Numerous studies in the literature have investigated the electrical transport properties of heavily Al-implanted 4H-SiC layers [1–5], Al-doped 4H-SiC epilayers grown by chemical vapor deposition (CVD), [6,7] and Al-doped 6HSiC wafers grown by physical vapor transport (PVT) [8,9]. Our group has previously investigated the temperature dependent resistivity  and temperature-dependent Hall coefficient  for heavily Al-doped 4H-SiC epilayers grown via CVD, PVT, and a solution-based method.10–18) For CVD-grown samples, we characterized the relationships between the conduction mechanisms (i.e., band, nearest-neighbor hopping (NNH), and variable range hopping (VRH) conduction) and both measurement temperature  and Al concentration  [11]. We found that, although the Al dopant makes 4H-SiC a p-type semiconductor, the values of   for all of the samples became negative at low temperatures. We experimentally elucidated the relationship between the sign of  and the band and NNH conduction regions [14,15], and we sought to explain the negative  from the perspective of electron and hole hopping [16,17]. Moreover, we found that the activation energies  associated with  in band and NNH conduction are similar to those  for the corresponding , respectively [14,15].

In the present study, we categorize the electrical properties into two types based on band and hopping conduction and mainly discuss physical models of and  for hopping conduction. First, in Sec. 2, we propose a physical model to explain the sign of  for hopping conduction. This model is developed from the viewpoint of the hopping of electrons and holes. Because negative  values have been reported in two cases (i.e., NNH conduction and VRH conduction according to Mott’s model) as hopping conduction in heavily Al-doped SiC [1,7,8,14–17], we briefly describe the application of our proposed model to these cases. Second, in Sec. 3, on the basis of our proposed model for in hopping conduction, we investigate the relationship between the net density of hopping charge carriers, from which NNH current arises, and the density of charge carriers determined by Hall effect measurements in NNH conduction, which leads to an explanation for why  becomes close to 

Reports on Anomalous Sign Of 

The sign of  has been experimentally reported to become negative at low temperatures for not only p-type-conducting single-crystalline SiC [1,7,8,14-17) but also Mg-doped InP [19], Mn-doped InP [20], and Mg-doped GaN [21].

In the case of lightly doped p-type narrow-bandgap semiconductors, the sign of  changes from positive to negative at high temperatures, at which the electron concentration in the conduction band approaches the hole concentration  in the valence band (i.e., where the conduction type becomes intrinsic). This occurs because the electron mobility is greater than the hole mobility  [22]. This scenario does not apply to heavily Al-doped wide-bandgap 4H-SiC.

In the case of Al-doped 6H-SiC, Krieger et al [8]. explained the negative  in NNH conduction on the basis of the amorphous semiconductor models proposed by Emin [23] and Grunewald et al [24]. Although positive and negative  values have been experimentally reported for n- and p-type hydrogenated amorphous silicon (a-Si:H), respectively [26-28], Street [25] has argued that these models are applicable to narrow-bandgap materials and materials whose mobility edge is located at the center of the band, which is not the case for a- Si:H because the conduction in this material occurs near the band edge. Thus, according to Street’s argument,the aforementioned models are not applicable to Al-doped single-crystalline SiC.

Kajikawa recently summarized a report of negative  values for p-type semiconductors at low temperatures and attempted to fit the corresponding experimental data curves for p-type 4H-SiC [29] using the Hall factor for NNH conduction in an impurity band [30,31] by including a transfer-integral-related parameter (J₃). Kajikawa found that to obtain a satisfactory curve fit to the experimental data, J₃ should be assumed to be negative [29]. Krieger et al [8]. likewise performed curve fitting of experimental data under the assumption of negative   for NNH conduction. Although Emin [23], Grunewald et al. 24, Holstein [32], and N´emeth and M¨uhlschlege [33] have discussed the Hall effect for hopping conduction on a quantum theoretical basis (i.e., from the perspective of electron wave functions), the question of why  or J₃ becomes negative for heavily doped crystalline semiconductors remains unanswered [29,34].

Physical Model for the Sign of For Hopping Conduction from the Viewpoint of Electron and Hole Hopping

We propose a physical model to explain the sign of  for hopping conduction. This model is developed from the viewpoint of electron and hole hopping. In hopping conduction, electrons or holes are restricted to seeking unoccupied sites to which to hop, indicating that the sign of  

is determined by the difference between the density of hopping sites for holes and the density of hopping sites for electrons under a given magnetic flux density (B). Under an electric field , an electron hops from a bandgap state (e.g., acceptor, donor, or localized state) to another bandgap state unoccupied by an electron. Analogous to the concept of holes in the valence band, the bandgap state unoccupied by an electron can be regarded as a bandgap state occupied by a hole. Therefore, the hopping of holes is the same phenomenon as the hopping of electrons, and the hopping directions are opposite to each other, indicating that the mathematical representation for the hopping probability of holes between two bandgap states is the same as that of electrons. Because the density of hopping holes is the same as that of hopping electrons under  , the net density of hopping charge carriers under  is referred to as n_Hopping(T). 

Under B, the direction of highest transfer probability for charge carriers hopping induced by , which is quantum-theoretically derived, has been reported to coincide with the direction indicated by the Lorentz force [35]. That is, for a left-to-right hopping current (I_Hopping) with B directed toward the back of the plane, holes and electrons (of which n_Hopping(T) consists) transfer in the upper direction, resulting in a change in the charge state of the upper bandgap sites. When the charge state becomes negative, which is defined as the hopping of electrons under B, this change in the charge state contributes to a negative Hall voltage. In the opposite case, which is defined as the hopping of holes under B, the change in the charge state contributes to a positive Hall voltage. These results indicate that the sign of  depends on the change in the charge state of hopping sites under B.

The measured Hall voltage due to hopping conduction (V_HHopping(T)) is defined as [35]

Where  the Hall coefficient due to hopping conduction and w is is the thickness of the sample along B.

Assuming that holes or electrons, of which  consists entirely, can hop under B, the absolute value of the ideal Hall voltage due to hopping conduction  is defined as 

Where  is the absolute value of the ideal Hall coefficient for hopping conduction, defined by

Figure 1: Schematic diagrams of NNH and VRH conduction. The hopping of a hole or an electron for NNH conduction (a) under  in an energy band diagram and (b) under B in real space. The inset shows the direction of F for I flowing from left to right and B directed toward the back of the plane. The hopping of a hole or an electron for VRH conduction (c) under  in an energy band diagram and (d) under B in real space.

Where q is the elementary charge and γ Hopping is the Hall factor for hopping conduction.

Because B affects hopping charge carriers with density of  the ratio between the hopping probability of holes   and that of electrons  due to B is the ratio between the density of hopping sites for holes and that for electrons under B. Here,

The ratio between the amount of holes and the amount of electrons, which accumulate at the electrode for the measurement of the Hall voltage when B is applied, is the ratio between  and , indicating that the ratio between the Hall voltage due to the hopping of holes  and that due to the hopping of electrons under B becomes the ratio between  and ;

And

Consequently,  can be expressed as

The physical implication of which is the recombination of the holes and electrons accumulated at the electrode.

Using Eqs. (1), (2), and (7), we obtain  as

This indicates that the sign of  is determined by the difference between the hopping probabilities of holes and electrons under B. 

In the following subsections, we discuss the mathematical representation of  and  for NNH conduction and VRH conduction according to Mott’s model.

Influence of an External Magnetic Field on Hopping Charge Carriers in NNH and VRH Conduction

Well-known models for hopping conduction include NNH conduction [8,9,36,38-44) and two models for VRH conduction.36–38, 40, 44–49) One of the VRH conduction models was proposed by Mott [36-38], and the other was proposed by Shklovskii and Efros and derived with consideration of the Coulomb interaction between charge carriers (i.e., the Coulomb gap) [36,37]. In the following subsections, we individually apply the proposed model to NNH conduction and Mott’s model because a negative  

has been reported for these two cases in heavily Al-doped SiC [1,7,8,14-17].

Figure 1 presents schematic diagrams of NNH conduction and Mott’s model of VRH conduction under  and B. Panels (a) and (b) depict the hopping of a hole or an electron for NNH conduction under  in an energy band diagram and under B in real space, respectively. The inset shows the hopping direction of the charge carriers (i.e., the force (F) affecting hopping charge carriers) for a left-to-right hopping current (I) with B directed toward the back of the plane. Panels (c) and (d) depict a hole or an electron hopping for VRH conduction under E in an energy band diagram and under B in real space, respectively. Here, I_NNH and I_VRH denote the currents for NNH conduction and VRH conduction according to Mott’s model, respectively.

In NNH conduction, as shown in Figure 1(a), under , an electron hops from a negatively ionized Al acceptor (Al⁻) site to its nearest-neighbor neutral Al acceptor (Al⁰) site or a hole hops from an Al⁰ site to its nearestneighbor Al− site. In VRH conduction according to Mott’s model, as shown on the left side of Figure 1(c), under E, an electron hops from a localized state to a localized state unoccupied by an electron (i.e., LS+h⁺) with a higher energy or a hole hops from a localized state to a localized state unoccupied by a hole (i.e., LS+e⁻) with a higher energy. The right side of Figure 1(c) shows that under , an electron hops from a localized state to LS+h+ with a lower energy or a hole hops from a localized state to LS+e⁻ with a lower energy.

Sign Of For NNH Conduction

 replaced by V_HNNH(T). Moreover,  γHopping, and  are replaced by the absolute value of the ideal Hall voltage  the absolute value of the ideal Hall coefficient  the Hall factor (γNNH), and the net density of hopping charge carriers  for NNH conduction, respectively.

The existence probabilities of Al⁻ and Al⁰ sites are  and  respectively, where  is the Fermi–Dirac distribution function for acceptors and can be expressed as

where k_B is the Boltzmann constant and is the acceptor degeneracy factor [50]. Because holes can hop from Al⁰ sites to their nearest-neighbor Al− sites and electrons can hop from Al− sites to their nearest-neighbor Al⁰ sites,  and  are replaced by  and  respectively. Consequently, using Eq. (8), we obtain  as 

Equation (10) indicates that  is positive when  > 0.5 and negative when   < 0.5. Consequently, in heavily Al-doped 4H-SiC,  becomes negative for NNH conduction because  has been reported to be located between  and , [51,52] that is,  < 0.5.

Sign Of For Mott’s Model of VRH Conduction

 is replaced by  Moreover,   and are replaced by the absolute value of the ideal Hall voltage  the absolute value of the ideal Hall coefficient  the Hall factor and the net density of hopping charge carriers  

for VRH conduction, respectively.

In Mott’s model of VRH conduction, holes or electrons can hop to unoccupied localized states for energies between E_F ± ΔE, [36] where

and  

 is the localization length of localized states around E_F. 

The rate  

of hole or electron hopping to an unoccupied localized state at a higher energy level (W), as shown on the left side of Figure 1(c), can be expressed as [38,53] 

where d is the distance between two localized states and is a prefactor that is approximately equal to the phonon frequency associated with the hopping process.

The rate  of hole or electron hopping to an unoccupied localized state at a lower energy level, as shown on the right side of Figure 1(c), can be expressed as [53]

Because

as is clear from Eqs. (12) and (13), under B, a large amount of holes at energy levels below E_F can easily hop to LS+e⁻ at energy levels higher than E_F; also, a large amount of electrons at energy levels higher than E_F can easily hop to LS+h⁺ at energy levels below E_F. For therefore, the approximate hopping probabilities of holes  and electrons  under B can be defined as [17]

And

Where  is the Fermi–Dirac distribution function for localized states, which can be expressed as

Here, the degeneracy factor for localized states at E is assumed to be 1. Using Eq. (8), we then obtain  as

In the case where  around  decreases with increasing E (e.g., a band tail above ), 

This inequality is valid because

And

Where  is a positive energy. Therefore,

As a result, Eq. (19) indicates that  becomes negative in the case where  (e.g., the band tail above E_V).

Consequently, in heavily Al-doped 4H-SiC, R_H(T) becomes negative in VRH conduction according to Mott’s model because  is located at the band tail above .

Typical Electrical Transport Properties in Heavily Al-Doped 4H-Sic

Figure 2 shows typical plots of ln  and ln  versus 1/T  on the same vertical scale for heavily Al-doped 4H-SiC grown by CVD [14,15]. The C_Al value and thickness of this sample were 5.2 × 10¹⁹ cm⁻³ and 90 μm, respectively.

Figure 2: Plots of ln  and ln  versus 1/T for the sample with C_Al of 5.2 × 10¹⁹ cm⁻³. The values of   at high and low temperatures are best approximated by the solid straight lines, which indicate band and NNH conduction, respectively. The values of  in the band and NNH conduction regions are best approximated by the broken straight lines.

The conduction mechanisms in semiconductors include band conduction, NNH conduction, [8,9,36,38-44] and VRH conduction [36,38,40,44-49]. If the currents due to band, NNH, and VRH conduction flow completely in parallel in the valence band, at E_Al, and around , respectively,  can be expressed as

Where 

And

Here,  and ρVRH(T) denote the temperature-dependent resistivity for band, NNH, and VRH conduction, respectively; and  

are the pre-exponential factors for band, NNH, and VRH conduction, respectively; T₀ is a constant associated with VRH conduction; and β is 1/4 for Mott’s model or 1/2 for the Shklovskii–Efros model of VRH conduction [36,37].

The data in the ln  plot were best approximated by two solid straight lines, indicating that, from Eqs. (25) and (26), the dominant conduction mechanisms at high and low temperatures are band and NNHconduction, respectively. The temperature at the point of intersection of the two solid straight lines extrapolated 

from the higher- and lower-temperature regions is denoted T_BH, and its value is 92 K.

In Figure 2,  becomes positive and negative at high and low temperatures, respectively. The sign inversion temperature for  is referred to as  and its value is 126 K.

Comparison of the  and Tinv values reveals that  is negative for NNH conduction. The data in the ln  plot were best approximated by the two broken straight lines, indicating that at high temperatures

And at low temperatures 

From the solid straight lines,  and  were estimated to be 113 and 35 meV, respectively. From the broken straight lines,  and  were estimated to be 105 and 35 meV, respectively. As a result,   and  are close to  and , respectively. In previously reported plots of ln  and ln  for samples with C_Al values between 2.4×10¹⁹ and 1.5×10²⁰cm⁻³,  and  were found to be close to  and , respectively [15].

For band conduction,  and  are respectively expressed as [50]

Where  is the Hall factor for holes for band conduction [50].

 In the freeze-out region,

Where  is the activation energy, which is usually between  Given that Eq. (32) has been reported to be valid even at high temperatures for heavily Al-doped SiC [6,10,51,52,54-56], the freeze-out region remains even in this temperature regime. Therefore,  varies with T much more sharply compared with

where n is equal to −1.5 for acoustic phonon scattering and 1.5 for ionized impurity scattering [50].

From Eqs. (25), (30), and (32)

whereas from Eqs. (28), (31), and (32),

indicating that

Consequently, at high temperatures, the activation energy of  becomes close to that of .

In addition, the values of ΔEHoleBand in Eq. (32) and n in Eq. (33) can be evaluated for the sample with C_Al of 5.2 × 10¹⁹ cm⁻³.

The triangles in Figure 3 show  Band estimated using  at high temperatures in Figure 2 and Eq. (31). The value of  was estimated to be 105 meV from the slope of the solid straight line in Figure 3. By contrast, the value of  was calculated to be 149 meV using the empirical equation reported previously.51, 52) Therefore, the  value satisfies the requirement of being between  and 

The triangles in Figure 4 show  estimated using Eq. (30),  shown in Figure 2, and  shown in Figure 3. As evident from Figure 4,  exhibits a weak temperature dependence. The value in Eq. (33), as evaluated from the slope of the straight line in Figure 4, was −1.00, indicating that acoustic phonon scattering may affect at high temperatures.

Net Density of Hopping Charge Carriers in NNH Conduction

The preceding discussion of  for hopping conduction leads to the following consideration: When the density of  

is greater than the density of Al⁻ (n_Al−), I_NNH is explained by the hopping of all of the electrons at Al⁻ sites to their nearest-neighbor Al⁰ sites. Viewed from a different perspective, I_NNH can also be described by the hopping of a portion of the holes at Al⁰, whose density is equal to n_Al−, to their nearest-neighbor Al⁻ sites. These descriptions indicate that the net density of hopping charge carriers can be expressed as

where N_Al is the density of Al acceptors, which is less than or equal to C_Al.

Figure 3: Plots of ln(p(T)/γ_Band)–1/T in the band conduction region and ln(n_NNH(T)/γNNH)^–1/T in the NNH conduction region. The data are best approximated by a straight line.

When  is less than n_Al−, by contrast, INNH can be explained by the hopping of all of the holes at Al⁰ sites to their nearest-neighbor Al⁻ sites or by the hopping of a portion of the electrons at Al⁻ sites, whose density is equal to  to their nearest-neighbor Al⁰ sites. Therefore,

In other words, for  

Whereas for  

Activation Energies for NNH Conduction

The density of charge carriers determined by Hall effect measurements  is defined as

From Eqs. (3), (10), and (41), we obtain the relationship between  and  as

Here, negative and positive values of  

may be regarded as the electron density and the hole density, respectively.

In heavily Al-doped 4H-SiC, because E_F is located between E_Al and E_V, [51,52] at low temperatures,

indicating that, from Eq. (42),

On the other hand,

Figure 4: Plots of ln(μ_h(T)γ_Band)–ln T in the band conduction region and ln(μ_NNH(T)γ_NNH)–ln T in the NNH conduction region.

indicating that, from Eq. (39),

Because the activation energy  is determined from

we obtain  as

 is derived from Eqs. (41), (44), and (46) as

Indicating that, from Eq. (29),

Consequently,

For NNH conduction, analogous to band conduction,  can be defined as

where  is the conductivity for NNH conduction and  is the mobility of hopping charge carriers for NNH conduction. From Eqs. (46), (47), and (52),

As a result, a difference between the temperature dependences (i.e.,  in Eq. (49) and in Eq. (53)) results from .

For ,  is proportional to (1) the density of hopping charge carriers (i.e., n_NNH(T)), (2) the probability of Al acceptor sites being unoccupied by charge carriers at E_Al (i.e., 1 − f_A(E_Al)), (3) the wave function overlap of two nearest-neighbor Al acceptor states, which can be expressed as exp(−2R_Al/α₀) [41], where R_Al is the average distance between nearest-neighbor Al acceptor sites, and (4) the thermally assisted transfer rate over the difference in energy (W_Al) between two nearest-neighbor Al acceptor levels, which can be expressed as ν_ph exp(−W_Al/k_BT). Therefore,

From Eqs. (52) and (54), because  

Because  at low temperatures,

Compared with ,  exhibits a very weak temperature dependence, similar to the temperature dependence of , indicating that, from Eqs. (26) and (53),

Consequently,

Thus, at low temperatures, the activation energy of negative is close to that of .The activation energies of  and  in NNH conduction, moreover, are close to 

In addition, the value of  in Eq. (47) can be evaluated for the sample with  of 5.2 × 10¹⁹ cm⁻³.

The circles in Figure 3 show  estimated using  at low temperatures in Figure 2 and Eqs. (41) and (44). The value of  was estimated to be 35 meV from the slope of the broken straight line in Figure 3, indicating that the value of  is 35 meV.

The circles in Figure 4 show  estimated using Eq. (52),  at low temperatures shown in Figure 2, and  shown in Figure 3. These results suggest that  may be independent of T.

We investigated  and  for thick heavily Al-doped p-type 4H-SiC epilayers grown by CVD. We found that the sign of  is positive at high temperatures (i.e., band conduction), whereas it is negative at low temperatures (i.e., NNH conduction or VRH conduction according to Mott’s model). The activation energies of negative  are close to those of  for band and NNH conduction, respectively.

We proposed and developed a physical model and mathematical representation to explain the negative  at low temperatures in hopping conduction from the viewpoint of the hopping of electrons and holes. In hopping conduction, electrons or holes are restricted to seeking unoccupied sites to which to hop, indicating that the sign of  is determined by the difference between the density of hopping sites for holes and the density of hopping sites for electrons under an applied magnetic flux density. Moreover, this model is found to be applicable to both NNH conduction and VRH conduction according to Mott’s model.

In NNH conduction, the density of charge carriers determined by Hall effect measurements has first been derived as  where  is the net density of hopping charge carriers from which the NNH current arises. In heavily Al-doped 4H-SiC, because  at low temperatures,  is found to be inversely proportional to . Therefore, we could elucidate the reason why the activation energy of  approaches that of  in NNH conduction. The activation energies of  and  in NNH conduction, moreover, were demonstrated to have values of .

Acknowledgments

This work was supported in part by JSPS KAKENHI Grant Number JP 20K04565 and by the Council for Science, Technology and Innovation (CSTI), the Cross-ministerial Strategic Innovation Promotion Program (SIP), and the “Next generation power electronics/Consistent R&D of next-generation SiC power electronics” project (funding agency: NEDO). The authors would like to thank S. Ji, K. Eto, Y. Ishida, K. Kojima, T. Kato, S. Yoshida, and H. Okumura in National Institute of Advanced Industrial Science and Technology, Japan, for sample preparation and discussion, and A. Takeshita, Y. Kondo, T. Imamura, K. Takano, K. Okuda, K. Ogawa, K. Ozawa, K. Iida, T. Nishijima, M. Ogami, and Y. Kubota in the Matsuura Laboratory for measurements of resistivity and Hall coefficients.

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