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 . 1 1 @k @kx @ky L À x 2 2 !1 m à 2 2 hvF kx ky : (3) However, when momentum is changed in the transverse direc- ðÞ @k @k @ ¼ h x y ky !þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tion by d~pT,themomentumandvelocitydirectionsboth change by the angle dh dp =p.Sincethevelocitymagnitude Straightforward evaluation of the derivatives yields T is unchanged, the velocity¼ vector changes by dv v dh ¼ F vFdpT=p.Therefore,theratioofvelocitytomomentum 2 ¼ ky kxky change is vF=p which corresponds to the transverse mass, m p=v .Thiscanalsobeseencomparingradial(longitudi- 2 2 3=2 À 2 2 3=2 T F 1 vF 0 kx ky kx ky 1 nal)¼ and transverse slices of the conical E k~ near the Dirac m à À þ þ (4) ðÞ¼ h 2 point (Fig. 2(b)). The linear E(k)alongthelongitudinalsliceð Þ B ÀÁkxky ÀÁkx C B C yields B À 2 2 3=2 2 2 3=2 C mL ,whiletheparabolicE(k)inthetransverseslice B kx ky kx ky C !1 B þ þ C yields finite mT.Thetransversecurvatureincreasesasthe @ ÀÁÀÁA 292 Am. J. Phys., Vol. 87, No. 4, April 2019 Ullal, Shi, and Sundararaman 292 Fig. 2. (a) Velocity (thick red arrows) is always parallel to momentum (thin black arrows) with constant magnitude vF. Therefore, velocity is unchanged for longitudinal changes in momentum d~pL yielding infinite longitudinal mass mL. Velocity changes direction for transverse changes d~pT , resulting in a small transverse mass mT which 0 as p 0. (b) Linear E(k) in the radial (longitudinal) slice of the conical! E k~ yields! m , while para- ð Þ L !1 bolic E(k) in the transverse slice yields a small mT that approaches zero as the slice gets closer to the Dirac point. slice gets closer to the Dirac point, resulting in mT 0near the Dirac point. ! The transverse mass is closely related to the cyclotron mass, which is the reason why the latter definition works for graphene.13 In a magnetic field, charged particles move in circles with centripetal force and acceleration perpendicular to the velocity. The cyclotron mass is the ratio of force to acceleration when they are both perpendicular to the momen- Fig. 3. (a) Fermi function and (b) its derivative for undoped and doped gra- tum (velocity) direction, which is exactly the case for the phene. Shaded regions indicate the contribution to conduction, proportional to the Fermi function derivative. (c) Transverse effective mass in graphene transverse mass as discussed above. as a function of electron energy, which is distributed around a non-zero When an electric field E~ is applied to a material, this value for doped graphene, but around zero for undoped graphene. applies a force eE~ on all the electrons, resulting in an accel- À 1 eration ~a e m à À E~. As the electrons move through the ¼À ð Þ Á analysis based on the Boltzmann transport equation and the material, they scatter against defects and lattice vibrations relaxation time approximation11 shows that the contribution (phonons), which cause the velocity to randomize due to colli- to conduction is proportional to the derivative of the Fermi sions over the Drude relaxation time scale s. With these two function (Fig. 3(b)). If the number density of electrons due to effects, the electrons pick up an average drift velocity doping is high enough that , then all the electrons 1 ~ EF kBT ~vd ~as es m à À E. The ratio of drift velocity to the that contribute to conduction have approximately the same applied¼ electric¼À ð fieldÞ definesÁ the mobility (excluding the sign magnitude of momentum, pF. Correspondingly, they all have due to negative charge) transverse mass m p =v E =v2 (Fig. 3(c)), while the T F F ¼ F F 1 longitudinal mass remains (as shown above for all gra- l es m à À : (10) 1 ¼ ð Þ phene electrons). When the effective mass is anisotropic, the well-known The total current density in the electron is ~j n e ~vd ¼ ðÀ Þ simplified expression for mobility l es=mà remains valid nel E~, where n is the number density of electrons. The ¼ provided an appropriate average of mà in all directions is conductivity¼ Á (tensor) is defined by ~j r E~, which implies used. In particular, mà should be the harmonic mean of all ne, elucidating the mobility to¼ be theÁ conductivity per r l directions, i.e., if the contributions are m , m , and m in unit¼ charge density. 1 2 3 ~ three perpendicular directions (principal axes), the net effec- In general, the effective mass mà varies with k (i.e., ~p), 1 1 1 1 tive mass should be m À m m m =3. For and hence so does the mobility. The experimentally deter- à 1À 2À 3À example, in Silicon, ðm Þ 0:¼ð89 andþ thereþ are twoÞ equal mined mobility is therefore an average over all charge car- L transverse values (in 3D)¼ m m 0:19. The corre- riers. First, consider the case of n-doped graphene, where a T1 T2 sponding average value for mobility¼ will¼ then be m 0:26. net excess of electrons over holes results in states being à For graphene, we now have m and m p =¼v (just occupied up to an energy E above the Dirac point. (The dis- L T F F F one in 2D), which yields ¼1 ¼ cussion for excess holes with EF below the Dirac point fol- lows in exactly the same way, with exactly the same results, 2 2pF 2EF due to the electron hole symmetry in the band structure.) mà ; (11) $ ¼ m 1 m 1 ¼ v ¼ v2 From the linear energy relation E vFp, we can see that this LÀ þ TÀ F F corresponds to a Fermi momentum¼ p E =v and wave- F ¼ F F vector kF EF= hvF (see Fig. 1(c)). which is twice the transverse value. Correspondingly, we ¼ ð Þ 2 Only electrons within a few kBT of the Fermi energy con- expect the mobility to be es=mà esvF= 2EF . tribute to electronic conduction in materials. Intuitively, only We can alternatively derive¼ this resultð byÞ averaging the these electrons have empty states available to “move” to, mobility contributions due to all electrons contributing to being close to the energy at which electronic states transition conduction on this “Fermi circle” of radius pF. This amounts from filled to empty (Fig. 3(a)). In fact, a more detailed to an average over /
293 Am. J. Phys., Vol. 87, No. 4, April 2019 Ullal, Shi, and Sundararaman 293 1 where we switched the integrals over momenta to polar coor- d/ m à À ~p l es ðÞðÞF dinates and substituted x v p= 2k T to simplify the inte- d F B Ð / gral over p. The integrals¼ overð x areÞ standard definite vF (12) d/ Ð M / integrals that evaluate to the constants 1 and ln 2 in the p ðÞ es F numerator and denominator, respectively, while the final ¼ Ð 2p term is exactly what we evaluated above to be 1=2. Putting that all together yields esv2 1 2p F d/M / 2 2 ¼ EF Á 2p 0 ðÞ esv 1 1 esv ð (13) l F F 1: (17) esv2 ¼ 2k T Á ln 2 Á 2 ¼ 4 ln 2 k T F 1; B ðÞB ¼ 2EF Note that the average mobility is isotropic (scalar) as by substituting Eqs. (7) and (6),andnotingthattheangularinte- expected and corresponds to an averaged effective mass 2p 2 2p 2 2p grals 0 d/ cos / 0 d/ sin / p and 0 d/ cos / sin / ¼ ¼ 4 ln 2 kBT 0. This also corresponds to an isotropic mobility es=mÃ,with mà ðÞ; (18) ¼ Ð Ð 2 Ð ¼ v2 the effective mà 2pF=vF 2EF=vF as argued above. F In pure (undoped)¼ graphene,¼ the electronic states switch from being occupied to unoccupied at the Dirac point (Figs. which is directly proportional to temperature. This is because the transverse mass m p, and the average magnitude of 1(d) and 3(a).) The contribution to conduction, proportional T / to the Fermi function derivative (Fig. 3(b)) as discussed momentum for electrons in graphene at finite temperature above, is centered near E 0. Unlike the doped case, the cor- T. This is in sharp contrast to conventional metals and ¼ semiconductors,/ and even doped graphene with E k T as responding effective mass is no longer of similar magnitude F B throughout the energy range with contributions to conduc- considered above, where the effective mass depends only tion, as shown in Fig. 3(c). In particular, the effective mass weakly on temperature. in the center of the distribution at the Dirac point is zero, while it is non-zero and linearly increasing away from the III. RESULTS AND DISCUSSION Dirac point. Therefore, we need to average over the carriers Table I compares the typical effective masses m , momen- proportional to the Fermi function derivative to estimate the tum relaxation time , and mobility of electronsà in a proto- mobility for undoped graphene. s l typical metal (silver), semiconductor (silicon), and both For undoped graphene with the Fermi energy at the Dirac doped and undoped graphene. The values for silver and sili- point, E 0, the occupation of electrons is given by the F con are based on experimental measurements, while that for Fermi function¼ graphene is based on the above derivation along with a first- 16 1 principles calculated value of s 2 ps for ideal undoped f E ; (14) graphene and 700 fs for graphene (ideally) doped to a ð Þ¼ E s 1 exp Fermi energy of 0.1 eV (limited only by electron-phonon þ kBT scattering). with derivative Metals have a short relaxation time because they have a large number of states at the Fermi level, which enhances E electron-phonon scattering. Semiconductors and graphene exp kBT 1 2 E have much smaller density of states at the energies of elec- f 0 E 2 sech : ð Þ¼ E ¼ 4kBT 2kBT trons that carry current, resulting in an increased relaxation k T 1 exp time by one and two orders of magnitude relative to the B þ k T B metal. The typical effective mass is somewhat smaller in (15) semiconductors than metals, but it is two orders of magni- tude smaller at room temperature in graphene because the We can therefore determine the average mobility of undoped transverse mass approaches zero near the Dirac point. graphene as Consequently, the mobility s=mà is smallest for metals. Semiconductor mobilities are/ one-to-two orders larger due 1 dpxdpyf 0 vFp m à À ~p both to larger s and somewhat smaller mÃ. However, in gra- ðÞðÞðÞ l es ð phene both factors contribute two orders making the mobility at room temperature four orders larger! dpxdpyf 0 vFp ðÞ ð 2p 1 vFp vF Table I. Comparison of typical relaxation time, effective mass, and electron pdp d/ sech2 M / 2k T p ðÞ mobility for metals, semiconductors, doped graphene and (undoped) gra- ð0 ð0 B phene at room temperature. es 2p ¼ 1 2 vFp pdp d/ sech 2 2k T Material s (fs) mÃ=me l (cm /V s) ð0 ð0 B Á 2p 1 Silver 30 1.0 50 dx sech2x d/M / esv2 ðÞ Silicon 200 0.26 1400 F 0 0 4 ð ð ; (16) Graphene with EF 0:1 eV 700 0.063 2 10 ¼ 2k T Á 1 Á 2p ¼  B dxxsech2x Undoped graphene 2000 0.018 2 105  ð0
294 Am. J. Phys., Vol. 87, No. 4, April 2019 Ullal, Shi, and Sundararaman 294 Note that despite the much higher mobilities in semicon- is critical for understanding the unusual electron transport in ductors and graphene, the number density n of electrons in graphene. metals is sufficiently larger that the conductivity r nel is ¼ still much larger in metals. Specifically, in graphene, the ACKNOWLEDGMENTS mobility is higher for undoped graphene due to the lower effective mass (and additionally because of a lowered C.K.U. acknowledges support from the National Science electron-phonon scattering rate16) than the doped case. Foundation under Grant No. EEC-1446038. J.S. acknowledges However, mobility is effectively the conductivity per carrier support from the National Science Foundation under Grant available for conduction, and the number of carriers is much Nos. CMMI-1635520 and CBET-1706815. J.S. and R.S. smaller for undoped graphene. Consequently, undoped gra- acknowledge start-up funding from the Materials Science and phene has a low conductivity despite the highest mobility, Engineering department at Rensselaer Polytechnic Institute. and graphene actually achieves a higher conductivity at an optimal doping level where the increasing effective mass a)Electronic mail: [email protected] and scattering rate are compensated by an increased carrier 1A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, density.1,2 183–191 (2007). 2 As temperature changes, the scattering time s is roughly A. Castro-Neto, F. Guinea, and N. M. Peres, “Drawing conclusions from inversely proportional to temperature near room temperature graphene,” Phys. World 19, 33–37 (2006). 3A. K. Geim and P. Kim, “Carbon wonderland,” Sci. Am. 298, 90–97 for pure materials, because the amplitude of lattice vibrations (2008). increases with temperature. The effective mass is mostly 4D. Cela, M. Dresselhaus, T. H. Zeng, M. Terrones, A. G. Souza Filho, and temperature dependent in metals and semiconductors, so that O. P. Ferreira, “Resource letter n-1: Nanotechnology,” Am. J. Phys. 82, the temperature dependence of mobility follows the scatter- 8–22 (2014). ing time. However for undoped graphene, the effective mass 5R. L. Pavelich and F. Marsiglio, “Calculation of 2d electronic band struc- is also temperature dependent causing an additional decrease ture using matrix mechanics,” Am. J. Phys. 84, 924–935 (2016). 6A. Matulis and F. M. Peeters, “Analogy between one-dimensional chain of mobility with increasing temperature and resulting in an 2 models and graphene,” Am. J. Phys. 77, 595–601 (2009). overall TÀ dependence near room temperature. The temper- 7T. R. Robinson, “On Klein tunneling in graphene,” Am. J. Phys. 80, ature dependence for doped graphene will be similar to con- 141–147 (2012). ventional metals and semiconductors because the average 8J. R€ossler, C. R€ossler, P. M€aki, and K. Ensslinmore, “Wave physics of the momentum and hence the average transverse mass is set by graphene lattice emulated in a ripple tank,” Am. J. Phys. 83, 761–764 the doping level and not by temperature, as derived above. (2015). 9M. Wilson, “Electrons in atomically thin carbon sheets behave like mass- less particles,” Phys. Today 59, 21–23 (2006). IV. CONCLUSIONS 10S. M. Sze, Physics of Semiconductor Devices (John Wiley & Sons, Hoboken, NJ, 1981), see discussion of Eq. (9) in Chap. 1. We have presented a simplified approach to explain the 11N. W. Ashcroft and N. D. Mermin, Solid State Physics (Cengage remarkable mobility of electrons in graphene that relies only Learning, Boston, MA, 1976), see Chap. 12 for discussion on conduction on the standard semiconductor theory definition, completely near Fermi surface. avoiding the conventionally invoked parallel to the Dirac 12K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, equation and corresponding relativistic explanation. We dis- I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless dirac fermions in graphene,” Nature 438, 197–200 (2005). cussed the calculation of the tensorial effective mass, the 13 V. Ariel and A. Natan, “Electron effective mass in graphene,” in 2013 emergence of a zero transverse mass (but infinite longitudi- International Conference on Electromagnetics in Advanced Applications nal mass) upon approaching the Dirac point and the corre- (ICEAA) (2013), p. 696. sponding temperature-dependent mobilities. The full 14P. Hoffman, Solid State Physics: An Introduction (Wiley, Weinheim, derivations require only basic concepts from calculus, ther- Germany, 2015), also see accompanying online note at
295 Am. J. Phys., Vol. 87, No. 4, April 2019 Ullal, Shi, and Sundararaman 295