Velocity saturation in La-doped BaSnO₃ thin films

Perovskite oxides continue to attract a lot of interest for fundamental and applied physics in view of the plethora of electronic properties they exhibit.^1,2 This also makes perovskites an ideal platform onto which diverse device functionalities can be integrated. However, the poor electron mobility of most perovskite oxides makes them a sub-optimal choice for electronic device applications. With a recorded room-temperature mobility of 320 cm²/V s in single-crystals,³ BaSnO₃ (BSO) is a notable exception to this rule. The cause for such high mobility in BSO thin films has been studied previously and attributed both to the low effective mass of electrons owing to the significant antibonding s-orbital-like character of the conduction-band minimum,⁴ and the low electron–phonon scattering rate due to the lower density of conduction band states.⁵ Such high mobilities in conjunction with the large and tunable doping densities (∼10²⁰ cm⁻³) achievable in this material, and the possibility of realizing heterostructures with carrier confinement make BSO promising from an electronic device perspective.⁶ Transistors fabricated from La-doped BaSnO₃ and SrSnO₃ channels (10–28 nm) are still in the early stages of development but have shown promising results compared to those based on other perovskite-oxides such as SrTiO₃ (STO).^7–9 However, in addition to the low-field transport properties such as mobility, the operation of electronic devices such as RF amplifiers, and highly-scaled transistors for logic, depends critically on high-field transport, i.e., saturation velocity of charge carriers in the high-mobility channel.^10,11 While low-field transport in BSO has been studied and modeled, there are no reports on high-field transport and the physical mechanisms limiting the saturation velocity in this material, despite its relative importance for device performance. Here, we report on the experimental measurement of saturation velocity in thin films of La-doped BSO and model its dependence on the dominant LO-phonon scattering process. We obtain a reasonable agreement between the theoretical values based on a simple optical phonon emission model and the measured saturation velocities, for an experimentally relevant range of doping densities in BSO thin films.

BSO films were grown via MBE with various La concentrations on insulating (100) SrTiO₃ (STO) and (110) DyScO₃ (DSO) substrates,¹² with and without BaTiO₃ (BTO) capping layers, as summarized in Table I. Due to the lattice mismatch between BSO and STO/DSO, all films are relaxed. For samples with a BTO capping layer, 20 nm of BTO was grown on the La-doped BSO film via hybrid MBE.¹³ The saturation velocity was measured on specially designed I-shaped test structures consisting of wide source and drain contact pads with a narrow mesa-defined channel of length L and width W [see Fig. 1(b)] connecting them. Such a structure not only confines the current flow and applied potential to the channel, but also reduces any extraneous voltage drop in the contact/access regions due to the large difference in areas between the channel and contact regions.¹⁴ The device fabrication consisted of i-line stepper lithography for all layers commencing with Ti/Al alignment markers, inductively coupled plasma-reactive ion etching (ICP-RIE) of the BTO cap layer using BCl₃/Ar, followed by ohmic contact deposition (Ti/Au, 50 nm/100 nm) using e-beam evaporation, and finishing with an ICP-RIE (BCl₃/Ar) mesa-etch of the BSO channel to define the I-shaped constrictions. Various constriction widths (2, 5 and 10 μm) and lengths (600 nm to 2 μm) were fabricated and tested on each sample. Pulsed I-V measurements (1 μs pulse width, 0.1% duty cycle) were performed using a Keithley 4200 semiconductor characterization system on the test structures. Contact resistances of 0.34, 0.49, and 0.3 Ω mm were measured on the three samples listed in Table I, respectively, using TLM test structures. The saturation velocity was extracted as v_sat = J_max/qn_s, where J_max and n_s are the measured maximum saturation current density (ampere per millimeter) and sheet carrier concentration (cm⁻²), respectively, and plotted as a function of electric field as shown in Fig. 2 for the sample with n_s = 2.65 × 10¹⁴ cm⁻² as an example (Fig. S2 of the supplementary material show electric field vs current density plots for the complete set of measured devices across all samples used in this study). Saturation current densities ranging from 5 to 50 A/mm were observed for the range of sample doping densities. We also observed device-device variation across each sample comparable to that shown in Fig. 2, which we attribute to the local doping/stoichiometric variations and structural defects in the deposited films. The critical thicknesses for strain relaxation in BSO films on STO and DSO substrates have been reported to be ∼1 nm leading to structural defects in these films (threading dislocation densities >10¹¹ cm⁻²).^15,16 In addition, pulsed I-V measurements were also performed for a subset of devices at 80 K. A comparison between the electric field vs current density plots at 300 and 80 K for two of the measured devices on the highest doping density sample is shown in Fig. 2(b). An increase in J_max (and hence extracted v_sat) of <18% can be observed for these two devices, which is comparable to the standard error of the mean (12%) measured across all devices at 300 K indicating that the current saturation is reflective of velocity and not mobility-limited. Figure S3 of the supplementary material also shows the electric field vs current density plot for the lowest doping density sample used in this study − n_s = 7.23 × 10¹³ cm⁻² showing <6% increase in J_max and v_sat at 80 K as compared to 300 K.

In order to explain the mechanism of velocity saturation in these films, it is important to note that strong LO-phonon induced carrier scattering occurs in such polar materials.^5,17 We performed first-principles calculations for the phonon spectrum and electron–phonon scattering rates in BSO based on the density functional theory (DFT)^18,19 and density functional perturbation theory (DFPT)²⁰ as implemented in the Quantum ESPRESSO code,²¹ with the generalized gradient approximation of Perdew-Burke-Ernzerhof for the exchange-correlation term.²² Optimized norm-conserving Vanderbilt pseudopotentials were employed.²³ The EPW code was used to interpolate the electron–phonon matrix elements and calculate the electron–phonon scattering rates.²⁴ The phonon spectra of BSO (Fig. S3 in the supplementary material) exhibits 15 branches with the three LO phonon branches (branches 6, 12, and 15) strongly affecting electron–phonon scattering for long-range Frölich interactions. This can be clearly seen in Fig. 3 which plots the scattering rate for the various phonon branches as a function of electron energy with reference to the Fermi level. It is evident that the total scattering rate rises steeply beyond electron energies of 120 meV which may be taken as the net effective LO phonon energy for electron scattering due to optical phonon emission in BSO.

Given the dominant optical phonon modes in BSO and strong LO-phonon induced carrier scattering shown above, the emission of LO-phonons by energetic carriers at the source is likely to act as the velocity saturation mechanism for BSO films. Such a model has been previously used to explain velocity saturation observed in GaN HEMTs, carbon nanotubes (CNTs), and layered materials.^25–28 To explain the model briefly, at high enough applied electric fields, it becomes energetically favorable for carriers injected into the channel from the source to emit LO phonons and backscatter into the source region. This causes a net balance of carriers for forward injection leading to current and hence velocity saturation. The key feature of this model is the existence of a density-dependent saturation velocity as illustrated in Fig. 4 for the case of a 2-D electron gas with carrier density lower than the crossover density (n₀) beyond which back-scattering into the bottom of the conduction band is Pauli blocked. At 0 K, the Fermi circle of radius k_F indicates the range of occupied states for a given electron density. For smaller densities, we see from Fig. 4 that the carriers can be accelerated to larger velocities (higher values of charge centroid, k₀) before the onset of optical phonon emission takes place (k_op = k_F + k₀) as compared to lower values for higher sheet carrier densities.

The total current in the device can be estimated by summing up all occupied energy levels in k-space and is given by J = qn_s(ħk₀/m^*) for electrons in a 2DEG, where m^* is the effective mass (0.2^*m₀ for BSO)²⁹ and the charge centroid k₀ is given by m^*ω_op/(2ħk_F) for n_s > n₀ and k₀ = k_op−k_F for n_s ≤ n₀.²⁸ This gives rise to a square-root dependence between the current density and sheet carrier concentration for high densities (n₀ > n_s) and since J = qn_sv_sat, v_sat decreases as n_s^−1/2 with increasing channel charge density. The present study, however, involved samples of fairly-thick La-doped BSO films (16–21 nm) which more closely resemble a 3D electron gas rather than a 2D case even after accounting for the depletion widths in these layers. The current density for 3-D carriers can be estimated by summing the occupied states across a Fermi sphere of radius k_F as

where n_0,3D (=k_op³/3π² = 1.68 × 10¹⁹ cm⁻³) is the crossover 3D density for which optical phonon emission to the bottom of the conduction band is not Pauli blocked and E_op is the optical phonon energy (120 meV as discussed above). The saturation velocity extracted (v_sat = J/qn_s) from these equations for the 3D case therefore has a n_3D^−1/3 dependence at higher carrier densities in contrast to the n_s^−1/2 dependence for 2D confinement which also leads to higher values of v_sat in comparison.

Figure 5 shows the predicted saturation velocity for a range of equivalent sheet carrier concentrations of the La:BSO 3DEGs along with the experimentally measured values for comparison (n_3D was estimated as n_s/t_La:BSO for each case as the depletion widths in the La:BSO layer are <2.5 and ∼5 nm, respectively, for the highest and lowest doping density films, and the error bars were extracted from the measured device-to-device variation on each sample, see Fig. 2(a) and S2 in the supplementary material). We see that the experimentally measured saturation velocity for a range of carrier concentrations shows a reasonable agreement with the predictions of the 3-D optical phonon model (at 0 K), indicating that optical phonon emission using an effective LO phonon energy of 120 meV is the likely cause for velocity saturation in La:BSO thin films. It can also be seen from Fig. 5 that the measured saturation velocity decreases for higher carrier densities as predicted by the model. The use of modulation-doped barriers to BaSnO₃ channels, to spatially separate out the 2DEG in the high-mobility channel from the impurity atoms in the barrier, is another promising approach that has been mooted for device applications.⁶ Hence, we also estimate the density-dependence of saturation velocity for the 2-D confinement case corresponding to δ-doped BSO channels. Since the carrier concentrations in La:BSO are typically much higher (>5 × 10¹³ cm⁻²) than those observed in other high charge density interfaces such as AlGaN/GaN, it is important to consider not just the occupancy of the first sub-band (the electrical quantum limit), but also that of higher order sub-bands which contribute to current densities and hence saturation velocity. In order to account for filling multiple sub-bands at realistic doping densities, a δ-doped BSO channel of thickness 5 nm was considered with confining barriers of 2 eV on either side, analogous to a finite potential well where each bound state represents a sub-band. The total current density was then obtained by summing up the contributions from each sub-band and the net saturation velocity is also shown in Fig. 5 (see the supplementary material for more details).

The saturation velocity for δ-doped BSO channels is estimated to drop from 1.8 × 10⁷ cm/s to a more or less constant value of 2 × 10⁶ cm/s for sheet carrier concentrations beyond 2 × 10¹⁴ cm⁻². It is evident that even though the saturation velocity drops with increasing carrier densities, the decrease is not as steep (proportional to n_s^1/2) as the corresponding rise in charge density (∼n_s and hence the current drive) of La:BSO thin film transistors. Hence, there is a definite motivation to have highly-doped BSO channels for high charge density switches with the modulation of such large charge densities being the primary challenge for device applications.

In conclusion, we have experimentally measured the saturation velocity of La:doped-BaSnO₃ thin films for a range of achievable doping densities. The experimental values show a good agreement with the predictions of a LO-phonon emission model with an effective phonon energy of 120 meV, indicating the likelihood of optical phonon emission being the mechanism limiting velocity saturation in La:BSO thin films. Delta-doped BSO channels exhibit a density-dependent saturation velocity decreasing proportional to n_s^−1/2 from 1.8 × 10⁷ cm/s to 2 × 10⁶ cm/s at sheet carrier densities ranging from 10¹³ to 2 × 10¹⁴ cm⁻², accounting for the contribution of multiple sub-bands necessary for such high carrier densities. The estimated and measured saturation velocities presented in this work would aid in informed device design for electronics based on thin film BSO channels.

See the supplementary material for SEM images of representative constriction structures fabricated on each sample, the electric field vs current density plots for the complete set of devices measured across all three samples used in this study along with the tabulated data, electric field vs current density plots at 80 and 300 K for the sample with n_s = 7.23 × 10¹³ cm⁻², the calculated phonon spectra of BaSnO₃, and the details of the optical phonon model for the 2-D case with and without sub-band contributions.

The authors acknowledge funding from the DARPA DREaM program (No. ONR N00014-18-1-2034, Program Manager Dr. Young-Kai Chen, monitored by the Office of Naval Research, Program Manager Dr. Paul Maki), the Office of Naval Research Grant No. N00014-18-1-2704 (Program Manager Dr. Brian Bennett), and Semiconductor Research Corporation Grant Nos. 2018-NC-2761-B and NSF ECCS-1740119. T.W. and A.J. acknowledge support from the National Science Foundation Faculty Early Career Development Program Grant No. DMR-1652994, and the Extreme Science and Engineering Discovery Environment (XSEDE) facility, National Science Foundation Grant No. ACI-1053575.