arXiv:submit/0473721 [cond-mat.other] 14 May 2012 Effective Mass and Energy-Mass Relationship * Viktor Ariel Department of Physical Electronics Tel-Aviv University The particle effective mass is often a challenging concept in solid state physics due to the many different definitions of the effective mass that are routinely used. Also, the most commonly used theoretical definition of the effective mass was derived from the assumption of a parabolic energy- momentum realtionship, E(p), and therefore should not be applied to non-parabolic materials. In this paper, we use wave-particle duality to derive a definition of the effective mass and the energy-mass approximation suitable for non-parabolic materials. The new energy-mass relationship can be considered a generalization of Einstein’s E = mc 2 suitable for arbitrary E(p) and therefore applicable to solid state materials and devices. We show that the resulting definition of the effec- tive mass seems suitable for non-paraboic solid state materials such as HgCdTe, GaAs, and graphene. I. INTRODUCTION The concept of mass has a long history in theoreti- cal physics and philosophy [1] and continues to be hotly debated until today [2]. In particular, in solid state ma- terials many different effective mass definitions are used such as the conductivity, density-of-states, and cyclotron effective mass [3], [4]. The most commonly used theoret- ical definition of the effective mass is m =(∂ 2 E/∂p 2 ) -1 , where p = k and k is the crystal momentum [3], [4]. We show that this definition is only limited to parabolic E(p) relationships while an alternative definition should be used for non-parabolic materials [5], [6], [7]. In this work, we use wave-particle duality by associ- ating a particle with a one-dimensional wave-packet [8]. First, we derive a definition of the effective mass, m, and an approximation for the energy-mass relationship, E(m). Then, we show that the new energy-mass rela- tionship can be considered a generalization of Einstein’s E = mc 2 suitable for any dispersion E(p) and there- fore applicable to solid state materials and devices. Fi- nally, we apply the definition of the effective mass to non- parabolic solid state materials such as HgCdTe, GaAs, and graphene. II. EFFECTIVE MASS DEFINITION AND ENERGY-MASS APPROXIMATION We begin with wave-particle duality by associating a particle with a wave-packet, and the particle velocity with the group velocity of the wave-packet, v g . This approach is commonly used in solid state physics for cal- culations of the energy-band structure and charge trans- port properties [3], [4]. Then based on the semi-classical definition of the particle momentum, the effective mass * Thanks to: Prof. Shlomo Ruschin, Dr. Amir Natan of Tel Aviv University, and Dr. Leon Altschul of SDL Ltd. for reviewing this work. appears as a proportionality factor between the particle momentum and the group velocity of the wave-packet p = mv g . (1) Consider a system of particles described by an isotropic energy-momentum relationship E(p). Assume that the this E(p) leads to a solution of the wave equation which can be represented by one-dimensional wave-packets. The group velocity is defined as the velocity of the wave- packet maximum and is usually approximated for one- dimensional packets as [8] v g ≃ ∂E ∂p . (2) Similarly, we use the traditional one-dimensional approx- imations for the phase velocity, v p , defined as the velocity of a single phase in the vicinity of the wave-packet max- imum and expressed as [8] v p ≃ E p . (3) By using definitions of momentum Eq. (1) and group velocity Eq. (2), we obtain the definition of the effective mass, which depends on the particle energy and momen- tum m ≡ p ∂E/∂p . (4) This result was previously demonstrated and applied to non-parabolic semiconductors [5], [7]. From Eq. (1) and Eq. (3), we can derive a relationship between total partical energy, E, and particle effective mass E ≃ mv g v p . (5) Note from Eq. (5) that the energy-mass approximation is a function of both group and phase velocities of the corresponding wave packet. This seems like a remark- able relationship between wave and particle properties of matter within the limits of the present assumptions.