1. Introduction

Indium antimonide has the highest electron mobility and saturation velocity of any known semiconductor, and is therefore a promising material for ultra-high-speed devices^{[1, 2]}. InSb quantum-well (QW) transistors formed by AlInSb/InSb heterostructure have been under the spotlight recently due to their remarkable potential in low power/voltage and high-speed circuits^[3]. It has been observed that to achieve the same performance as the state-of-the-art silicon MOSFETs, InSb QW-FETs consume about 10% in power owing to extremely high electron mobility (as high as 30000 cm$^{2}$/(V$\cdot$s)) and saturation velocity (5.107 cm/s) in the channel region of the III-V semiconductors, InSb has the lightest effective mass and narrowest band gap along with a large dielectric constant ($\varepsilon$) and an exceptionally large Lande $g$-factor of $\sim$$-51$ in bulk and as large as $\sim$$-60$ in quantum well heterostructures^[4]. These properties make it an exciting candidate for many different applications including high speed electronics and spintronics. Hence InSb heterostructure is chosen for high speed electronics, where significant advancements have been made in both n- and p-channel field effect transistors (FET) for digital and analogue applications, quantum and spintronics applications where the large $g$-factor may give rise to all electron spin manipulation and ballistic transport devices where the large ballistic length may give rise to ultra sensitive magnetic field sensors^[5]. These have been widely investigated in III-V semiconductors such as GaAs, but development in InSb-based systems is not so far progressed. AlInSb/InSb high electron mobility transistor (HEMT) devices are considered to be very promising candidates for high-speed and high-power applications^[6]. These devices offer an advantage such as high breakdown voltage, high charge density and good electron mobility. The formation of the 2-D electron gas (2DEG) in these devices is the heart during the device operation. For accurate and fast simulation of circuits based on these devices, an analytical expression for 2DEG density $n_{\rm s}$ is required. The expression should be also physics based, which obviates the use of a large number of empirical parameters. The currently available models for charge density are primarily based on numerical calculations, semi empirical model expressions, or simplifying approximations^{[7, 8, 9]}. Models relying on numeric calculations will be slow because of the iterative nature. Semi empirical models are fast but do not provide the necessary insight into the device operation. They also tend to have a large number of empirical parameters, which have to be extracted from experimental data^{[10, 11]}. In contrast to these approaches, we have obtained an accurate physics-based analytical model for charge density and Fermi level in the AlInSb/InSb HEMT device with minimal empirical parameters in this paper. For this purpose, the sheet charge density equation of the 2DEG region is used and it is assumed that the AlInSb layer is completely ionized in the device.

The rest of the paper is organized as follows: Section 2 presents the schematic device structure of the AlInSb/InSb HEMT device. Section 3 deals with the model formulation of charge density and Fermi level for the AlInSb/InSb HEMT device. Also, the unified charge density model for the AlInSb/InSb HEMT device is discussed in Section 4. Results and the discussions are presented in Section 5. The conclusion is given in Section 6.

2. Device structure for AlInSb/InSb HEMT

The cross-sectional view of the AlInSb/InSb HEMT device for two different experimentally determined positions of the 2DEG channel in the device structure is shown in Figure 1. The layer sequence, from bottom to top, is InSb-undoped/UID-AlInSb/n-AlInSb/metal, with 2DEG channel formed in the interface between InSb-undoped/UID-AlInSb.

A doped AlInSb barrier layer saves the carriers from Coulomb scattering caused by ionized impurities and provides carriers a channel at a zero gate voltage. The device thus works in a depletion mode with carriers confined in the InSb quantum well. The device structure is also known as a modulation doped field-effective transistor (MODFET)^{[12, 13]}. The quantum well formed of a ternary compound with mole fraction of Al$_{0.3}$In$_{0.7}$Sb/InSb HEMT is used. It has been demonstrated that when the mole fraction of Al increases to 30%, the electron wave function is perfectly confined within the well. Figure 2 shows the sub bands energy levels $E_{\rm 1}$, $E_{\rm 0}$ along with the conduction band in the 2DEG quantum well.

3. Development of 2DEG charge density model

The difficulty in modeling the 2DEG charge density arises from the complicated variation of $E_{\rm f}$ with $n_{\rm s}$ in the quantum well^[14]. This relationship is given by

$$\begin{split} n_{\rm s}={}& DV_{\rm th} \left[ \ln \left( \exp \frac{E_{\rm f}-E_0}{V_{\rm th}}+1 \right)\right. \\[2mm]& \left. + \ln \left(\exp \frac{E_{\rm f}-E_1}{V_{\rm th}}+1 \right) \right], \end{split}$$

(1)

where $E_0=\gamma_0 n_{\rm s}^{2/3}$ and $E_1=\gamma_1 n_{\rm s}^{2/3}$ are the levels of the two lowest subbands, $E_{\rm f}$ is the Fermi level expressed in volts, and $V_{\rm th}=kT/q$ is the thermal voltage.

$$n_{\rm s}=DV_{\rm th}\left[\ln \left( \exp \frac{E_{\rm f}-E_0}{V_{\rm th}}+1 \right)\right].$$

(2)

Assuming that the AlInSb layer is completely ionized, we can write

$n_{\rm s}=\frac{\varepsilon}{qd(v_{\rm g0}-E_{\rm f})},$

(3)

where $v_{\rm go}=v_{\rm g}-v_{\rm off}$, and $v_{\rm off}$ are the cutoff voltage. The variation of $n_{\rm s}$ with respect to $v_{\rm g}$ is a complicated transcendental function. Hence, we propose a few assumptions in order to simplify these equations, which allow us to develop a precise analytical expression for $n_{\rm s}$ versus $v_{\rm g}$ for the typical operating range of gate voltages and temperatures and drain current of the AIInSb/InSb device.

Figure 3 shows important quantities $E_{\rm f}$, $E_0$ and $E_1$ as obtained from numerical calculations based on Equations (1) and (2) for a wide range of gate voltages from close to $v_{\rm off}$ and above. It is interesting to note the relative positions of the sub band levels when compared with $E_{\rm f}$ in this range. The charge density increases linearly with the energy bands. Upper-level $E_1$ is larger than $E_{\rm f}$ for the complete range of gate voltages. Hence contribution to the charge density from the second sub band can be safely ignored. Here $E_0$ have two distinct regions namely region I and region II. In region I, $E_0$ is larger than $E_{\rm f}$ corresponding to $V_{\rm go}$ < 0.9 V. In region II, $E_0$ is lower than $E_{\rm f}$ corresponding to $V_{\rm go}$ $>$ 0.9 V. We consider the full-scale range of gate voltages except very close to cutoff. The Fermi level equation is derived as follows for the regions I and II.

The second term can be ignored in Equation (2) for the calculation of Fermi's level^[14]. This is written in a simplified form as,

$$\frac{n_{\rm s}}{DV_{\rm th}}={\rm e}^{\frac{E_{\rm f}-E_0}{v_{\rm th}}}.$$

(4)

Taking ln on both sides

$$\ln \frac{n_{\rm s}}{DV_{\rm th}}=\frac{E_{\rm f}-E_0}{v_{\rm th}}.$$

(5)

By solving we get the equation for Fermi's level as follows,

$$E_{\rm f}=\gamma_0 n_{\rm s}^{2/3}+v_{\rm th}\ln \frac{n_{\rm s}}{DV_{\rm th}}.$$

(6)

3.1 Charge density model in region I

By considering the full-scale range of gate voltages, except very close to cutoff, $E_{\rm f}$ will be much smaller than $v_{\rm go}$. The model for $n_{\rm s}$ is separately developed in regions I and II. These models are then combined to give a unified model covering both regions.

The Fermi level in region I is given by

$$E_{\rm f}^{\rm I}=\gamma_0 n_{\rm s}^{2/3}+v_{\rm th}\ln \frac{n_{\rm s}}{DV_{\rm th}}.$$

(7)

It is important to point out here that, for vertically scaled devices with $d$ < 5 nm, the 2DEG distance from the interface, i.e., $\delta d$, will become comparable with $d$. In such cases, $d$ should be replaced by $d_{\rm eff} = d + \delta d$ in the model expressions.

3.2 Fermi level and sheet carrier density for region I

The Fermi level formulation for region I is obtained by substituting the charge density equation $n_{\rm s}$ to the general Fermi level equation as follows,

$$\gamma_0 \left[ \frac{\varepsilon}{qd}(v_{\rm go}-E_{\rm f}) \right]^{2/3}+v_{\rm th} \ln \left[ \frac{\varepsilon}{Dv_{\rm th}qd}(v_{\rm go}-E_{\rm f}) \right]=E_{\rm f}^{\rm I}.$$

(8)

Also substituting for $C_{\rm g}= \varepsilon /d,$ and $\beta$ $=$ $C_{\rm g}/qDV_{\rm th}$ and solving we get Fermi's level in region I as,

$$E_{\rm f}^{\rm I}=v_{\rm go}\frac{\gamma_0\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}+v_{\rm th} \ln \beta v_{\rm go}} {v_{\rm go}+v_{\rm th}+\dfrac{2}{3}\gamma_0\left( \dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}}.$$

(9)

Using Fermi level in region I we can derive the sheet carrier density equation in region I. Assuming that the AlInSb layer is completely ionized, we can write,

$$n_{\rm s}=\frac{\varepsilon}{qd}(v_{\rm go}-E_{\rm f}).$$

(10)

Substituting Fermi level in region I in the above equation

$$n_{\rm s}=\frac{\varepsilon}{qd} \left[ v_{\rm go}- v_{\rm go}\frac{\gamma_0\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}+v_{\rm th} \ln \beta v_{\rm go}} {v_{\rm go}+v_{\rm th}+\dfrac{2}{3}\gamma_0\left(c_{\rm g}v_{\rm go}/q \right)^{2/3}}\right],$$

(11)

$$n_{\rm s}=\frac{\varepsilon v_{\rm go}}{qd} \left[1- \frac{\gamma_0\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}+v_{\rm th} \ln \beta v_{\rm go}} {v_{\rm go}+v_{\rm th}+\dfrac{2}{3}\gamma_0\left( \dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}}\right],$$

(12)

$$n_{\rm s}=\frac{c_{\rm g} v_{\rm go}}{q} \left[1- \frac{\gamma_0\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}+v_{\rm th} \ln \beta v_{\rm go}} {v_{\rm go}+v_{\rm th}+\dfrac{2}{3}\gamma_0\left( c_{\rm g}v_{\rm go}/q \right)^{2/3}}\right].$$

(13)

We develop an analytical model for $n_{\rm s}$ versus gate voltage in region I, the sheet carrier density in region I is given by,

$$n_{\rm s}^{\rm I}=\frac{c_{\rm g} v_{\rm go}}{q} \frac{ \left[ v_{\rm go}+v_{\rm th}(1-\ln \beta v_{\rm go})-\dfrac{\gamma_0}{3}\left( \dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3} \right] } {v_{\rm go}+v_{\rm th}+2 \dfrac{\gamma_0}{3} \left( \dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}}.$$

(14)

Charge density model in region II

The relationship approximate expression for $n_{\rm s}^{\rm II}$ and it is given by,

$$n_{\rm s}^{\rm II} \approx D \left(E_{\rm f}^{\rm II}-E_0 \right),$$

(15)

$$\frac{n_{\rm s}^{\rm II}}{D} = E_{\rm f}^{\rm II}-E_0,$$

(16)

$$\frac{n_{\rm s}^{\rm II}}{D}+E_0 = E_{\rm f}^{\rm II}.$$

(17)

Since the AlInSb layer is completely ionized, we can substitute $n_{\rm s}$ in the above equation.

3.4 Fermi level and sheet carrier density for region II

The Fermi level formulation for region II is obtained by substituting the charge density equation to the general Fermi level.

$$\frac{1}{D} \left[ \frac{\varepsilon}{qd}(V_{\rm g0}-E_{\rm f}) \right]+\gamma_0n_{\rm s}^{2/3}=E_{\rm f}^{\rm II},$$

(18)

$$\frac{c_{\rm g}}{qD}[(v_{\rm go}-E_{\rm f})]+\gamma_0\left[\frac{\varepsilon}{qd}(V_{\rm g0}-E_{\rm f}) \right]^{\frac{2}{3}} =E_{\rm f}^{\rm II},$$

(19)

By solving the above equation, we get the Fermi level in region II.

$$E_{\rm f}^{\rm II}=\frac{v_{\rm go}\beta v_{\rm go}v_{\rm th}+\dfrac{\gamma_0}{3}\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}} {v_{\rm go}(1+\beta v_{\rm th})+\dfrac{2}{3}\gamma_0\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}}.$$

(20)

Since the AlInSb layer is completely ionized, we can use $n_{\rm s}$ equation for substitution of Fermi level in region II. Substituting Fermi's level $E_{\rm f}$ of region II in Equation (10),

$$n_{\rm s}= \frac{\varepsilon}{qd} \left[v_{\rm go}- \frac{v_{\rm go}\beta v_{\rm go}v_{\rm th}+\dfrac{\gamma_0}{3}\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}} {v_{\rm go}(1+\beta v_{\rm th})+\dfrac{2}{3}\gamma_0\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}} \right],$$

(21)

$$n_{\rm s}= \frac{\varepsilon v_{\rm go}}{qd} \left[1- \frac{\beta v_{\rm go}v_{\rm th}+\dfrac{\gamma_0}{3}\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}} {v_{\rm go}(1+\beta v_{\rm th})+\dfrac{2}{3}\gamma_0\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}} \right].$$

(22)

The sheet carrier density in region II is given by

$$n_{\rm s}^{\rm II}= \frac{c_{\rm g} v_{\rm go}}{q} \frac{v_{\rm go}-\dfrac{\gamma_0}{3}\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}} {v_{\rm go}(1+\beta v_{\rm th})+\dfrac{2}{3}\gamma_0\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}}.$$

(23)

4. Unified charge density model

In order to find a unifying expression for $E_{\rm f}$ and $n_{\rm s}$ applicable for the full-scale range of gate voltages, we observe that the respective expressions for regions I and II differ only in one term to the numerator and one term to a denominator, both with a factor of $V_{\rm th}$. Hence, both the regional models are expected to provide a fairly reasonable approximation for full scale within the voltage range, except near cutoff. To improve accuracy, we propose to use a unified form that combines the two regional models. Specifically the expression for $n_{\rm s}$ is given by

$n_{\rm s}= \frac{c_{\rm g} v_{\rm go}}{q} \frac{v_{\rm go}+v_{\rm th}\left[1-\ln \beta v_{\rm go}-\dfrac{\gamma_0}{3}\left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}\right] } {v_{\rm go} \left(1+\dfrac{v_{\rm th}}{v_{\rm god}} \right) +\dfrac{2}{3}\gamma_0 \left(\dfrac{c_{\rm g}v_{\rm go}}{q} \right)^{2/3}},$

(24)

Here, $V_{\rm gon}$ and $V_{\rm god}$ are functions of $V_{\rm g}$ given by the interpolation expression.

$V_{\rm gox}=\frac{V_{\rm go}\alpha_x}{\sqrt{V_{\rm go}^2+\alpha_{\rm x}^2}},$

(25)

where $\alpha_{\rm n}=\frac{e}{\beta}$ and $\alpha_{\rm d}=\frac{1}{\beta}$. We observe that the unified model has the proper limitation for charge density of regions I and II. A similar unified expression can be also easily established for the Fermi level by applying the unified charge density in the fundamental charge density equation.

5. Results and discussion

In this model, we have discussed the AlInSb/InSb HEMT devices with two different regions for charge density, by considering the various temperatures, Fermi level regions and validated with experimental data. The model and experimental data comparisons are done for the Al$_{0.3}$In$_{0.7}$Sb/InSb HEMT devices by accounting corresponding permittivity and thickness are 16$\varepsilon_0$ and 20 nm with carrier densities of 3.28 $\times$ 10$^{15}$~m$^{-2}$ and also considering the parameters $\gamma_0$, $\gamma_1$ and cutoff voltage.

Figure 3 shows the Fermi level of first sub band and second sub band with the gate voltage variation. The parameters used are given in Table 1. The charge density increases linearly with the energy bands. Upper-level $E_1$ is larger than $E_{\rm f}$ for the complete range of gate voltages. Hence contribution to the charge density from the second sub band can be safely ignored. Here $E_0$ have two distinct regions namely regions I and II. In region I, $E_0$ is larger than $E_{\rm f}$ corresponding to $V_{\rm go}$ < 0.9 V. In region II where $E_0$ is lower than $E_{\rm f}$ corresponding to $V_{\rm go}$ $>$ 0.9 V. We consider the full-scale range of gate voltages except very close to cutoff. Figure 4 shows the relative positions of Fermi levels with the first sub band and second sub band with the gate voltage. The Fermi level of the first sub band will not increase with charge density, but the conduction pattern of the triangular quantum well of the second sub band will cause the charge density to increase linearly with applied gate voltage.

Figures 5 and 6 show the comparison of charge density model for region I with thermal voltage for different values of gate voltage. From this plot, we can say that, the first sub band energy level increases with respect to gate voltage, which causes the energy level less than 0.9 V in region I and greater than 0.9 V in region II.

The observations are shown in Figures 5 and 6 such that the charge density for electron decreases while accounting for the thermal voltage and increases by leaving the thermal voltage respectively. The model results are validated with experimental data.

Figures 7 and 8 show the comparison of the charge density model for region II with thermal voltage for different values of gate voltage. From this plot, we can say that, second sub band energy level decreases with respect to gate voltage, which causes the energy level greater than 0.9 V in region I and less than 0.9 V in region II. The observations are shown in Figures 6 and 7 such that the charge density for electron decreases while accounting for the thermal voltage and increases by leaving the thermal voltage respectively. The model results are validated with experimental data.

Figure 9 shows the comparison of a unified model for charge density of AlInSb/InSb with the gate voltage. It is applicable for the full-scale range of gate voltage, we observe that the respective expressions for regions I and II differ only in one term to the numerator and one term to a denominator both with a factor of thermal voltage. Hence both the regional models are expected to provide a fairly reasonable approximation for full-scale voltage range, except near cutoff. For achieving accuracy, we relate the unified two regional models expression. Here the function of gate voltage is to conduct electron for on and off states. Electrons in region I perform quite well with respect to region II, electrons in region II models performs relatively poorly in region I, with a rapid increase in full-scale gate voltage.

Figure 10 shows the comparison of charge density with different temperature versus the gate voltage ($T=$ 200, 300, 400, 500 K). From the graph, we conclude that the InSb operates at high power at elevated temperatures. In Figure 9, the unified model for charge density is compared with experiment data for different temperatures up to 500 K. It is apparent from the results that the model agrees quite well with the experimental data throughout the temperature range. The increase in error is found to be almost linear with increasing temperature for bias point gate voltage is 0.2 V.

6. Conclusion

In this work, a compact model has been derived for the AlInSb/InSb devices and to vary the structural parameters of the device by considering the variation of Fermi level ($E_{\rm f}$), the first subband $E_0$, the second subband $E_1$ and sheet carrier charge density ($n_{\rm s}$) with applied gate voltage ($V_{\rm g}$) and compare the performance from the device with various temperatures. The results generated by this model agree with the experimental results.