We report on device physics modeling, assisted by experimental measurement results, for all-electronic transduction of resonant electromechanical motion of nanoscale β-Ga2O3 vibrating channel transistors (VCTs), including both direct readout of high frequency transmission current (without frequency down-conversion) and frequency modulation (FM) down-conversion techniques. The β-Ga2O3 VCTs under consideration have fundamental mode resonance frequencies at ∼25 MHz–1 GHz, with quality factors (Qs) of ∼100–5000 and transconductance gm at ∼0.2 nS–5 μS. In analysis of signal transduction with varying device parameters, the transistor's gate trench depth z0 and transconductance gm play key roles in improving the electromechanical coupling and device performance. Reduction of trench depth z0 can engender a major enhancement in readout current. Reducing channel thickness h and increasing channel length L can improve the readout current while downshifting the resonance frequency at the same time. We design β-Ga2O3 VCT with a suspended modulation doped field effect transistor structure for efficient operation in the GHz range. This study paves the way for future engineering of all-electronic transduction and integration of high frequency β-Ga2O3 resonators on chip with β-Ga2O3 electronics and optoelectronics.

Beta gallium oxide (β-Ga2O3), a semiconductor with an ultra-wide bandgap (UWBG, Eg ≈ 4.8 eV)1,2 has recently attracted increasing interest for future miniature yet high-voltage power and radio frequency (RF) electronics,3–7 thanks to its high internal critical breakdown strength (ℰbr = 8 MV cm−1 predicted,8 and ℰbr= 5.7 MV cm−1 measured9). β-Ga2O3 field effect transistors (FETs) with lateral channel structures have been demonstrated to fully utilize the potential of the β-Ga2O3 crystal for miniaturization and high frequency operation. Among the demonstrated lateral β-Ga2O3 FETs, on current up to 580 mA/mm,10 on-off ratio up to ∼1010 range,6 subthreshold swing (SS) as low as 72 mV/dec,11 and cutoff frequency up to 27 GHz (Ref. 12) have been demonstrated. In parallel, the UWBG of the β-Ga2O3 crystal offers a photon absorption edge (270–280 nm) right at the cutoff wavelength (280 nm) of the solar-blind regime, perfectly appealing for solar blind ultraviolet (SBUV) optoelectronics.13–16 Beyond electronic and optoelectronic properties, β-Ga2O3 also exhibits excellent mechanical properties (Young's modulus EY = 261 GPa).17 The ensemble of compelling attributes in the β-Ga2O3 crystal makes it well suited for new UWBG resonant micro/nanoelectromechanical systems (M/NEMS) with electromechanical coupling and electrical tunability, thus can enable physical sensors and RF functions to integrate with, and greatly supplement, the rapidly emerging β-Ga2O3 electronics and optoelectronics. β-Ga2O3 nanomechanical devices have been recently demonstrated,17 with their SBUV light sensing capabilities tested through optical detection schemes.18,19 However, only limited effort has been initiated in coupling the electrical and mechanical properties of the β-Ga2O3 crystal into devices.20β-Ga2O3 FET with suspended, movable channel can be a simple yet highly efficient structure in realizing a β-Ga2O3 vibrating channel transistor (VCT), a device that can integrate RF mechanical motion with electrical output current through electrostatic and transistor gating effects, providing pronounced electromechanical coupling. Further, the development of such devices can be fueled by the surge of the β-Ga2O3 lateral FET research for enhanced performance, on-chip integration, and wafer-scale manufacturing.

In this Letter, we analyze the all-electronic transduction (i.e., both excitation and detection) of the mechanical motion of β-Ga2O3 VCT by both directly probing the transmission current output (without frequency conversion or mixing) from the β-Ga2O3 VCT and using frequency modulation (FM) down-mixing techniques. We use the derived models to evaluate the response of a β-Ga2O3 VCT by employing both analytical calculations and Cadence simulations. In addition, we study the device parameters that can enhance the all-electronic transduction of β-Ga2O3 resonant NEMS. The results can provide guidelines for developing future β-Ga2O3 resonant NEMS with efficient electromechanical transduction capabilities on chip.

Figure 1(a) illustrates a β-Ga2O3 suspended channel transistor on an insulating substrate with a local gate. This structure provides low parasitic capacitance and features the benefit of individual device addressability. To fabricate such devices, we start with an insulating sapphire substrate and deposit bottom local gate electrode (2 nm titanium and 23 nm platinum). Then, we grow SiO2 (300 nm) and Al2O3 (22 nm) dielectric layers and expose the local gate area by plasma etching to define microtrenches. The typical microtrench depth z0 is around 300 nm (for the first batch of such devices). Afterward, top source/drain (S/D) electrodes are deposited. We transfer mechanically cleaved β-Ga2O3 nanoflakes onto the predefined substrate to form a suspended structure across the trench. Finally, additive deposition of S/D electrodes is conducted to further ensure electrical contact and mechanical clamping.20 

FIG. 1.

(a) Schematic illustration of the β-Ga2O3 VCT established on an insulating substrate with a local gate. (b) Crystal structure of β-Ga2O3 with orientation corresponding to the β-Ga2O3 thin film in (a). (c) Signal transduction diagram for electronically reading out the motion of β-Ga2O3 VCT overlaid on the cross-sectional view of the device. Measurement schemes: (d) direct readout of current without frequency down-conversion; (e) frequency modulation (FM) down-mixing. TIA is the transimpedance amplifier, vRF is the magnitude of the RF signal, ϕ(t) = ωt + (ωΔ/ωLF)sin(ωLFt), ωΔ is the frequency deviation, and ωLF is the low-frequency (LF) modulation.

FIG. 1.

(a) Schematic illustration of the β-Ga2O3 VCT established on an insulating substrate with a local gate. (b) Crystal structure of β-Ga2O3 with orientation corresponding to the β-Ga2O3 thin film in (a). (c) Signal transduction diagram for electronically reading out the motion of β-Ga2O3 VCT overlaid on the cross-sectional view of the device. Measurement schemes: (d) direct readout of current without frequency down-conversion; (e) frequency modulation (FM) down-mixing. TIA is the transimpedance amplifier, vRF is the magnitude of the RF signal, ϕ(t) = ωt + (ωΔ/ωLF)sin(ωLFt), ωΔ is the frequency deviation, and ωLF is the low-frequency (LF) modulation.

Close modal

To detect the motion of the β-Ga2O3 resonator through current output of the VCT, a DC gate voltage VG and a DC drain voltage VD are applied to set the operating point of the transistor. In addition, an RF signal vg is imposed onto the gate to electrostatically drive the resonant motion of the suspended channel. Figure 1(c) depicts the intricate signal transduction paths in transducing resonator motion into output current. First, with oscillating gate voltage vg, the carrier density n = n0 + δn in the VCT channel is altered. In parallel, varying vg also modulates the displacement z = z0 + δz, inducing the gate-channel capacitance change Cch = Cch0 + δCch and thus changes n. Then, the change of strain ε = ε0 + δε in the channel induced by z modifies the bandgap Eg = Eg0 + δEg of the crystal, and therefore also alters n. Variation of both n and ε can contribute to an electron mobility μ = μ0 + δμ. The combined effect modulates the channel conductance G = G0 + δG. Finally, alternating current output I + i translated from G upon applied constant VD can be drawn from the source electrode. Thus, device motion can be resolved from current i. Direct readout of the transmission current [Fig. 1(d)] and FM down-mixing techniques [Fig. 1(e)] are both investigated for motion transduction of β-Ga2O3 VCT.

Unless otherwise noted, we first use a set of experimentally verified [from the device in Fig. 2(c)], realistic device parameters to model the β-Ga2O3 VCT in this study. The suspended channel has a length of L = 10 μm, width of w = L/3, thickness of h = 350 nm, gate microtrench depth of z0 = 300 nm, and a built-in stress of σ = 50 MPa. Such parameters provide a fundamental mode resonance frequency ω1/2π ≈ 25 MHz. We assign a quality factor Q = 100 and a transconductance gm = 0.2 nS to the device as typical values. The device operates at VG = −30 V and VD = 10 V.

FIG. 2.

Direct electronic readout of motion using β-Ga2O3 VCT with local gate structure. Schematic illustration of capacitances for β-Ga2O3 suspended channel transistors with (a) global gate and (b) local gate structures. (c) Microscope image of a β-Ga2O3 VCT with local gate. Calculated current readout of β-Ga2O3 VCT for (d) gm = 0.2 μS with global gate, and (e) gm = 0.2 nS with local gate. (f) Measured transfer characteristics of the β-Ga2O3 suspended channel transistor in (c) in vacuum. (g) Scaling of gm with respect to z0. (h) Calculated i and (i) iz from β-Ga2O3 VCT with different device parameters.

FIG. 2.

Direct electronic readout of motion using β-Ga2O3 VCT with local gate structure. Schematic illustration of capacitances for β-Ga2O3 suspended channel transistors with (a) global gate and (b) local gate structures. (c) Microscope image of a β-Ga2O3 VCT with local gate. Calculated current readout of β-Ga2O3 VCT for (d) gm = 0.2 μS with global gate, and (e) gm = 0.2 nS with local gate. (f) Measured transfer characteristics of the β-Ga2O3 suspended channel transistor in (c) in vacuum. (g) Scaling of gm with respect to z0. (h) Calculated i and (i) iz from β-Ga2O3 VCT with different device parameters.

Close modal

The motion of the β-Ga2O3 resonator is driven electrostatically by the applied vg. The vibrational displacement can be expressed as

δz=CchVGvRFMeff1ωn2ω2+jωnω/Q,
(1)

where Cch=dCch/dzεpA/z02 is the displacement derivative of gate-channel capacitance CchεpA/z0, εp is the permittivity, A = w × L is the gate-channel coupling area, vRF is the signal magnitude of vg, Meff and ωn are the effective mass and frequency of the nth resonance mode, respectively, and ω is the sampling frequency.

To read out the motion of the β-Ga2O3 resonator directly without frequency conversion or mixing, we applied vg = vRFsin(ωt) to the gate [Fig. 1(d)]. The RF current that can be probed from the source terminal of the device consists of four components,21 

i=jωCtotvRFjωCchVGδzz0+gmvRFgmVGδzz0,
(2)

where Ctot is the total capacitance of the device, including Cch and parasitic capacitance Cpara = CGD + CGS, and the transconductance of the suspended channel transistor gm = dI/dVG. The first two terms are introduced by capacitive effects, icap = (CtotvRFCchVGδz/z0), and the last two terms depend on field effect in the transistor, iFE = gm(vRFVGδz/z0). The mechanical oscillation of the device contributes to the second and fourth terms, so mechanical motion induced current iz = (–jωCchVGgmVG)δz/z0.

The suspended channel transistor with global gate [Fig. 2(a)] is the most straightforward way to realizing β-Ga2O3 VCT. However, the parasitic capacitance of such device could contribute significant background current [jωCtotvRF in Eq. (2)], dominating over the current introduced by device motion. Given a typical parasitic capacitance for global gate, i.e., CtotCpara ≈ 4 pF (two 150 μm × 100 μm bonding pads with 290 nm SiO2 dielectric), and using a reasonable boosted gm = 1 μS (estimated using gm = μεpwVD/(Lz0), based on the highest measured mobility μ ≈ 100 cm2/(V·s) reported for β-Ga2O3 MOSFETs7,22,23), the device resonance is still completely overwhelmed by, and hidden in, the background current [Fig. 2(d)]. To overcome this, we design and study the β-Ga2O3 VCT with a local gate structure [Fig. 2(b)]. The fabricated device shows clear switching behavior with gm ≈ 0.2 nS in vacuum [Fig. 2(f)]. We can estimate that the typical total capacitance for the local gate structure is Ctot = Cpara + Cch ≈ 2 fF (Cpara obtained from simulation in COMSOL Multiphysics 5.5, Cpara = 1 fF now for the local gate configuration, ∼4000 times smaller than the earlier case of global gate), then the predicted current spectrum of the fundamental resonance for the typical β-Ga2O3 VCT is shown in Fig. 2(e). The signal-to-background ratio (SBR) is significantly improved by using the local gate structure [from Figs. 2(d) to 2(e)]. According to Eq. (2), the output current i can be modulated by gm and z0 can change gm. Figure 2(g) demonstrates the scaling of gm by sweeping z0.

While maintaining the same resonance frequency, we can modulate the output current by varying the gm and z0, and gmz0−1. The results show that capacitive effect contributes more to i than the effect of gm does [Fig. 2(h)], i.e., halving z0 introduces more significant change in i than increasing gm by three orders of magnitude can attain. Same trend can also be found in iz [Fig. 2(i)].

FM down-mixing scheme is often used to convert signal at high frequency to related signal at lower frequency for enhanced sensitivity in detection,24e.g., a lock-in amplifier (LIA, Stanford Research SR830) has a noise floor of 0.13 pA/√Hz at 1 kHz and 0.013 pA/√Hz at 100 Hz.

To conduct FM down-mixing measurement, instead of using pure sinusoidal signal, a frequency modulated voltage is applied through the gate of the device, vg = vFM(t) = vRFsin(ϕ(t)), as shown in Fig. 1(e). Then, we can define the instantaneous frequency: ωi = ∂ϕ(t)/∂t = ω + ωΔcos(ωLFt). The β-Ga2O3 resonator acts as a mixer to down convert the motion signal at ω to a LF current iLF at ωLF. We can derive iLF,

iLF(VG+12vRF)gmωΔz0Re[δz]ωi,
(3)

where Re[δz] is the real part of device displacement [Eq. (1)] and

dRe[δz]dωi=CchVGvRFMeff2ωi(ωi2ωn2ωn2/Q)(ωi2ωn2+ωn2/Q)[(ωi2ωn2)2+(ωiωn/Q)2]2.
(4)

(See supplementary material for detailed derivation.) Using Eqs. (3) and (4), we can plot the iLF for devices with different parameters [Fig. 3(a)]. It follows the similar trend as that of the non-mixing output current i (or iz) from VCT, where reducing z0 [Fig. 3(b)] provides better enhancement of iLF than increasing gm does [Fig. 3(c)]. For example, reducing z0 from 500 nm to 50 nm engenders four orders of magnitude enhancement on iLF. Although, increasing gm to 1 μS still introduces significant iLF enhancement to 1.14 μA at resonance [Fig. 3(c)]. We can also boost iLF by increasing Q of the device [Fig. 3(d)]. For in-vacuum operation in tens of MHz range, the resonator Q is limited by thermoelastic damping (TED), clamping loss, surface dissipation, and other mechanisms. Through simulations (COMSOL Multiphysics 5.5), the Qs, if only limited by TED and clamping loss, respectively, are in the range of 60 000–100 000. Meanwhile, doubly clamped β-Ga2O3 resonators from previous work have exhibited Qs of 870–1660 (Ref. 19), well above the Q measured in this work. Thus, the Q of the current device is likely limited by surface and other damping processes, which clearly warrant further studies and may be minimized through thermal annealing and surface engineering for future devices.

FIG. 3.

Response of the β-Ga2O3 VCT using FM down-mixing technique. (a) Calculated iLF spectra with different gm and z0. Contribution of (b) z0 reduction, (c) gm increase, and (d) Q boost to the enhancement of iLF at resonance (insets are in log –log scale). Color plots of iLF resonance spectra at various VG from (e) measurement and (f) analytical model, and iLF resonance spectra with varying vRF from (g) measurement and (h) analytical model.

FIG. 3.

Response of the β-Ga2O3 VCT using FM down-mixing technique. (a) Calculated iLF spectra with different gm and z0. Contribution of (b) z0 reduction, (c) gm increase, and (d) Q boost to the enhancement of iLF at resonance (insets are in log –log scale). Color plots of iLF resonance spectra at various VG from (e) measurement and (f) analytical model, and iLF resonance spectra with varying vRF from (g) measurement and (h) analytical model.

Close modal

We measure a fabricated β-Ga2O3 VCT [Fig. 2(c) with transfer characteristics shown in Fig. 2(f)] using the FM down-mixing technique. Panels 3(e) and 3(g) of Fig. 3 show the spectra of iLF with different VG and vRF. The L, w, and h of the device are 7.4 μm, 2.4 μm, and 350 nm, respectively. Due to nonidealities in fabrication, the length of the suspended region is ∼10.2 μm, resulting in a fundamental mode resonance frequency ω1/2π ≈ 26.3 MHz. The z0 is around 300 nm and the measured Q is around 82. Using these parameters, Eqs. (3) and (4) model the response of the device very well [Figs. 3(f) and 3(h)]. The discrepancy in magnitude of the current between measurement and analytical model can be attributed to the contact resistance between the electrode and the β-Ga2O3 crystal, which is not included in the analytical model due to its complexity.

In favor of RF applications, a higher frequency of β-Ga2O3 VCT is desired, specifically operation in the GHz range. The resonance frequency of the β-Ga2O3 resonator can be tuned through varying the size and mode number of the resonator. Such variation can also modify response current in both direct high frequency readout and FM down-mixing schemes. We can estimate the resonance frequency of nth flexural mode of the doubly clamped structure using the equation25 

ωn=[π(n+1/2)]2L2EYIρwh1+0.97σwhL2(n+1)2π2EYI,
(5)

where n is the mode number, I = wh3/12 is the moment of inertia, EY = 240 GPa (measured from a different device19) and ρ = 5950 kg/m3 are the Young's modulus and the mass density of β-Ga2O3, respectively, and σ is the built-in stress of the suspended channel. We can scale ωn by changing the n and sweeping the h and L of the suspended channel [Figs. 4(a) and 4(b)]. The ωn of the β-Ga2O3 doubly clamped resonator scales with ha (0 ≤ a ≤ 1) and with Lb [−2 ≤ b ≤ −1, and b ≈ −2 for the L range in Fig. 4(b)]. For a 10 μm long (w = L/3) device, the fourth out-of-plane flexural mode resonance reaches 1 GHz when h ≈ 1.7 μm. Based on Eqs. (2)–(5), the variation of h and L (w scales along with L to keep the top-view aspect ratio of the device) could also contribute to the change of current response of the β-Ga2O3 VCT [Figs. 4(c) and 4(d)]. Opposite to the frequency scaling, the response current scales as izhp (−2 < p < −1) and iLFhq (−4 < q < −1) with varying thickness h, and scales as izLr and iLFLs [r ≈ 4 and s ≈ 6 for the L range in Fig. 4(d)] to length L variation. As such, the trade-off is that higher resonance frequency comes with the cost of lower output current. In addition, the higher mode number contributes to more dramatic reduction in iLF than in iz. To mitigate the effect of lowered current at higher frequency, we envision β-Ga2O3 VCT in suspended modulation-doped FET (MODFET, also known as a type of high-electron-mobility transistor, HEMT) structure. Therefore, the degradation of gating effect of the transistor caused by the thick channel can be reduced, thanks to the two-dimensional electron gas (2DEG) close to the bottom of the suspended structure [Fig. 4(e)]. Based on concurrent results on β-Ga2O3 MODFETs [μ up to 180 cm2/(V·s)],26 we estimate a gm = 5 μS for a L = 6 μm, w = 2 μm, h = 1 μm, and z0 = 100 nm β-Ga2O3 device with suspended MODFET. The resonance frequency of the fourth out-of-plane resonance mode of this device is ω4 = 2π × 1.62 GHz. Based on this device, we sweep the h and L of the resonator and calculate ωn and iz for both the first and the fourth modes [Figs. 4(f) and 4(g)]. When the resonance frequency is around 1 GHz, the MODFET device generates an iz ≈ 110 nA, which is around two orders higher than the values for normal β-Ga2O3 VCT [Figs. 4(c) and 4(d)].

FIG. 4.

Scaling of resonance frequency and output current of multimode resonances. (a) Scaling of ωn with varying h. Curves are with built-in stresses of 5, 50, and 500 MPa, respectively, for each mode. (b) Linear and log –log (inset) scaling of ωn with respect to L. (c) and (d) Scaling of iz and iLF with respect to h and L, respectively, for first and fourth modes. (e) Schematic of β-Ga2O3 VCT with suspended MODFET structure. (f) and (g) Scaling of ωn and iz with respect to h and L, respectively, for the first and fourth out-of-plane flexural modes of a β-Ga2O3 VCT with suspended MODFET structure (with gm = 5 μS).

FIG. 4.

Scaling of resonance frequency and output current of multimode resonances. (a) Scaling of ωn with varying h. Curves are with built-in stresses of 5, 50, and 500 MPa, respectively, for each mode. (b) Linear and log –log (inset) scaling of ωn with respect to L. (c) and (d) Scaling of iz and iLF with respect to h and L, respectively, for first and fourth modes. (e) Schematic of β-Ga2O3 VCT with suspended MODFET structure. (f) and (g) Scaling of ωn and iz with respect to h and L, respectively, for the first and fourth out-of-plane flexural modes of a β-Ga2O3 VCT with suspended MODFET structure (with gm = 5 μS).

Close modal

Equivalent circuit modeling of electromechanical devices plays crucial roles in evaluating the device operation in complex circuits, in simulating their performance when interfacing with electronic devices, and toward electronic design automation (EDA) for large scale integration and manufacturing. We develop the small signal equivalent circuit model of the β-Ga2O3 VCT and simulate the circuit in Cadence design tool. Figure 5 shows the equivalent circuit and comparison of i between Cadence simulation and analytical calculation. Given the results for the device in Fig. 2(c), the current from capacitive branch of the circuit icap contributes to the majority of i [Figs. 5(c) and 5(d)]. The field-effect current iFE can gain dominance by significantly boosting gm (= 1 μS) of the transistor [Fig. 5(e)], which could be achieved by using suspended MODFET structure.

FIG. 5.

Analytical model and Cadence simulation of the β-Ga2O3 VCT. (a) Small signal equivalent circuit model used for Cadence simulation, with gate-channel capacitance Cch, parasitic capacitances CGD and CGS, motional components Cm = η2/keff, Rm=keffMeff/(Qη2), and Lm = Meff/η2, where keff is the effective spring constant and η = CchVG/z0, current in capacitive branch icap, and field effect current iFE = gm(vRFδzVG/z0). (b) Comparison of i by analytical model and Cadence simulation. (c) Magnitude of icap and (d) iFE from analytical model and Cadence simulation, respectively. (e) iFE dominates in the VCT with boosted gm = 1 μS.

FIG. 5.

Analytical model and Cadence simulation of the β-Ga2O3 VCT. (a) Small signal equivalent circuit model used for Cadence simulation, with gate-channel capacitance Cch, parasitic capacitances CGD and CGS, motional components Cm = η2/keff, Rm=keffMeff/(Qη2), and Lm = Meff/η2, where keff is the effective spring constant and η = CchVG/z0, current in capacitive branch icap, and field effect current iFE = gm(vRFδzVG/z0). (b) Comparison of i by analytical model and Cadence simulation. (c) Magnitude of icap and (d) iFE from analytical model and Cadence simulation, respectively. (e) iFE dominates in the VCT with boosted gm = 1 μS.

Close modal

In conclusion, we have demonstrated the measurement and analysis for electrical readout of β-Ga2O3 VCTs. The analysis suggests that increasing transconductance gm, reducing microtrench depth z0, enhancing the quality factor Q, and thinning and elongating the suspended channel can improve the all-electronic transduction although some of these are at the cost of lowered resonance frequency ωn. Among these methods, z0 reduction provides the most efficient improvement of readout current without affecting ωn. GHz operation can be achieved by using higher order resonance modes, and the MODFET structure can be used to compensate output current reduction at increased frequency. With future optimization, direct all-electronic transduction of high frequency β-Ga2O3 resonators, without FM mixing and downconversion, shall be achievable in experimental measurements. With proper impedance matching, such devices can be integrated with rapidly emerging β-Ga2O3 electronics and optoelectronics for sensing and signal processing applications.

See the supplementary material for characterization of β-Ga2O3 suspended channel transistor, detailed derivation of equations on low frequency current analysis in the frequency modulation (FM) down-mixing scheme, and discussions on frequency scaling.

We thank the financial support from the Defense Threat Reduction Agency (DTRA) Basic Scientific Research Program (Grant No. HDTRA1-19-1-0035).

The data that support the findings of this study are available within the article and its supplementary material.

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