Base Transit Time in Abrupt GaN/InGaN/GaN and AlGaN/GaN/AlGaN HBTs

Abstract

Base transit time, τ_b, in abrupt npn GaN/InGaN/GaN and AlGaN/GaN/AlGaN double heterojunction bipolar transistors (DHBTs) is reported. Base transit time strongly depends not only on the quasi-neutral base width, but also on the low field electron mobility, μ_n, in the neutral base region and the effective electron velocity, S_c, at the edge of base-collector heterojunction. μ_n and S_c are temperature-dependent parameters. A unity gain cut-off frequency of 10.6 GHz is obtained in AlGaN/GaN/AlGaN DHBTs and 19.1 GHz in GaN/InGaN/GaN DHBTs for a neutral base width of 0.05um. It is also shown that non-stationary transport is not required to study τ_b for neutral base width in the range of 0.05um for GaN-based HBTs.

Introduction

Wide bandgap group III-nitride semiconductors are currently being pursued for possible high temperature and high power applications. Current gain as high as 10⁵ is reported for GaN/SiC HBT [1]. In order to investigate high frequency performance the behavior of the base transit time, τ_b , needs to be investigated. Mohammad et al. [2] has reported the dependence of τ_b on base doping concentration in a graded GaN/InGaN HBT.

The double integral formulation of τ_b by Kroemer [3] for HBTs is based upon the assumption that excess minority carrier concentration at the edge of base-collector depletion layer is negligible. Roulston [4] emphasized upon the use of a finite carrier velocity (saturation velocity) at the edge of the base-collector depletion region, thereby, introducing the component of base transit time due to velocity saturation. A more general formulation for carrier velocity at b-c heterojunction was used by Hafizi et al. [5]. Jahan et al. [6], based upon a self-consistent calculation of thermionic and tunneling components of the total current, proposed a tunneling factor for the determination of carrier velocity at the b-c junction. In this paper, the method formulated by Jahan et al. [7] is used in the determination of the effective electron velocity at base-collector junction which affects the electrons transport across base-collector junction.

In this paper, τ_b in abrupt GaN/InGaN/GaN and AlGaN/GaN/AlGaN HBT is reported. The computation of τ_b includes the effects of bandgap narrowing, carrier saturation and partitioning of the total current into thermionic and tunneling components. Results obtained from an ensemble Monte Carlo simulation are used in the determination of low field mobility, μ_n.

Theory

Base transit time, τ_b , can be expressed as a sum of τ_{b 1} + τ_{b 2}, where

$${\tau _{b1}} = \int_0^{{W_B}} {{{{n_{ib}}^2\left( x \right)} \over {{N_{AB}}\left( x \right)}}\int_x^{{W_B}} {{1 \over {{D_n}\left( y \right)}}{{{N_{AB}}\left( y \right)} \over {{n_{ib}}^2\left( y \right)}}dydx} } $$

((1))

and

$${\tau _<Emphasis Type="BoldDoubleUnderline"></Emphasis>} = {1 \over {{S_c}}}{{{N_{AB}}\left( {{W_B}} \right)} \over {{n_{ib}}^2\left( {{W_B}} \right)}}\int_0^{{W_B}} {{{{n_{ib}}^2\left( x \right)} \over {{N_{AB}}\left( x \right)}}dx} $$

where N_AB (x) is the base doping concentration, n_ib ² (x) = n_ie ² exp(Δ _g/kT) is the effective intrinsic carrier concentration in base region. The effective bandgap narrowing across the emitter-base heterojunction, Δ _g (In_xGa_1−xN)=x·E_g(InN)+(1−x)·E_g(GaN)−x· (1−x)E_gb (E_gb=1.0 eV)[8] where x is the In-mole fraction in In_xGa_1−xN and n_ie is the effective intrinsic carrier concentration in emitter region.τ_{b 1} is the component of the base transit time due to diffusion in the neutral base region and τ_{b 2} accounts for the finite base-collector junction velocity, S_c. The effective minority carrier velocity, S_c, characterizing electron transport across the space-charge region at the base-collector junction is formulated as S_c = v_th ·γ·exp[(qv_jp −ΔE_c)/kT] [9], where ΔE_cis the conduction band discontinuity, v_jp is the applied voltage drop at the collector, γ is the tunneling transmission factor, and v_th is the thermal velocity. The factor γ is determined by invoking the proper partitioning of the total current into tunneling and thermionic components [7]. WKB method is used to compute the transmission probability required in calculating the tunneling component of the total current.

Results and Discussions

5 × 4 μ m² single finger npn GaN/In_xGa_1−xN and Al_0.2Ga_0.8N HBTs are investigated. The material parameters used in the simulation are as follows: m_e(GaN)=0.2m₀, m_h(GaN)=0.6m₀ [10], m_e(InN)=0.115m₀ [11], m_h(InN)=1.6m₀ [12], m_e(AlN)=0.314m₀, m_h(AlN)=0.71m₀ [13], E_gGaN(T)=3.503+5.08 × 10⁻⁴ × T²/(T-996) eV, E_gInN(T)=2.01−1.8 × 10⁻⁴ ×T eV, E_gAlN(T)=6.118−1.799 × 10⁻³T²/(T+1462) eV [14], ε _GaN=9.5ε _0, ε _InN=9.50 ε ₀ and ε _AlN==8.5ε ₀ [13] _where ε is static dielectric constant, m₀ is electron rest mass, T is absolute temperature (K), and ε ₀ is permittivity in vacuum. Emitter and collector doping concentrations equal 5 × 10¹⁷ cm⁻³. The base is 0.05um wide and is doped uniformly 10¹⁹ cm⁻³. The conduction band offset, ΔE_c, is assumed to be 75% of the difference in bandgaps of the constituting semiconductor alloys [15].

In Fig. 1(a) the low field mobility, μ_n, and the diffusion coefficient, D_n , are plotted as a function of temperature for In_0.2Ga_0.8N and GaN at a doping concentration of 10¹⁹ cm⁻³. The low field mobility data is obtained from an ensemble Monte Carlo simulation which accounts for piezoelectric, ionized impurity, alloy, intervalley, acoustic and polar optical phonon scattering [16].The diffusion constant D_n=kT/q·μ_n ·[F _1/2 (ζ_n)/F _−1/2 (ζ_n)] where ζ_n = (E_fn −E_c)/kT is the reduced Fermi level for electrons and F _{±1 2} (ζ_n) is the Fermi-Dirac integral. D_n initially increases followed by gradual decrease beyond T=200K. In Fig. 1(b) the base-collector electron junction velocity, S_c, is plotted as a function of base-collector bias, V_bc, for varying In-mole fraction with temperature as a parameter. Due to a smaller band offset at the b-c junction S_c approaches thermal velocity at lower V_bc for x=0.1 as compared to that at x=0.2. The behavior of S_c for AlGaN/GaN/AlGaN is similar to that of GaN/InGaN/GaN.

Fig. 1(a)

Low field mobility _μn and diffusion coefficient D_n for In_0.2Ga_0.8N and GaN as a function of temperature at a doping concentration of10¹⁹ cm⁻³. Solid triangles represent D_n and μ_n for GaN

Fig. 1(b)

The effective electron velocity at base-collector junction, S_c , versus V_bc for In_xGa_1−xN/GaN.

In Fig. 2(a) the base transit time τ _b is plotted as a function of V_be for V_bc = 3.0 V. τ _b decreases with increasing In-mole fraction in GaN/In_xGa_1−xN/GaN HBTs, irrespective of temperature and this behavior can be explained with the aid of Fig. 1. A higher S_C at x=0.1 results in a lower τ _b2 as compared to that at x=0.2 for V_bc = 3V. The magnitude of τ _b1 at x=0.1 is slightly greater than that at x=0.2. This may be attributed to a slightly higher low field mobility at x=0.1 that results in a lower diffusion constant as compared to that at x=0.2. The relative magnitudes of τ _b1 and τ _b2 reflects the role of the transport processes namely drift-diffusion versus thermionic field emission. As seen from Fig 2(b), the diffusion controlled component of the base transit time τ _b1 is always the dominating time constant at elevated temperatures. However, at higher D_n at 300K results in a lower τ _b1 and a higher τ _b2. Or in other words, at 300K the contribution of thermionic field emission related component of the base transit time τ _b2 becomes significant. τ _b for Al_.2Ga_.8N/GaN HBTs at V_bc =3.0 V is controlled by τ _b2 and is due to the small electron effective velocity.

Fig. 2(a)

τ_b is plotted as a function of V_be at various temperature for GaN/In_xGa_1−xN HBTs with base doping concentration of 10¹⁹ cm⁻³, V_bc=3V. Base width equals 0.05um. The inset shows τ_b for Al_0.2Ga_0.8N/GaN DHBT. Lines represent x=0.1 and lines with open triangles represent x=0.2.

Fig. 2(b)

Shows τ_{b 1} and τ_{b 2} for GaN/In_0.2Ga_0.8N DHBT. Lines represent τ_b 1 and lines with open squares represent τ_{b 2} .

The dependence of τ_b upon base width for a base doping concentration of 10¹⁹ cm⁻³ at room temperature is shown in Fig.3. V_be=2V and V_bc=3V are assumed. As expected, τ_b increases with increasing base width. For extremely narrow base widths, the dominant component of τ_b is τ _b2. As compared to In_xGa_1−xN-based HBTs, GaN-based HBTs have higherτ_b due to lower low field mobility and lower effective electron velocity at base-collector junction. The above calculation is performed using stationary transport. The validity of the above approximation is shown in Fig. 4, where the average velocity is plotted as a function of distance. As noted, the average velocity remains constant over a base width variation from 0.02μm to 0.8μm for both GaN and In_.2Ga_.8N in the presence of an applied field of 500KV/cm. For comparison a similar plot for GaAs is also shown where the applied electric field is 10KV/cm. For GaAs the effect of non-stationary transport is clear and for base widths less than 1τm assuming stationary transport data becomes questionable.

Fig. 3

τ_b is plotted as a function of base width with In-mole fraction as a parameter for GaN/In_xGa_1−xN and Al_0.2Ga_0.8N/GaN HBTs (solid triangles). A base doping of 10¹⁹ cm⁻³, T=300K, V_be=2.0V and V_bc=3.0V are considered.

Fig.4

Average velocity is a function of distance at high electric field for GaAs, GaN, and In_0.2Ga_0.8N.

In Fig. 5, the unity gain cutoff frequency f_T=1(2 · π · τ_ec) is plotted as a function of collector current density, J_c, where τ_ec = τ_e + τ_b + τ_c is the total transit time from emitter to the collector.τ_e is emitter charging time, expressed as τ_e = r_e·C_jewhere r_e is the dynamic resistance and C_je is the base-emitter junction capacitance.τ_c = R_c· C_jcis the collector charging time where R_c is the collector resistance and C_jc is the collector junction capacitance. For collector current densities below 100 A/cm² τ _e dominant time constant, however for current densities above 100 A/cm² τ _b dominates. An f_T of 19.1 GHz is obtained at 500K for GaN/In_.2Ga_.8N/GaN DHBT. A higher low field mobility and b-c junction velocity in Al_0.2Ga_0.8N/GaN DHBT produce a f_T of 10.6 GHz at 300K.

Fig. 5

f_T is shown as a function of collector current density at various temperatures for GaN/In_0.2Ga_0.8N and Al_0.2Ga_0.8N/GaN HBTs. Base width equals 0.05um with N_AB=10¹⁹ cm⁻³. Solid lines represent T=500K and dashed lines represent T=300K

Conclusion

Base transit time in abrupt GaN/InGaN/GaN and AlGaN/GaN/AlGaN HBTs are determined by accounting for bandgap narrowing, carrier degeneracy and the proper low field mobility. A narrow base width with a consequently large base resistance is recommended along with a lower In-mole fraction to realize superior unity gain current cut-off frequency, f_T .f_T=10.6 GHz at T=500K for Al_0.2Ga_0.8N/GaN HBT and f_T = 19.1 GHz at T=300K for GaN/In_0.2Ga_0.8N have been demonstrated.