Broken Symmetry Effects due to Polarization on Resonant Tunneling Transport in Double-Barrier Nitride Heterostructures

The phenomenon of resonant tunneling transport through polar double-barrier heterostructures is systematically investigated by a combined experimental and theoretical approach. On the experimental side, $\mathrm{Ga}\mathrm{N}$/$\mathrm{Al}\mathrm{N}$ resonant tunneling diodes (RTDs) are grown by molecular beam epitaxy. In situ electron diffraction is used to monitor the number of monolayers incorporated into each tunneling barrier of the RTD active region. Using this precise epitaxial control at the monolayer level, we demonstrate exponential modulation of the resonant tunneling current density as a function of barrier thickness. At the same time, both the peak voltage and the characteristic threshold bias exhibit a dependence on barrier thickness as a result of the intense electric fields present within the polar heterostructures. To get further insight into the asymmetric tunneling injection originating from the polar active region, we present an analytical theory for tunneling transport across polar heterostructures. A general expression for the resonant tunneling current that includes contributions from coherent and sequential tunneling processes is introduced. After the application of this theory to the case of $\mathrm{Ga}\mathrm{N}$/$\mathrm{Al}\mathrm{N}$ RTDs, their experimental current-voltage characteristics are reproduced over both bias polarities, with tunneling currents spanning several orders of magnitude. This agreement allows us to elucidate the effect of the internal polarization fields on the magnitude of the tunneling current and broadening of the resonant tunneling line shape. Under reverse bias, we identify new tunneling features originating from highly attenuated resonant tunneling phenomena, which are completely captured by our model. The analytical form of our quantum transport model provides a simple expression that reveals the connection between the design parameters of a general polar RTD and its current-voltage characteristics. This new theory paves the way for the design of polar resonant tunneling devices exhibiting efficient resonant current injection and enhanced tunneling dynamics as required in various practical applications.

Figure 1

Different magnitudes of resonant tunneling current are engineered in $\mathrm{Ga}\mathrm{N}$/$\mathrm{Al}\mathrm{N}$ RTDs by systematically varying the thickness of the $\mathrm{Al}\mathrm{N}$ barriers. (a) The equilibrium conduction-band diagram ($E_C$) of the double-barrier active region of each sample is calculated with the use of a self-consistent Schrödinger-Poisson solver [47]. The energy of the ground-state inside the quantum well rises toward higher values with increasing barrier thickness. The inset schematically depicts the complete device structure, including the unintentionally doped (UID) spacer layers and doping levels of the contact regions. When the devices are biased, asymmetric current-voltage characteristics are observed. (b) Under forward bias, the characteristic resonant peak and negative differential conductance shift toward larger voltages for thicker $\mathrm{Al}\mathrm{N}$ barriers. Because of the strong tunneling attenuation, the magnitude of the peak current is modulated over 2 orders of magnitude when the barrier thickness is increased from 1.5 to 2.4 nm. The currents in (b) are scaled by different factors to facilitate direct comparison. (c) Under reverse bias, the polarization-induced threshold voltage ($V_{\mathrm{th}}$) exhibits a linear dependence on the barrier thickness. This result, consistent with theoretical predictions, allows measurement of the internal polarization fields [36]. The magnitude of this critical voltage is experimentally determined with the linear-interpolation procedure described in Sec. 2. The dashed lines indicate the linear fits, with $V_{\mathrm{th}}$ as the voltage intercept.

Figure 2

Quantum transport model for resonant tunneling injection across polar double-barrier heterostructures. The current-voltage characteristics of a $\mathrm{Ga}\mathrm{N}$/$\mathrm{Al}\mathrm{N}$ RTD are calculated using analytical expressions for the various tunneling current components. under reverse bias are displayed in panels (a)–(c). (a) When the device is biased at the threshold voltage $V_{\mathrm{th}}=-2t_b{F}_{\pi }$, the flatband configuration enhances the direct-tunneling-current component $J_{\mathrm{DT}}$, which becomes the dominant transport mechanism. At this critical voltage, both $\mathrm{Al}\mathrm{N}$ barriers sustain a finite electric field, which originates from the internal polarization charges. (b),(c) For voltages below the critical threshold bias (i.e., $V_{\mathrm{th}}<V_{\mathrm{bias}}<0$), transport is also mediated by resonant tunneling injection. In this regime, the large asymmetry between the tunneling barriers results in strongly attenuated resonant-tunneling-current components ($J_{\mathrm{RT}}^{(n)}$). Panel (d) displays each of each these tunneling-current components. Their sum, the total tunneling current $J_{total}$, is displayed by the dashed black line. (e) As a result of the internal polarization fields, polar RTDs exhibit a strong asymmetry between the single-barrier transmission probabilities. Under reverse bias the increasing asymmetry between $T_E^{(1)}$ and $T_C^{(1)}$ leads to a highly attenuated resonant tunneling current [See $J_{RT}^{(1)}$ in d]. In contrast, under forward bias, $T_E^{(1)}$ and $T_C^{(1)}$ become more symmetric, resulting in an enhancement of the maximum electron transmission $T_{max}^{(1)}$. Panels (f)–(h) show the RTD band diagrams at equilibrium and forward bias. (f) Under equilibrium conditions, a wide depletion region builds up next to the collector barrier and the ground-state energy rises approximately $1$ eV above the equilibrium Fermi level (dashed black line). (g) Under forward bias, the current is supported by resonant tunneling electrons injected across the double-barrier structure ($J_{\mathrm{RT}}^{(1)}$). The transparency of the active region increases as the tunneling transmission of each barrier becomes more symmetric. (h) Maximum resonant injection is reached at the resonant bias, when the ground-state eigenenergy aligns with the Fermi energy of the emitter reservoir. The resonant tunneling component $J_{\mathrm{RT}}^{(2)}$ injected through the first excited state is also calculated.

Figure 3

Comparison between experimental results and theoretical calculations obtained with our analytical resonant tunneling transport model. (a) Room-temperature $J$-$V$ characteristics as a function of the barrier thickness measured from the fabricated RTDs. (b) Theoretical $J$-$V$ characteristics calculated with Eq. (7) for different levels of incoherent tunneling transport, determined by the total width ${\mathrm{\Gamma }}_{e+i}$. A parasitic resistance ${\rho }_s$ is also included in series with the ideal RTD to capture the effects of the access and contact resistances [see the inset in (c)]. The values of ${\rho }_s$ for each diode are presented in Table (1). (c) Differential resistance under reverse-bias conditions for the RTD with $t_b=1.5$ nm. The arrows indicate regions of resistance modulation corresponding to the peak currents of the different resonant tunneling components. These features are completely captured by our analytical model, demonstrating good quantitative agreement. Under high reverse bias ($V_{\mathrm{bias}}<V_{\mathrm{th}}$), the differential resistance is controlled mainly by the series resistance ${\rho }_s$. The inset shows the equivalent-circuit model of the RTD including the parasitic series resistance.