semiconductors. We discuss the more intuitive concept of transverse effective mass, which emerges naturally from these basic derivations, and which approaches zero in the limit of undoped graphene at low temperature and is responsible for its extremely high mobility. VC 2019 American Association of Physics Teachers. https://doi.org/10.1119/1.5092453 I. INTRODUCTION correctly, will likely lead to confusion especially at the intro- ductory undergraduate level. Moreover, it is not an intuitive Graphene is often described in superlatives, with a multitude explanation for students from related fields in chemistry, of extreme electronic, mechanical, and chemical properties of materials science, or electrical engineering, who are all 1–3 interest in disparate fields of research. This increasingly increasingly likely to encounter graphene in their careers. motivates exposure to graphene science at the undergraduate Pedagogical descriptions that try to avoid the relativistic / 4 level, with excellent pedagogical resources introducing the cal- Dirac explanation often rely on alternate definitions of the 5,6 culation of its unique Dirac-point band structure, explaining mass that work correctly for graphene, e.g., cyclotron effective 7 novel transport phenomena such as Klein tunneling, and even mass,13,14 quarternion effective mass,15 etc. While these defini- outlining experimental demonstrations of the unique wave tions work for a reason, as we will discuss below, they do not 8 mechanics of honeycomb lattices in ripple tanks. provide an intuitive picture of how electrons in graphene con- Of graphene’s extreme properties, its exceptional electrical duct remarkably well. Most importantly, some of these alter- conductivity and mobility, arising from the effective massless- nate approaches attempt to redefine the effective mass as the 13 ness of electrons in the Dirac band structure are often dis- ratio of the momentum to velocity, mà p=v,ratherthanthe cussed.9 ¼ Explaining the high mobility from a low effective ratio of force to acceleration, mà F=a @[email protected] mass is easily accessible at an undergraduate level with stan- nate definitions are correct only for¼ a parabolic¼ band structure, 10 dard semiconductor physics derivations of Drude theory. where the mass is independent of the momentum and the two However, explaining why the Dirac band structure corresponds expressions above become equivalent. to massless carriers is somewhat more challenging, and has not Here, we describe an alternate pedagogical approach to yet been discussed clearly in a pedagogical context. explain massless electrons in graphene, where we retain the Specifically, the Dirac band structure contains a linear disper- standard definition of effective mass from semiconductor the- sion relation E vFp,correspondingtoaconstantelectron ory, albeit the full tensorial version. We demonstrate how to ¼ speed v @E=@p vF, independent of the momentum. Here, work through this definition to understand how graphene’s ¼ 5 ¼ vF 8:3 10 m/s is the Fermi velocity of graphene, the effective mass and mobility vary with doping and temperature,  velocity of electrons at the Fermi energy up to which states are and how it contrasts for conventional metals and semiconduc- 11 filled with electrons. Naive application of the conventional tors. We show how to arrive at the concept that the transverse semiconductor definition of effective mass, which corresponds 1 effective mass, rather than the usual longitudinal one, domi- to mà À @v=@p 0 as we discuss below in further detail, nates transport in graphene, and is the mass that approaches ð Þ ¼ ¼ leads to the opposite result of infinite mass! (See Fig. 1.) zero near the Dirac point in graphene. The approach presented 12 Research papers invoke seminal work that demonstrated here should be suitable for intuitively explaining the remark- that electron transport in graphene is essentially governed by able electronic properties of graphene—a topic of continuing the Dirac equation, with the charge carriers mimicking rela- research interest—at the senior undergraduate level. tivistic particles with zero rest mass. In the relativistic pic- ture, linear dispersion corresponds to massless carriers by II. DERIVATIONS recognizing that E2 mc2 2 pc 2 for a particle of mass m reduces to E pc¼ðwhen Þm þð0. However,Þ this is only an Electrons in materials with a band structure or dispersion analogy6 and must¼ be applied very¼ carefully to graphene. For relation E(k) have group velocity v @E=@p and momentum ¼ graphene, the electrons have a constant velocity vF 8.3 p hk, using only basic principles in quantum mechanics. 105 m/s c/400, as discussed above, instead of the speed When¼ an electric field is applied to the material, the electric Âof light c. Importantly, there is no Lorentz invariance for car- force accelerates the electrons and generates a current. For riers in graphene: the frame in which the carbon nuclei are at the same force, lighter electrons will be accelerated more, rest is special! Therefore, explaining masslessness of gra- and will result in higher mobility and conductivity. phene carriers using relativity, though valid when done Consequently, the mass relevant for determining conduction 291 Am. J. Phys. 87 (4), April 2019 http://aapt.org/ajp VC 2019 American Association of Physics Teachers 291 2 vF p p p y À x y ; (5) ¼ p3 p p p2 À x y x ! using ~p hk~. With the definition sin2/ sin / cos / M / À ; (6) ð Þ sin / cos / cos2/ À the inverse effective mass tensor can be written in polar coordinates as 1 vF m à À M / : (7) ðÞ¼ p ðÞ Fig. 1. Schematic band structure of (a) a free-electron metal, (b) a parabolic- As with any symmetric tensor, the inverse mass tensor is band semiconductor, (c) doped graphene and (d) undoped graphene, with best characterized in its principal axes or eigenvectors, so band velocity (E vs p slopes) and effective mass (inverse E vs p curvature) that it becomes diagonal. Solving the characteristic equation annotated. The shading denotes the Fermi occupation factors of electrons. 1 det m à À k1 0 yields the two eigenvalues Naively, the linear band structure yields zero curvature and an infinite effec- ½ð Þ À ¼ tive mass in graphene, rather than zero or a low value. vF k 0; ; (8) ¼ p by electrons is the ratio of force F to acceleration a (exactly &' as in Newton’s second law). Now F dp=dt and a dv=dt, which yield ¼ ¼ and their corresponding eigenvectors can be derived to be 1 p 1 p 1 dv=dt @v @@E=@p x y (9) À ~x ; : mà ðÞ ¼ p py p px ðÞ dp=dt ¼ @p ¼ @p &'À @2E @2E ; (1) The first eigenvector is exactly p^, the unit vector along the @p2 ¼ h2@k2 momentum. This principal direction therefore corresponds to changes in momentum parallel to the momentum direction, the well-known expression in semiconductor theory that the which is a “longitudinal” change. The corresponding inverse effective mass is the inverse of the curvature of the band mass eigenvalue is 0, therefore implying that the longitudinal 10,11 structure E(k). The curvature and effective mass are both mass mL , which is exactly the result we obtained in the finite (non-zero and not infinite) for metals and semiconduc- non-tensorial!1 analysis. tors, as shown in Fig. 1. However, for graphene, the linear However, now we have the second eigenvector which is band structure has seemingly zero curvature corresponding perpendicular to p^, corresponding to changes in momentum to an infinite effective mass, in stark contrast to the massless perpendicular to the momentum direction, which is a carrier explanation for its high mobility. “transverse” change. The corresponding inverse mass eigen- The simplest correct explanation for massless electrons in value is vF=p, corresponding to a transverse mass mT p=vF. graphene lies within the standard definition, but necessitates As we approach the Dirac point p 0, the transverse¼ mass 10,11 ! the full tensorial version, mT 0. Therefore, at the Dirac point, electrons in graphene have! an infinite longitudinal mass, but are massless in the 1 1 transverse direction. m à À ~ ~E k~ : (2) ðÞ h2 rk rk ðÞ This result can also be understood intuitively, directly from the linear dispersion relation, E ~p vF ~p .Thecorresponding The mass tensor is just a matrix that connects how changes velocity ~v E ~p v p^,isalwaysparalleltothemomen-ð Þ¼ j j r~p ð Þ¼ F of momentum and velocity are related, d~p m à d~v, or tum ~p,buthasaconstantmagnitudevF independent of p.As 1 ¼ Á equivalently d~v m à À d~p, which are not in the same shown in Fig. 2(a),whenthemomentumischangedinthelon- direction for a general¼ð ÞE k~Á . For two-dimensional graphene gitudinal direction, the momentum direction p^ is unchanged, so near the Dirac point, the aboveð Þ definition reduces to the velocity direction and magnitude remain unchanged. No change in velocity with changing momentum yields zero 2 inverse mass and an infinite longitudinal mass, m .