3.1.6.2 Schottky Contact

A Schottky barrier refers to a metal-semiconductor contact having a large barrier height (i.e. $\phi_{\mathrm{B}}>kT)$ and low doping concentration that is less than the density of states in the conduction band or valence band. The potential barrier between the metal and the semiconductor can be identified on an energy band diagram. To construct such a diagram we first consider the energy band diagram of the metal and the semiconductor, and align them using the same vacuum level as shown in Fig. 3.2 (a). As the metal and semiconductor are brought together, the Fermi energies of the two materials must be equal at thermal equilibrium Fig. 3.2 (b).

Figure 3.2: Energy band diagram of a metal adjacent to n-type semiconductor under thermal noneqilibrium condition (a), metal-semiconductor contact in thermal equilibrium (b).

The barrier height $\phi_{\mathrm{B}}$ is defined as the potential difference between the Fermi energy of the metal and the band edge where the majority carrier reside. From Fig. 3.2 one finds that for n-type semiconductors the barrier height is obtained from

$$\phi_\mathrm{Bn} = \phi_{\mathrm{m}}- \chi,$$ (3.40)

where $\phi_{\mathrm{m}}$ is the work function of the metal and $\chi$ is the electron affinity. The work function of selected metals as measured in vacuum can be found in Fig. 3.3 [108].

For p-type material, the barrier height is given by the difference between the valence band edge and the Fermi energy in the metal,

$$\phi_\mathrm{Bp} = \frac{E_{\mathrm{g}}}{{\mathrm{q}}}+\chi-\phi_{\mathrm{m}},$$ (3.41)

A metal-semiconductor junction will therefore form a barrier for electrons and holes

Figure 3.3: Energy band diagram of the selected metals and 4H-SiC.

if the Fermi energy of the metal is located between the conduction and the valence band edge.

In addition, we define the work function difference as the difference between the work function of the metal and that of the semiconductor. For n-type material it reads

$$\phi_{\mathrm{wf}}= \phi_{\mathrm{m}}- \chi - \frac{\ensuremath{E_\mathrm{c}}-E_{\mathrm{F,n}}}{{\mathrm{q}}},$$ (3.42)

similarly, for p-type material

$$\phi_{\mathrm{wf}}= \chi + \frac{\ensuremath{E_\mathrm{c}}-E_{\mathrm{F,p}}}{{\mathrm{q}}} - \phi_{\mathrm{m}}.$$ (3.43)

The work function difference energy becomes

$$E_{\mathrm{w}}= {\mathrm{q}}\cdot\phi_{\mathrm{wf}}.$$ (3.44)

The measured barrier height for selected metal/4H-SiC junction is listed in Table 3.1 [109,110]. These experimental barrier heights depend on the surface polarity of SiC (Si- and C-face), and often differ from the ones calculated using (3.40) and (3.41).

Table 3.1: Work function of selected metals and their measured and calculated barrier height on n-type 4H-SiC.

	Al	Ti	Zn	W	Mo	Cu	Ni	Au	Pt
$\phi_{\mathrm{m}}$	4.28	4.33	4.33	4.55	4.60	4.65	5.10	5.15	5.65
$\phi_{\mathrm{B}}$(Si-face)		1.12					1.69	1.81
$\phi_{\mathrm{B}}$(C-face)		1.25					1.87	2.07
$\phi_{\mathrm{B}}$(calculated)	1.01	1.06	1.06	1.28	1.33	1.38	1.63	1.68	2.08

This is due to the detailed behavior of the metal-semiconductor interface. The ideal metal-semiconductor theory assumes that both materials are pure and that there is no interaction between the two materials nor any interfacial layer. Chemical reactions between the metal and the semiconductor alter the barrier height as do interface states at the surface of the semiconductor and interfacial layers. Furthermore, one finds the barrier heights reported in the literature to vary widely due to different surface cleaning procedures.

The current density is calculated according to the thermionic emission condition [111] neglecting tunneling currents:

$${\mathbf{n}}\cdot{\mathbf{J}}_n = -{\mathrm{q}}\cdot v_n \cdot (n-n_\mathrm{s}),$$ (3.45)

$${\mathbf{n}}\cdot{\mathbf{J}}_p = {\mathrm{q}}\cdot v_p \cdot (p-p_\mathrm{s}),$$ (3.46)

here, the thermionic recombination velocities $v_n$ and $v_p$ for electrons and holes, respectively are given by

$$v_\nu = \sqrt{\frac{{\mathrm{k_B}}\cdot T_\mathrm{L}}{2\pi \cdot ... ...dot({\mathrm{k_B}}\cdot T_\mathrm{L})^2}{N_\nu \cdot h^3},\hspace{1cm}\nu = n,p$$ (3.47)

and are usually represented by the expression

$$v_\nu = A^* \cdot \frac {T_\mathrm{L}^2}{{\mathrm{q}}\cdot N_\nu },\hspace{1cm}\nu = n,p$$ (3.48)

where

$$A^* = \frac{4\pi\cdot{\mathrm{q}}\cdot m_\nu \cdot{\mathrm{k_B}}^2}{h^3},\hspace{1cm}\nu = n,p$$ (3.49)

is known as the effective Richardson constant. It is dependent on the effective mass and has a theoretical value of 146 and 72 $\mathrm{A cm^{-2} K^{-2}}$ for n-type 4H- and 6H-SiC, respectively [108].

The effective carrier masses are explicitly described in Section 3.2.2.

The carrier concentrations at the surface are given by

$$n_\mathrm{s}=N_\mathrm{c}\cdot\exp\left(\frac{-\ensuremath{E_\mathrm{c}}-E_{\mathrm{w}}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right),$$ (3.50)

$$p_\mathrm{s}=N_\mathrm{v}\cdot\exp\left(\frac{E_\mathrm{v}-E_{\mathrm{w}}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right).$$ (3.51)

Note that the expressions (3.45) and (3.46) are equivalent to the most commonly used expressions [111]

$$J_n = \frac{4\pi{\mathrm{q}}\cdot m_n \cdot ({\mathrm{k_B}}\cdot ... ...hi_\mathrm{m}}{{\mathrm{k_B}}\cdot T_\mathrm{L}} \right) - 1\right]\hspace{5mm}$$ (3.52)

$$J_p = - \frac{4\pi{\mathrm{q}}\cdot m_p \cdot ({\mathrm{k_B}}\cdo... ...}}\cdot \phi_\mathrm{Bp}}{{\mathrm{k_B}}\cdot T_\mathrm{L}} \right) - 1\right].$$ (3.53)

The Schottky contact boundary conditions for the carrier temperatures $T_n$ and $T_p$ and the lattice temperature $T_\mathrm{L}$ are similar to the ones which apply for the Ohmic contact, i.e. (3.36) and (3.37), or respectively (3.38).

T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation