Equations of the Model

Let us consider a multiple-GL structure with the side Ohmic contacts to all GLs and two split gates (isolated from GLs) on the top of this structure as shown in Fig. 3.1a. Applying the positive (7_n = V_g > 0) or negative (U_p = -V_g < 0) voltage between the gate and the adjacent contact (gate voltage), one can obtain the electrically-induced n or p region. In the single- and multiple-GL structures with two split top gates under the voltages of different polarity, one can create lateral n-p or n-i-p junctions. Generally, the source-drain voltage V can be applied between the side Ohmic contacts to GLs. Depending on the polarity of this voltage, the n-p and n-i-p junctions can be either direct or reverse biased. We assume that the potentials of the first

(source) contact and the pertinent gate are tp_s = 0 and g = V_g > 0, respectively, and the potentials of another gate and contact (drain) are cp_g = -V_g < 0 and d = V = 0 (or d = V 0. If the slot between the gates 2Lg is sufficiently wide (markedly exceeds the thickness of the gate layer W_% separating the gates and the topmost GL), there are intrinsic / regions in each GL under the slot. Thus the n-i-p junction is formed. The pertinent band diagrams are shown in Figs. 3.1b and 3.1c. Since the side contacts are the Ohmic contacts, the electron Fermi energy in the k-th GL sufficiently far from the contacts are given by fJ_k = tetfa. Here e is the electron charge and % = 2 = kd >^s the potential of the k-th GL, к = 1, 2,..., К, where К is the number of GLs in the structure, d is the spacing between GLs, and the axis z is directed perpendicular to the GL plane with z = 0 corresponding to the topmost GL and z = z_K = Kd to the lowest one.

Figure 3.1 Schematic view of (a) a multiple-GL structure and its band diagrams at (b) zero bias voltage (V = 0) and (c) at reverse bias (for GL with к = 1). Arrows indicate the directions of propagation of injected electrons and holes as well as those thermogenerated and generated due to interband tunneling.

Focusing on the n region (the p region can be considered in a similar way) and introducing the dimensionless potential р= 2 <р/ V_g, one can arrive at the following one-dimensional Poisson equation governing the potential distribution in the z direction (in the n region):

Here ae is the dielectric constant, 8(z] and S^ and S^ are the equilibrium sheet densities in the k-th GL of electrons and holes, respectively. These densities, taking into account the linear dispersion low for electrons and holes in graphene, are expressed via the electron Fermi energy as

where St = (тг/6)(/с_вТ/Лv_F)² is the electron anwd hole density in the intrinsic graphene at the temperature T, v_F ^ 10^s cm/s is the characteristic velocity of electrons and holes in graphene, and Ti and k_B are the Planck and Boltzmann constants, respectively. Here it is assumed that the electron (hole) energy spectrum is e = v_¥p, where p is the absolute value of the electron momentum. The boundary conditions are assumed to be as follows:

Equations (3.1)-(3.3) yield for 1,

for К =2, and

for Л"> 2. Here

where Г = [8n/se){eW_gI^/V_g) °= T²/V_g and U_g = eV_g/2k_BT.

Numerical Results

Equations (3.4)-(3.7) were solved numerically. The results of the calculations are shown in Figs. 3.2-3.5. In these calculations, we assumed that ae = 4, d = 0.35 nm, and W_g = 10 nm.

Figure 3.2 shows the dependences of the electron Fermi energy

as a function of the GL index к calculated for multiple-GL structures with different number of GLs К at different gate voltages and temperatures. One can see that the Fermi energy steeply decreases with increasing GL index. However, in GLs with not too large k, the Fermi energy is larger or about of the thermal energy. As one might expect, the electron Fermi energies in all GLs at T = 77 К are somewhat larger than at T = 300 К (see also Fig. 3.5). The obtained values of the electron Fermi energies in topmost GLs are = 92 meV and ^ 77 meV for V_g = 1000 mV at T = 77 К and T = 300 K, respectively.

Figure 3.3 shows the voltage dependences of the electron Fermi energies in some GLs in multiple-GL structure with К = 2, К = 10, and /О 50 at Г=300 К.

Figure 3.2 The electron Fermi energy as a function of the GL index к calculated for multiple-GL structures with different number of GLs К for different gate voltages V_gatT= 300 К (upper panel) and T = 77 К (lower panel).

Figure 3.4 shows the electron densities in the structures with different number of GLs К at different temperatures. One can see that the calculated electron densities in GLs with sufficiently large indices (k > 15 at T = 77 К and T = 300 K) approach to their values in the intrinsic graphene (Sj = 0.59 x Ю¹⁰ cm'² and 8.97 x Ю¹⁰ cm"²). The electron densities in GLs in the structures with different К are rather close to each other, particularly, in GLs with small and moderate indices.

Figure 3.3 Voltage dependences of electron Fermi energies in some GLs in multiple-GL structure with different KatT= 300 K.

Figure 3.4 Electron density vs GL index in multiple GL-structures with different number of GLs {K = 10 and К = 50) at different temperatures and V₆ = 1000 mV.

Figure 3.5 presents the Fermi energies in GLs with different indices at different temperatures. One can see from Fig. 3.5 (as well as from Fig. 3.2) that the higher T corresponds to lower /л_к. This is due to an increasing dependence of the density of states on the energy and the thermal spread in the electron energies.

Figure 3.5 Comparison of the vs к dependences calculated for different temperatures using numerical and simplified analytical (solid line) models.