In this paper, we study the Schottky transport in narrow-gap semiconductor and few-layer graphene in which the energy dispersions are highly non-parabolic. We propose that the contrasting current-temperature scaling relation of J ∝ T 2 in the conventional Schottky interface and J ∝ T 3 in graphene-based Schottky interface can be reconciled under Kane's k · p non-parabolic band model for narrow-gap semiconductor. Our new model suggests a more general form of J ∝ T 2 + γkBT 3 , where the non-parabolicty parameter, γ, provides a smooth transition from T 2 to T 3 scaling. For few-layer graphene, it is found that N -layers graphene with ABC-stacking follows J ∝ T 2/N+1 while ABA-stacking follows a universal form of J ∝ T 3 regardless of the number of layers. Intriguingly, the Richardson constant extracted from the Arrhenius plot using an incorrect scaling relation disagrees with the actual value by two orders of magnitude, suggesting that correct models must be used in order to extract important properties for many novel Schottky devices.
INTRODUCTIONTranslating the unusual physical properties of novel nanomaterial-based heterostructures into functional device applications has become one of the major research goals in recent years [1]. One important heterostructure is the metal/semiconductor interface, commonly known as the Schottky interface [2], where novel applications such as broadband ultrasensitive photodetector [3], gatetunable Schottky barrier [4], promising solar cell performance [5] and ultrafast phototransistor [6] have recently been demonstrated. The current transport across a Schottky interface is mainly due to majority carriers. In general, there are three different transport mechanisms, namely diffusion of carriers from the semiconductor into the metal, thermionic emission of carriers across the Schottky barrier and quantum-mechanical tunneling through the barrier [7]. For the thermionic emission, the Schottky diode equation is written as [8] J =J e eV ηk B T − 1 ,whereJ is the reverse saturation current density determined by the thermionic emission process, V is the bias voltage and η is an ideality factor. For bulk materials with parabalic energy disperson (E k ∝ k 2 ), the reversed saturation current densityJ takes the well-known Richardson form of [9,11] where Φ denotes the magnitude of the Schottky barrier's height. The exponential term, e −Φ/kB T , in Eq. (??) originates from the classical Boltzmann statistics and is universal regardless of the form of the transport electron energy dispersion while theJ ∝ T 2 current-temperature scaling relation is a signature of the parabolic energy dispersion of the transport electrons.For novel materials with non-parabolic energy dispersion [see Figs. 1(a)-(d) for examples of non-parabolic energy dispersions], the validity ofJ ∝ T 2 should be verified. Although it is well-known that the energy dispersion plays an important role in governing the Schottky transport, the traditionalJ ∝ T 2 model is still widely used in the vast majority of recent experimental wor...