3.6 Incomplete Ionization

At low temperatures the thermal energy within a semiconductor is not high enough to fully activate all of the donor and acceptor impurity atoms. As a result the carrier concentration will not reach the concentration of dopant atoms [154,155]. SiC distinguishes from narrow bandgap semiconductors, such as silicon, in that common doping impurities in SiC have activation energies larger than the thermal energy ${\mathrm{k_B}}T$ even at room temperature. This causes the incomplete ionization of such impurities, which leads to strong temperature and frequency dependence of the semiconductor junction differential admittance [156].

In Chapter 2 the fundamentals of SiC technology have been briefly discussed. Impurity doping in SiC primarily accomplished through the introduction of nitrogen (N), phosphorous (P) and arsenide (As) for n-type, and aluminum (Al), boron (B) and gallium (Ga) for p-type. Among these dopants which could in theory be utilized over a wide doping range, the most important dopant centers for 4H- and 6H-SiC are N, acting as donor as well as Al and B, acting as acceptors.

Many dopants of SiC preferentially incorporate into either Si lattice site or C sites. N preferentially incorporate into the lattice sites which are normally occupied by carbon atoms. Al prefers the Si-site of SiC, whereas B may substitute on both sites [35]. Inequivalent sites of $\alpha$-SiC, C (or Si) sites, one with cubic (k) surrounding and the other with hexagonal (h) surrounding are expected to cause site-dependent impurity levels [157]. In the case of 4H-SiC there are equal numbers of cubic and hexagonal sites, while there is one hexagonal site and two kinds of inequivalent quasi-cubic sites denoted k1 and k2 sites in 6H-SiC (see Section 2.1.1, Fig. 2.3). Doping atoms substituting on these sites therefore experience different surroundings, and give rise to different ionization energies.

At present, only few research data on shallow impurity levels in SiC polytypes is available [158]. The macroscopic parameters are accessible by several measurement methods such as infrared absorption, the Hall effect, thermal and optical admittance spectroscopy, and deep level transient spectroscopy [33]. The experimental values of the ionization energy reported so far [159,160,73,158,33,161,162,163] vary within a range of about 5-15% (see Table 3.9). It should be mentioned that various possible sources of error and the uncertainty about several material constants will probably result in comparable scatter when comparing the data of different publications and different measurement methods.

The ionization energies of Al/B acceptors are only weakly sensitive to the particular polytype and to inequivalent lattice sites. However, they decrease with increasing acceptor concentration or increasing compensation [73]. This finding may be the reason for the scatter of published data listed in Table 3.9.

Table 3.9: Experimental values of the average ionization energy (in meV) level for Al, B, and N in 4H- and 6H-SiC.

	Al	B	N $_{\mathrm h}$	N $_{\mathrm k_1}$	N $_{\mathrm k_2}$
4H-SiC	$220\pm 20$	$330\pm 30$	$50\pm 5$	$90\pm 5$	-
6H-SiC	$220\pm 20$	$330\pm 30$	$80\pm 5$	$140\pm 5$	$145\pm 5$

The difference in the donor energy levels in different polytypes must arise from different total potentials which are the sum of the host and the impurity potentials. In terms of band-structure description in Section 3.2, these large differences in the donor ground state energies in different polytypes are due to substantially different band structures, particularly the location and the effective masses of the conduction band edges. The site dependence of ionization energy in a given polytype (4H or 6H-SiC) can be attributed to the spatial variation of the conduction band bottom wave functions as viewed from inequivalent donor sites [158].

The donor impurity ionization energies are

$$\Delta E_\mathrm{d}{_\mathrm{h}} = \ensuremath{E_\mathrm{c}}-E_\mathrm{d}{_\mathrm{h}},$$ (3.115)

$$\Delta E_\mathrm{d}{_\mathrm{k}} = \ensuremath{E_\mathrm{c}}-E_\mathrm{d}{_\mathrm{k}},$$ (3.116)

$$\Delta E_\mathrm{d}{_\mathrm{k_2}} = \ensuremath{E_\mathrm{c}}-E_\mathrm{d}{_\mathrm{k_2}},$$ (3.117)

where $\ensuremath{E_\mathrm{c}}$ denotes the conduction band minimum and $E_\mathrm{d}{_\mathrm{h}}$ ( $E_\mathrm{d}{_\mathrm{k}}$) the ground state energy level of the hexagonal (cubic) N donor. If one is not concerned with the dynamic effects of incomplete ionization, these donor levels can be lumped together and replaced by a single effective level $E_\mathrm{d}$ at

$$\Delta E_\mathrm{d}= \ensuremath{E_\mathrm{c}}- E_\mathrm{d}.$$ (3.118)

As in the case of donors, also acceptors should in principle show two different energy levels corresponding to the inequivalent sites. However, this energy difference seems to be too small in Al-doped $\alpha$-SiC to be readily detectable [73]. Therfore, the acceptor impurity ionization energy is given by

$$\Delta E_\mathrm{a}= E_\mathrm{a}-E_\mathrm{v},$$ (3.119)

here, $E_\mathrm{v}$ is the energy at the top of the valence band and $E_\mathrm{a}$ is the ground state energy of the acceptor.

Note that the relatively large binding energies lead to incomplete ionization of the dopants in 4H- and 6H-SiC that will affect the device behavior within a wide range of operation conditions. The most important effect is a substantial increase of the bulk resistance dependent on the temperature and the doping concentration which can be deduced from equilibrium considerations.

At thermal equilibrium, that is, the individual electron and hole current flowing across the junction are identically zero, and the Fermi level is constant. Hence, for this condition the Poisson equation (3.10) can be simplified to

$$n+N_\mathrm{A}^-=p+N_\mathrm{D}^+.$$ (3.120)

Solving equations (3.69) and (3.120) yields the equilibrium electron concentration in an n-type material

$$n=\frac{1}{2}\cdot \left[N_\mathrm{D}^{+}-N_\mathrm{A}^{-}+\sqrt{\left(N_\mathrm{D}^{+}-N_\mathrm{A}^{-}\right)^2-4n^2_i}\right],$$ (3.121)

similarly we obtain the concentration of holes in a p-type semiconductor

$$p=\frac{1}{2}\cdot \left[N_\mathrm{A}^{-}-N_\mathrm{D}^{+}+\sqrt{\left(N_\mathrm{D}^{+}-N_\mathrm{A}^{-}\right)^2-4n^2_i}\right].$$ (3.122)

The concentration of ionized impurity atoms is given by a steady-state Gibbs distribution [131]:

$$N_\mathrm{D}^{+} = \displaystyle\frac{N_\mathrm{D}}{1+g_\mathrm{D... ...nsuremath{E_\mathrm{c}}-E_\mathrm{d}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right),$$ (3.123)

$$N_\mathrm{A}^{-} = \displaystyle\frac{N_\mathrm{A}}{1+g_\mathrm{A... ...(- \frac{E_\mathrm{a}-E_\mathrm{v}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right),\$$ (3.124)

where $N_\mathrm{D}$ and $N_\mathrm{A}$ are the substitutional (active) dopant concentration for donor and acceptor respectively, $g_\mathrm{D}$ with typical value of 2 and $g_\mathrm{A}$ with typical value of 4 are the degeneracy factors for the impurity levels of donors and acceptors in $\alpha$-SiC, respectively.

Now we can obtain from (3.118), (3.120) and (3.123) an explicit relation for the ionization degree of a single donor level in n-type material

$$\xi _\mathrm{D}=\displaystyle\frac{N_\mathrm{D}^+}{N_\mathrm{D}}=... ...playstyle\frac{\Delta E_\mathrm{d}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right) },$$ (3.125)

and similarly, in p-type material from (3.119), (3.120) and (3.124)

$$\xi _\mathrm{A}=\frac{N_\mathrm{A}^-}{N_\mathrm{A}}=\displaystyle... ...playstyle\frac{\Delta E_\mathrm{a}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right) }.$$ (3.126)

Using the values of the effective ionization energies listed in Table 3.9, the ionization level of the dopants N and accepter Al in 4H- and 6H-SiC at different temperature have been calculated in Fig. 3.14 and 3.15, respectively.

For n-type it has been assumed that the number of k-type donors is the same as the number of h-type donors. Thus, the nitrogen donor level was set to the average value for all sites, $\Delta E_\mathrm{d}$=70 and $\Delta E_\mathrm{d}$=100 meV for 4H and 6H-SiC, respectively. The p-type results were calculated assuming the aluminum acceptor value $\Delta E_\mathrm{a}$=200 meV. The result clearly show that the carrier concentration ionization level decreases with increasing doping concentration and decreasing temperature.

From the mobility data described in Section 3.3 and the carrier concentration level mentioned above, the resistivity $\varrho_\nu$ in n-type $\alpha$-SiC material can be calculated by

$$\varrho_n = \frac{1}{{\mathrm{q}}\cdot\mu_n\cdot N_\mathrm{D}^+},$$ (3.127)

and respectively for p-type material

$$\varrho_p = \frac{1}{{\mathrm{q}}\cdot\mu_p\cdot N_\mathrm{A}^-}.$$ (3.128)

At high doping levels and increasing temperature, the increasing dopant ionization overcompensates the decreasing mobility. The resistivity then decreases with increasing temperature, as can be seen in Figs. 3.16 and 3.17 for n-type and p-type $\alpha$-SiC, respectively. The resistivity in p-type SiC is higher than in n-type SiC due to the deep energy levels of the p-type dopants in $\alpha$-SiC.

Fig. 3.18 through Fig. 3.21 illustrate the carrier concentration ionization degree of a donor (N) and acceptor (Al, B) in $\alpha$-SiC calculated from (3.125) and (3.126), respectively. Comparison of the influence of the temperature on the ionization degree at three different doping concentrations and the doping dependence of incomplete ionization at three different temperatures were depicted.

From the last Fig 3.21, one can see that in 4H-SiC at room temperature and an acceptor concentration of $N_\mathrm{A}=10^{15}$cm$^{-3}$ only $70\%$ of Al and $20\%$ of B are ionized. $\xi _\mathrm{A}$ decreases with increasing doping concentration and decreasing temperature, which finally leads to the freeze-out of holes at low temperatures. It should be noted that the ionization degree for aluminum is considerably higher than boron. In order to achieve similar conductivity with boron to that with aluminum, almost an order of magnitude higher concentration has to be implanted.

Figure 3.14: Ionization level of the donor (N) and acceptor (Al) as a function of doping concentration in 4H-SiC for different temperatures.

Figure 3.15: Ionization level of the donor (N) and acceptor (Al) as a function of the doping concentration in 6H-SiC for different temperatures.

Figure 3.16: Resistivity of n-type SiC for incomplete ionization calculated from the doping-and temperature dependent mobility.

Figure 3.17: Resistivity of p-type SiC for incomplete ionization calculated from the doping-and temperature dependent mobility.

Figure 3.18: Ionization degree of N in $\alpha$-SiC as a function of the doping concentration for different temperatures.

Figure 3.19: Ionization degree of N in $\alpha$-SiC as a function of the temperature for different doping concentrations.

Figure 3.20: Influence of the temperature on the ionization degree of acceptors Al and B in 4H-SiC.

Figure 3.21: Ionization degree of acceptors Al and B in 4H-SiC at different doping concentrations.

As mentioned earlier, in contrary to p-type SiC material, where the small polytype dependence of the acceptor energies observed, the ionization energies of the donors in n-type SiC vary substantially with the SiC polytype (Table 3.9). The reason is the difference in the location of the conduction band edge and the effective electron mass. It has been determined optically and electrically that inequivalent sites of $\alpha$-SiC, one with cubic (k) surrounding and the other with hexagonal (h) surrounding cause site-dependent impurity levels. In that case the single effective level assumption, $\Delta E_d$ for donor impurities cannot be valid any more in both polytypes. Thus, the electron carrier concentration determined from the neutrality condition (3.120) has the form

$$n = \sum_{i=1}^x N_\mathrm{D}^{+}{_\mathrm{h_i}}+ \sum_{j=1}^y N_\mathrm{D}^{+}{_\mathrm{k_j}},$$ (3.129)

where $x$ and $y$ are the number of inequivalent hexagonal and cubic sites in $\alpha$-SiC, respectively. For 4H-SiC $x=y=1$, while $x=1$ and $y=2$ for 6H-SiC. Rewriting the donor impurity atom equation (3.123) in order to account for the exact impurity levels in 4H-SiC yields

$$N_\mathrm{D}^{+} = \frac{\frac{1}{2}\cdot N_\mathrm{D}} {1+g_\mat... ...ac{\Delta E_\mathrm{d}{_\mathrm{k}}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right)}.$$ (3.130)

In electrothermal equilibrium, substituting (3.129) into (3.130) gives the exact ionization degree of a single donor in 4H-SiC

$$\xi_\mathrm{D}=\frac{N_\mathrm{D}^{+}}{N_\mathrm{D}} = \frac{-1+\... ...c{\Delta E_\mathrm{d}{_\mathrm{k}}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right) }.$$ (3.131)

By analogy, one can formulate the corresponding ionization degree of a donor in 6H-SiC.

Fig. 3.22 and Fig. 3.23 illustrate the carrier concentration ionization degree ( $\xi _\mathrm{D}$) as a function of concentration and temperature, respectively after (3.131). It is important to note that the incomplete ionization of N becomes only relevant at low temperatures and high doping concentrations. Here, for temperatures of only 100 K and $N_\mathrm{D}=10^{15}$ cm$^{-3}$, $60\%$ and $40\%$ of N is ionized in 4H- and 6H-SiC, respectively. Generally, more than $90\%$ of N is ionized for temperatures above 250 K and $N_\mathrm{D}<10^{16}$ cm$^{-3}$ in both polytypes.

Figure 3.22: Ionization degree of donor (N) with site-dependent activation energy in $\alpha$-SiC at different temperatures.

Figure 3.23: Ionization degree of donor (N) with site-dependent activation energy in $\alpha$-SiC for different doping concentrations.

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