1. Introduction
Since the thickness of TMDs can be reduced to less than a few nanometers, TMDs are a promising alternative channel of materials for achieving ultra-thin body field effect transistors (FETs) that are robust to short channel effects. TMDs can be classified into two types based on their crystal structures: trigonal prismatic (2H-TMD type) and octahedral (1T-TMD type). MoS
2 is a typical TMD with a 2H-TMD type structure. TMDs with a 1T-TMD type (group space P3m1) structure have much lower lattice thermal conductivity and better thermoelectric performance at room temperature. This implies that HfS
2 is a good 2D thermoelectric material. Hao Wang et al. predicted that the lattice thermal conductivity of bulk HfS
2 is much lower than that of MoS
2, making the HfS
2 system a promising thermoelectric material [
1]. They also investigated the thermoelectric properties of monolayer HfS
2 and demonstrated that it is an excellent n-type thermoelectric material, with much improved thermoelectric performance when compared with the bulk. Furthermore, the valence band valleys of monolayer HfS
2 can be tuned by external biaxial tensile strain owing to the degeneracy of the valence band valleys reaching a maximum at a strain value of 6%. Monolayer HfS
2, like the CdI
2-type structure, has become well-known due to its low lattice thermal conductivity resulting in promising applications in thermoelectric engineering [
2]. Hongyan Guo et al. performed a systematic first-principles study of the effect of tensile strains on the electronic properties of early monolayer TMDs and suggested that tensile strain could significantly affect the electronic properties of many early TMDs in general, and the electronic bandgap in particular [
3]. The bandgap of group IVB TMDs such as HfS
2 increased as biaxial tensile strain and started to decrease at a strain of 6~8%. Their investigation suggested that strain engineering was an effective approach for modifying the electronic properties of TMDs such as monolayer HfS
2, thereby opening an alternative way for future electronic applications. Recently, Mayuri Sritharan et al. pointed out that transition metal dichalcogenide (TMD) monolayers with native high-κ oxides presented a new avenue for the development of new generation field-effect transistor (FET) devices [
4]. These TMD materials were experimentally shown to form a natively compatible high-dielectric oxide layer, which could enhance both transistor device scaling and supply voltage shrinking. They found that monolayer hafnium disulphide (HfS
2) exhibited isotropic transport at a channel length of 10 nm with ON currents over 1000 μA/μm and good subthreshold swing and identified monolayer HfS
2 as a superior material within the TMD family for ultra-scaled FETs.
Dr. Afzalian et al. from imec developed an atomistic-modelling solver (ATOMOS) to assess the physics and fundamental performance potential of a monolayer HfS
2 FET down to a gate length of 5 nm, including the effect of carrier phonon scattering [
5,
6]. They predicted that monolayer HfS
2 FET had good scalability down to a gate length of 5 nm, with a promising high on the current level. They also proposed the concept of a dynamically doped 2D FET, which scaled better than other 2D FET counterparts. Used in combination with a best mobility material such as monolayer HfS
2, it allowed for the maintenance of excellent performance on the drive current and competitive energy-delay. In Reference [
7], Dr. Jiwon Chang investigated the ballistic transport characteristics of monolayer HfS
2 FET via quantum transport simulation. He focused on the role of degenerate conduction band (CB) valleys in monolayer HfS
2 and predicted that the effect of channel orientation on device performance was much weaker in monolayer HfS
2 FET owing to the degenerate CB valleys of monolayer HfS
2.
Kanazawa et al. first experimentally evaluated the FET performance of HfS
2 and revealed its potential for transistor channels [
8]. Although the other properties of HfS
2 were not thoroughly investigated, they also reported some basic characteristics of HfS
2 essential for FET device fabrication. Continuous efforts to seek other 2-D materials led to the discovery of anisotropic 2-D materials. Unlike Mo-based TMDs, HfS
2 is characterized by its highly anisotropic CB. Theoretical investigations of HfS
2 FETs have been performed [
5,
6,
7]. Anisotropic 2-D materials other than phosphorene, such as Hf-based-TMDs, have been explored with density functional theory (DFT) calculations [
9,
10]. Monolayers of Hf-based TMDs such as HfS
2 are quite different from those of Mo-based TMDs because of different crystal symmetry and atomic bonding. They have indirect bandgaps with three-fold degenerate conduction band (CB) valleys whose dispersions around the minimum CB are highly anisotropic. Among Hf-based-TMDs, monolayer HfS
2 exhibits sizable bandgaps, larger than 1 eV, suitable for nanoscale FET applications. In this study, we present a comprehensive computational study of the carrier mobility of monolayer HfS
2 based on the first principle method, compact band model, and Kubo–Greenwood mobility approach. We discuss the influence of the band structure anisotropy of monolayer HfS
2 on its carrier transport performance, as well as the strain engineering of monolayer HfS
2.
As a member of 1T-MX2, the carrier transport performance of monolayer HfS2 deserves to be further explored and improved. Although its carrier transport performance has been reported, few studies have investigated the effects of strain on this material. Strain effects on materials can tune its electronic band degeneracy and broaden its carrier transport properties. In the present study, we use DFT accompanied by Boltzmann transport theory and deformation potential theory to study the strain effect on the carrier transport properties of monolayer HfS2. The results indicate that the carrier effective mass is highly shrunken under the biaxial compressive strain reaches 4%. As a consequence, the carrier mobility of monolayer HfS2 reaches the highest value for both electron and hole. In this case, we surprisingly discovered that the highest mobility value was due to the carrier effective mass shrinking.
The effect of strain on the band structure and bandgap of monolayer HfS
2 has been investigated in previous studies [
11,
12], showing that the application of tensile and compressive biaxial strain results in the bandgap increasing and decreasing, respectively. Manouchehr Hosseini et al. studied the effect of strain on the mobility of monolayer MoS
2 and found that tensile biaxial strain could significantly enhance its mobility [
13]; however, an investigation of strain effect on the mobility of other 1T-TMDs such as HfS
2 was still missing. The conduction and valence band structures of HfS
2 are different owing to indirect bandgap and the conduction band and valence band having different effective masses.
In this study, we present a comprehensive analysis of the effect of biaxial strain on the mobility of monolayer HfS
2, employing the first principle method to perform band structure calculations and then the Kubo–Greenwood mobility approach for evaluating carrier mobility [
14,
15]. The methodology for the first principle method, compact band model, and Kubo–Greenwood mobility approach is briefly presented in
Section 2. The effects of biaxial strain on the band structure and mobility of monolayer HfS
2 are discussed in
Section 3, and the concluding works are presented in
Section 4.
3. Results and Discussion
In unstrained monolayer HfS
2, the lowest minima in the CB are denoted as M’ valley and M valley in
Figure 1b. One M’ valley and two M valleys are degenerate in the unstrained and biaxial strain conditions, respectively. However, upon the application of biaxial strain, they do not split into one M’ valley and two M valleys with different effective masses and energy minima. For the VB of the unstrained monolayer HfS
2, the top two bands are denoted as the HH band and LH band, respectively. Ab-initio band structure calculation using GGA with SOC. We further investigated the band structure close to the CB and VB via contour plot, which shows the constant energy lines as shown in
Figure 2. The CB resembles a valley with a parabolic elliptic energy dispersion, known as the M’ and M valleys. For the valence bands, the LH looks like an isotropic parabolic band while the HH is a six-fold symmetry band. The contours around the HH and LH bands are almost isotropic and circular at low energy, whereas the M’ and M valleys demonstrate anisotropic behavior. While the valence bands, including the HH and LH, show almost isotropic dispersion, a strong anisotropic one occurs in the CB around the M’ and M points owing to the difference in the orbital structures of the VB (p orbit of S atom) and CB (d orbit of Hf atom).
In
Figure 3, we present the DOS and velocity square of monolayer HfS
2, as obtained using the compact band models and ab-initio calculations. Our model fits very well with the ab-initio results, owing to the use of an energy-dependent effective mass function. Our model is better than the EMA model (which used a constant effective mass) for carrier mobility calculation.
The key parameters including the effective masses and non-parabolicity factors of the CB and VB compact models under biaxial strain are shown in
Figure 4. The effective mass and the non-parabolicity factor of the Rudenko model are extracted from the DFT-calculated band structure and reported in
Figure 4a and
Figure 4b, respectively. In the CB, two paths in the M’ valley have different effective masses. We denote the effective mass in the armchair direction (M’–Г) as
my0 and the effective mass along the zigzag direction (i.e., M’–K) as
mx0. As illustrated in the figure, the
my0 in the M’ valley is ~10× larger than
mx0. The non-parabolicity factor α
y is extremely small compared to α
x, and it slightly increases as tensile strain increases. The difference of two effective masses in the M’ valley increases as the strain increases, and the M’ valley exhibits more anisotropy under tensile strains. Meanwhile, the effective mass of the M’ valley decreases as the strain increases and becomes the lowest effective mass at a compressive biaxial strain of 4%. In
Figure 4c,d, the key parameters of the new compact band model are also extracted from the ab-initio valence band HH structure. In
Figure 4e,f, the key parameters of the new compact band model are also extracted from the ab-initio valence band LH structure. The effective mass decreases as biaxial compressive strain increases for the CB and VB models, while it increases as biaxial tensile strain increases. The hole effective mass was explored under various strains. Two HH and LH bands contribute to the major VB over most of the applied strain range. The HH and LH exhibit the lowest effective mass under compressive biaxial strain, and the top VB corresponds to the HH band in this strain range. The effective mass of the HH band increases as the biaxial strain changes from a compressive to the tensile regime.
Figure 5 illustrates the band structure of the unstrained and strained monolayer HfS
2, including the lowest M valley in the CB and the two highest VBs (HH and LH). The band structure is plotted for five strains: −4%, −2%, 0%, 2%, and 4%. The M valley HfS
2 is lower than the other valleys and located at the conduction band minimum (CBM). Note that the M valley is still dominant across all strain regions. As can be inferred from this figure, the strain can modify the shape and energy of each valley. Under biaxial strain, the CB and the VB of the edges shift upward and downward as strain increases, respectively. The energy distance between the M valley and the Г valley of unstrained material is evaluated to be ~720 meV, which is consistent with the values reported in References [
4,
11]. Biaxial compressive strain increases this energy distance, tensile strain decreases it. This implies that the effective mass along the M’–Г direction decreases with increasing biaxial compressive strain. For the VBs, the HH and LH energy splitting at the Г point is due to SOC not being sensitive to strain. In particular, a relatively large biaxial compressive strain shifts the band edge energy of the HH band so that it becomes the top VB, as shown in
Figure 5b. Here we can anticipate that, while under compressive strain, one can neglect the scattering between the HH band and the Δ
ГM and Δ
ГK bands, and, under tensile strain, this type of scattering can significantly degrade hole mobility.
Strain ranging from −4% to +4% is applied to the band structure of monolayer HfS
2 in
Figure 6. The bandgap of monolayer HfS
2 as a function of biaxial strain is shown in
Figure 6a. The bandgap of unstrained HfS
2 is 1.25 eV, and in agreement with References [
3,
6,
25,
26]. The strain dependence of the bandgap from our work is in agreement with References [
1,
11,
12]. It is well known that strain can modify band structure. In this study, we found that biaxial tensile strain led to an increase in the bandgap, and that biaxial compressive strain led to a decrease in the bandgap. Under the biaxial stress effect es of −4 to 4%, the bandgap of the system increases from 0.65 to 1.67 eV. In
Figure 6b, the absolute conduction band edge (
Ec) and the valence band edge (
Ev) for ab-initio calculations performed using GGA with and without SOC as a function of the biaxial strain.
Ec denotes the M valley and
Ev denotes the HH band. The energy at the bottom (top) of CB (VB) increases (decreases) with respect to strain increase. Both
Ec and
Ev almost change linearly with respect to biaxial strain, but the slope is opposite. SOC will caused an upward shift in the energy of
Ev. Moreover, the strain-induced modulation of
Ec was relatively smaller than that of
Ev.
Figure 6c shows the variation in energy difference between the Г and M valleys denoted as Δ
1, and the valence band difference between the HH band and the Δ
ГK band (Δ
3) and the HH–LH band splitting (Δ
2) under different biaxial strains, respectively.
According to Reference [
6], dynamic doping with a gate voltage of the 2D FET electrostatically induces a high carrier concentration and maintains the high carrier mobility of monolayer HfS
2. We used a carrier density of 6 × 10
12 cm
−2 in the following results for calculated carrier mobility.
Figure 7a shows the room temperature electron mobility without polar optical phonon (POP) scattering versus the electron density for the unstrained monolayer HfS
2 in two compact models. The calculated electron mobility calculated using the Rudenko model is lower than that using EMA model owing to the conduction band non-parabolicity effect. This result implies that the non-parabolicity effect is important and cannot be neglected in electron mobility calculation. The SOC causes almost no change in the CB according to the ab-initio GGA calculation, while the result in calculated electron mobilities identical to those obtained using the Rudenko model which fits the GGA ab-initio conduction bands with and without SOC. At low carrier concentrations, mobility is strongly limited by POP scattering. However, as the carrier concentration increases, the screening effect reduces the impact of POP scattering. As can be seen in
Figure 7b, the electron mobility with POP scattering increases with the electron density for monolayer HfS
2 because of screening, and our calculations are in agreement with imec’s work [
6]. Square symbol is imec’s calculated data from Reference [
6]. The biaxial strain dependency of the intrinsic phonon-limited mobility is presented in
Figure 8a. Enhancement factors due to −Δ
m/
m and −Δ
τ/
τ versus biaxial strain are shown in
Figure 8b.
Apparently, the effects of compressive and tensile strain on electron mobility are distinct, and this difference can be mainly be attributed to the roles of electron effective mass and scattering rate. For example, under biaxial compressive strain, the electron effective masses of the M’ and M valleys are much lower compared to the unstrained condition. A reduced equivalent effective mass
mc implies a decrease in the density of states. According to Equation (19), the scattering time increases owing to the reduced density of states, which in turn enhances electron mobility. Under biaxial tensile strain, however, the electron effective mass of the CB higher is higher than that of the unstrained one. With a biaxial compressive strain of 4%, the phonon-limited mobility becomes 90% higher than that of the unstrained material. In contrast, a biaxial tensile strain of 4% reduces the electron mobility by ~40% as compared to that under the unstrained condition, owing to the increase in both effective mass and scattering rate. Moreover,
Figure 8 indicates that the biaxial strain-induced electron mobility enhancement with compressive strain is larger than that with tensile strain.
Figure 9a shows the room temperature hole mobility without POP scattering versus hole density for the unstrained monolayer HfS
2 in two compact models. As can be inferred from these plots,, the hole mobility determined using the new compact band model is lower than that determined using the EMA model, owing to the valence band non-parabolicity effect. The non-parabolicity effect in the band structure alters the carrier dynamics within the band, which in turn affects carrier mobility and electrical conductivity. This implies that the non-parabolicity effect is important in hole mobility calculation and cannot be ignored. The enhanced screening effect of POP scattering weakens the electrostatic interactions between holes. As illustrated in
Figure 9b, the hole mobility with POP scattering increases with the hole density for monolayer HfS
2 because of screening, and our calculations are in agreement with the imec’s work [
6]. Square symbol is imec’s calculated data from Reference [
6]. Note that SOC strongly affects HH and LH valence band degeneracy at the Г point and causes valence band splitting. The calculated hole mobility calculated using the new compact model, which fits the GGA ab-initio valence band structure with SOC, is higher than the one without SOC owing to the HH and LH band splitting effect. Most holes are in the HH band in the GGA ab-initio valence band structure, and with effective mass shrinkage of the HH due to SOC. The findings from the calculations for hole mobility can be attributed to these trends, as shown in
Figure 9b. The biaxial strain-dependency of the intrinsic phonon-limited hole mobility change ratio is presented in
Figure 10a. The enhancement factors of hole mobility due to −Δ
mc/
mc and −Δ<
τ>/<
τ> versus biaxial strain are shown in
Figure 10b. The relationship between the hole mobility change ratio and the enhancement factors is included. Apparently, the effects of compressive and tensile biaxial strain on hole mobility are remarkably different, which can mainly be attributed to the role of hole effective mass and scattering rate. For example, under biaxial compressive strain, the effective masses of the HH and LH bands are much lower than in an unstrained condition, which enhances hole mobility. By contrast, under biaxial tensile strain, the hole effective mass of VB is higher than that of the unstrained one. In the biaxial strain range of −4% to 4%, hole inter-band scattering due to the HH band and satellite bands such as Δ
ГM and Δ
ГM is not serious and can be ignored. This is because high mobility generally means that carriers can traverse longer distances within the material (with a longer mean free path) without being affected by scattering. Consequently, the probability of carriers interacting with defects, lattice vibrations, or other scattering in the material is decreased. Moreover, when biaxial tensile strain is >4%, it increases the valence band degeneracy of monolayer HfS
2 even more, and the degenerate valence bands and higher inter-band phonon scattering rate reduce the hole mobility. Under biaxial compressive strain, the hole mobility increases because of the reduction in the hole effective mass and results in the reduction of the scattering rate. Under a biaxial compressive strain of 4%, the phonon-limited mobility increases by ~13% compared to that of the unstrained material. In contrast, a biaxial tensile strain of 4%, the hole mobility decreases by ~11% compared to that of the unstrained, owing to increases in the hole effective mass and scattering rate.