The hot-electron effect on electron transport in nominally undoped ZnO epilayers is studied at electric fields up to $430\xb150$ kV/cm applied with 3 ns voltage pulses. The transient measurements do not demonstrate any change in the electron density at the fields up to 320 kV/cm. The deviation from Ohm’s law can be approximated by the linear dependence of the current on the electric field valid at the moderate fields (from 50 to 250 kV/cm). The model calculations based on the Boltzmann kinetic equation are used to demonstrate that the differential mobility at the moderate fields is almost independent of the scattering on the charged point defects. The electron drift velocity is estimated from the experimental values on the differential conductivity and the current density together with the calculated differential mobility as a reference for calibration. The drift velocity reaches $2.7\xb10.3\xd7107$ cm/s at the field of $\u223c320$ kV/cm and approaches the theoretical limit predicted by the known Monte Carlo simulations.

## I. INTRODUCTION

Zinc oxide (ZnO) is a wide bandgap semiconductor^{1} that has already found successful applications when a p-n junction is not required, such as transparent electrodes for light-emitting devices and solar cells, transparent thin-film transistors, and others.^{2–4} The transistor high frequency performance strongly depends on electron transport and scattering at elevated electron energies achievable at high electric fields (hot-electron effects). The expected high theoretical values of the electron drift velocity in wurtzite ZnO^{5} endorse efficient operation of ZnO-based transistors at centimeter- and millimeter-wave frequencies.^{6}

The hot-electron effect on current is often treated in terms of the dependence of the electron drift velocity on the applied electric field, $vdr(E)$ curve. The conduction band model is a prerequisite for predictions of the hot-electron effects. The model for wurtzite ZnO takes into account the lowest $\Gamma 1$ valley and the upper satellite valleys located 4.4 eV and 4.6 eV above at $\Gamma 3$ and at U points, respectively. The corresponding electron effective masses of 0.17, 0.42, and 0.70 m$e$ are assumed.^{5} The nonparabolic dependence of the energy on the momentum is taken into account in the lowest valley. Other authors^{7,8} consider the mass tensor in the $\Gamma 1$ valley.

According to Monte Carlo simulations,^{5,7} a second linear $vdr(E)$ dependence appears at moderate fields (from 50 to 250 kV/cm) at room temperature. The velocity reaches the peak value of^{5} $3.2\xd7107$ cm/s or^{7} $2.2\xd7107$ cm/s at $\u223c270$ kV/cm field and decreases to the saturation value at higher fields. The simulations show that the intervalley transitions are rare: even for an applied electric field of 1000 kV/cm, the lowest $\Gamma 1$ valley remains 92% populated.^{9} More electrons stay in the $\Gamma 1$ valley if a nonparabolic dependence of the electron energy on the momentum is taken into account. The stronger nonparabolicity shifts the velocity peak toward higher electric fields.^{10} The peak value diminishes if the thermal bath temperature is increased. The saturation velocity remains almost independent of the doping^{10,11} except for doping levels exceeding $1019$ cm$\u22123$.

The combined scattering on charged centers (point defects) and phonons takes place at low electric fields.^{12–14} The measured low-field electron mobility value of 400 $cm2$/(V s) is achieved in nominally undoped epilayers at room temperature;^{15} it approaches the calculated one^{10} for the electron density of 10$17$ cm$\u22123$. The effect of charged centers weakens at the moderate fields where the second linear $vdr(E)$ dependence develops. According to the simulations,^{10} the differential mobility is almost independent of the center densities in the range from $1015$ to $1018$ cm$\u22123$ at room temperature.

While there are many reports on the Monte Carlo simulations of hot electron effects in ZnO,^{5,7–11,16} the related experimental studies remain scarce.^{17–19} In particular, the ZnO epilayers grown on sapphire did not withstand the applied electric field above $\u223c150$ kV/cm at room temperature.^{19} The highest value for the drift velocity of $1.5\xd7107$ cm/s was obtained at a field of $\u223c100$ kV/cm at room temperature for the Ga-doped ZnO epilayers with the electron density of $\u223c1.4\xd71017$ cm$\u22123$. The same epilayers demonstrated a weak hot-phonon effect caused by optical phonon accumulation.^{18}

This paper discusses the velocity–field dependence for nominally undoped ZnO epilayers at room temperature. The study aims at measuring the hot-electron effects at high electric fields, including those above 150 kV/cm where the experimental data have been absent. The field strength of $430\xb150$ kV/cm is reached with 3 ns pulses. Transient measurements help to make sure that the electron density is independent of the electric field. The measured current–field dependence is interpreted in terms of the electron drift velocity under the assumption of a uniform electric field. Scattering on charged point centers is taken into account in addition to scattering on acoustic and optical phonons. The Boltzmann kinetic equation is solved within the spherical harmonics expansion (SHE) approach for interpretation of the experimental results; an excellent fitting of the estimated drift velocity is obtained at low and moderate electric fields.

## II. SAMPLES

The wurtzite ZnO epilayers were grown by plasma-assisted molecular beam epitaxy (PA-MBE) on highly-resistive (60 k$\Omega $mm) carbon-compensated 2.5 $\mu $m-thick Ga-polar (0001) GaN templates prepared by metal-organic chemical vapor deposition (MOCVD) on c-plane sapphire substrates. The low-temperature (LT) nucleation layer (20 nm in thickness) was deposited at $300\xb0$C, and the high-temperature ZnO structures were grown at $670\xb0$C substrate temperature. The oxygen gas flow rate of 1.6 sccm and 400 W plasma power led to the resultant growth rates of 115 nm/h for ZnO sample #1153 and 135 nm/h for #1161; the epilayers were grown to thicknesses $d$ of 350 nm and 375 nm, respectively. More details on the substrate preparation (including the deposition of the LT-ZnO nucleation layer), the ZnO epilayer growth, and the structural and morphological analysis can be found elsewhere.^{20}

The transmission line model (TLM) patterns were processed with evaporated Ti/Au (25 nm/30 nm ) stacks acting as Ohmic contacts. The channel width was 300 $\mu $m and the interelectrode distances $L$ were 1.7, 3.9, 6.9, and 9.9 $\mu $m. The contact resistance of $Rc=23\xb110\Omega $ was estimated at low electric fields from the dependence of the sample resistance on the interelectrode distance. The transfer lengths and sheet resistances were obtained as 1.0 $\mu $m and $6720\Omega /\u25a1$ for sample #1153 and 1.2 $\mu $m and 5530 $\Omega /\u25a1$ for sample #1161. The channel resistance, the length, and the cross-sectional area were used to estimate the low-field conductivity $\sigma 0$ (Table I). The electron Hall mobility was measured for as-grown wafers with soldered In contacts in the Van der Pauw configuration. The electron magnetoresistance mobility is measured for the TLM structures.

Wafer | #1153 | #1161 |

Epilayer conductivity σ_{0} | 4.7 S/cm | 5.5 S/cm |

Electron magnetoresistance mobility | 188 cm^{2}/(V s) | 146 cm^{2}/(V s) |

Electron drift mobility μ_{0} | 154 cm^{2}/(V s) | 125 cm^{2}/(V s) |

Electron density n_{0} | 1.9×10^{17}cm^{−3} | 2.7×10^{17}cm^{−3} |

Hall electron mobility | 53 cm^{2}/(V s) | 50 cm^{2}/(V s) |

Wafer | #1153 | #1161 |

Epilayer conductivity σ_{0} | 4.7 S/cm | 5.5 S/cm |

Electron magnetoresistance mobility | 188 cm^{2}/(V s) | 146 cm^{2}/(V s) |

Electron drift mobility μ_{0} | 154 cm^{2}/(V s) | 125 cm^{2}/(V s) |

Electron density n_{0} | 1.9×10^{17}cm^{−3} | 2.7×10^{17}cm^{−3} |

Hall electron mobility | 53 cm^{2}/(V s) | 50 cm^{2}/(V s) |

## III. METHODOLOGY OF TRANSIENT MEASUREMENTS

The electron transport measurements at high electric fields were carried out on the two-electrode TLM samples. The nanosecond voltage pulses were applied in order to minimize the sample self-heating due to the current. The detailed description of the experimental procedure and the measurement setup can be found elsewhere.^{21,22}

The accuracy of the pulsed measurement is better when longer pulses are used unless the sample self-heating interferes. Typically, the pulse rise lasts about 1 ns in our setup, and the rising edge tends to overlap with the falling edge as the pulse duration is reduced below 2–3 ns.

The transient studies are carried out with short voltage pulses. The transmitted waveform for sample #1161 is shown in the inset of Fig. 1. The current $I(t)$ and the voltage $U(t)$ are obtained from the waveforms of the transmitted and the reference signals. Based on the current-voltage estimation procedure,^{21} the peak value of 2 V at the oscilloscope input and at certain attenuation (26 dB) corresponds to the voltage of 78 V (320 kV/cm) across the sample. The obtained current-voltage dependence is illustrated in Fig. 1 (symbols). The current deduced during the rise (squares) nearly coincides with that available from the falling edge (triangles). The predicted negative differential mobility at fields exceeding 290 kV/cm (Ref. 5) is not confirmed by our experimental data (symbols). The coincidence suggests that neither the acoustic phonon temperature nor the electron density changes within the considered time and field domain. In other words, the sample self-heating as well as possible excess generation of electrons by the electric field, the hot-electron capture, and the thermal release do not play an important role within the time-frequency domain of the short voltage pulses. This is a good base for interpretation of the experimental results on the current in terms of the electron drift velocity.

Under short-pulse measurements, the acoustic phonon temperature changes substantially less than the electron temperature. In addition to the described transient current measurements, this can also be illustrated by the hot-electron noise thermometry.^{23} The main principle is that the noise temperature measured at zero field immediately after the voltage pulse can provide with the sample temperature at the end of the voltage pulse. The technique has been illustrated for ZnO samples with similar electron densities.^{18} Figure 2 shows the excess noise temperature in the #1153 ZnO sample at different time moments during the voltage pulse (circles) and shortly after the switch off (square). The average electric field applied to the ZnO channel was 59 kV/cm at room temperature (293 K). The lattice temperature is measured in the absence of the electric field (vertical line in Fig. 2 stands for the end of the voltage pulse). One can see that the excess noise temperature (excess sample temperature) is 210 K after several dozens of nanoseconds needed for the gated measurements of the noise temperature. The heat accumulates during the voltage pulse, and the sample temperature is considerably lower when the voltage pulse is shorter.^{18} According to estimations, the excess temperature would be several degrees if we diminish voltage pulse duration from 400 ns to a few nanoseconds.

## IV. BACKGROUND FOR ESTIMATION OF THE DRIFT VELOCITY

The Hall electron mobility was substantially lower than that obtained from the magnetoresistance measurements (Table I), and a new method based on the hot-electron effect was developed for estimation of the electron drift velocity. The proposed method for estimating the electron drift velocity is based on solving the Boltzmann kinetic equation within the SHE approach.^{24} The SHE effectively yields the small-signal response mobility at a constant applied electric field.

Let us discuss the results obtained for the electrons located inside a nonparabolic $\Gamma 1$ valley at room temperature. The model assumes a nonparabolicity coefficient of 0.4 eV$\u22121$ and an effective mass of 0.21 m$e$.^{7} The details of the model are given in Ref. 24, and all required material parameters are the same as in Ref. 7. The Pauli exclusion principle is not taken into account. A reliable accuracy is obtained when eight spherical harmonics are used for the expansion of the electron distribution function. The interaction with phonons is considered within the spherical model in the standard way. The phonons are assumed to remain in equilibrium with the thermal bath. The nonelastic acoustic scattering caused by deformation-potential and piezoelectric mechanisms is taken into account. The electron scattering by screened point defects is included in the Brooks and Herring model. The related scattering rate^{25} is proportional to the product of the point defect density $N$ and the number $Z$ of the elementary charges per defect,

where $k\u2192$ and $k\u2032\u2192$ are the initial and the final states of the electron, $e$ is the elementary charge, $\epsilon 0$ is the static permittivity of ZnO, $V0$ is the volume of the simulated system, and $q0\u22121$ is the screening length. The electron density is kept constant ($1\xd71017$ cm$\u22123$) in order to exclude the electron gas degeneracy from consideration, while the product $N\xd7Z2$ is varied from 10$17$ cm$\u22123$ to 10$20$ cm$\u22123$.

Results of the calculations are illustrated in Fig. 3. The small-signal response mobility $\mu (E)$ at field $E$ is calculated in the Green function approach.^{24} At the moderate electric field of 50 $<Em<250$ kV/cm, the density variation of the charged point centers leads to a less significant change in the mobility, and one can assume $\mu (Em)=(80\xb110)$ $cm2$/(V s) if $N\xd7Z2<4\xd71018$ cm$\u22123$ (triangles in Fig. 3). Essentially, the same result follows for the differential mobility estimated from the slope of the simulated electron drift velocity (Fig. 3, a diamond^{5} and squares^{10}). These results confirm that the optical phonon scattering dominates at the moderate electric fields.

There is a several percent decrease in the calculated mobility (star) at 100 kV/cm electric field when the hot-phonon effect is taken into account in the relaxation time approximation.^{24} For the assumed value of $\tau ph=1$ ps, the hot-phonon effect is weak at the chosen electron density of $1017$ cm$\u22123$ in agreement with the hot-electron energy relaxation study for doped wurtzite ZnO.^{18}

The calculated mobility at the moderate electric fields, $\mu (Em)$, for $N\xd7Z2<4\xd71018$ cm$\u22123$, can serve as a reference for calibration of the drift velocity. The procedure can be illustrated as follows. Supposing that the electron density $n0$ is independent of the bias, the following relations hold: The current density $i(E)$ at any field $E$ equals

and the differential conductivity at $Em$ is

where $\mu (Em)$ is the calculated small-signal mobility at $E=Em$ and $\sigma (Em)=didE$ is the differential conductivity measured as the slope of the $i(E)$ dependence at $E=Em$. The slope is almost independent of the electric field in the range of moderate fields. After simple transformations, one obtains

The data on the current density $i(E)$ and the differential conductivity $\sigma (Em)$ are available from the hot-electron experiment. Supposing that the calculated mobility $\mu (Em)$ is used as the reference, the drift velocity $vdr(E)$ follows according to Eq. (4).

## V. RESULTS AND DISCUSSION

The interelectrode distance was varied in order to find out the optimal conditions for resolving the hot-electron effect.

The choice of the sample length for resolving the hot-electron effect is a compromise. Figure 4 compares the results for various samples subjected to 3 ns voltage pulses. The voltage pulses are short enough to minimize other contributions except for the hot-electron effect on the electron mobility. Shorter channels withstand higher electric fields. Indeed, the measurements for 1.7 $\mu $m samples are carried out at fields up to $430\xb150$ kV/cm (Fig. 4, squares). On the other hand, the error in the contact resistance introduces an uncertainty in estimating the electric field strength because the channel resistance becomes comparable with the contact resistance in the short channels.

Moreover, the channel is often longer than the interelectrode distance because the channel extends under the electrodes. Therefore, a systematic error is introduced because the average electric field is estimated as $E=(U\u2212IRc)/L$. If the channel length exceeded $L$, the results for the shortest channel (squares) would correspond to lower electric fields, and, therefore, the squares would shift toward the bullets, triangles, and diamonds in Fig. 4.

To reiterate, the soft damage is observed through measuring the resistance at a low electric field before and after the high-field experiment. A thermoelectric breakdown is triggered if the electric field is increased above the threshold for the soft damage. Thus, short voltage pulses are preferable.

Figure 5 presents the results for the samples subjected to short pulses of voltage in the field range below $\u223c320$ kV/cm. The uncertainty in the channel length as well as the effects of sample self-heating and electron trapping are best minimized in the channels of moderate length (bullets and diamonds). This allows one to consider the experimental results in terms of the electron drift velocity $vdr$ and the field-independent electron density $n0$. As demonstrated by the transient experiments (Figure 1), the electron density is independent of the electric field if the field increases and decreases fast enough.

Let us apply the proposed hot-electron method for estimating the electron drift velocity, which is based on Eq. (4) that is valid up to 320 kV/cm electric field. In this range, as discussed, the experimental $i(E)$ dependence can be approximated with a dashed line (Fig. 5) different from that available at the low fields (solid lines).

In the next step, once the electron mobility $\mu (Em)$ is known from the model, the electron drift velocity can be estimated according to Eq. (4) where the current density $i(E)$ is measured, and the experimental value for the differential conductivity $\sigma (Em)$ is available from the slope of the dashed line for ZnO sample #1153 in Fig. 5. The experimental results are illustrated in Fig. 6 together with the calculated curves. A reasonably good agreement of the experimental results (symbols) with those of our model calculations (dashed curve) is obtained. The highest experimental value for the electron drift velocity of$\u223c2.7\xb10.3\xd7107$ cm/s is reached at 320 kV/cm. The similar procedure to estimate the drift velocity according to Eq. (4) is performed for the sample #1161, and lower electron velocities are attained at $\u223c2.3\xb10.3\xd7107$. The Monte Carlo results by Albrecht *et al.*^{5} (solid curve) give slightly higher values of the electron drift velocity mainly because of a higher low-field electron mobility, 300 cm$2$/(V s).

The electron low-field drift mobility $\mu 0$ can be estimated from the slope of $vdr(E)$ in the Ohmic range of Fig. 6. After this procedure, the electron density $n0$ can be estimated from the conductivity $\sigma 0=en0\mu 0$. The results for the drift mobilities of both samples, $\mu 0=154cm2$/(V s) for #1153 and $\mu 0=125cm2$/(V s) for #1161, and the electron densities $n0=1.9\xd71017$ cm$\u22123$ and $n0=2.7\xd71017$ cm$\u22123$, respectively, are presented in Table I. The obtained electron drift mobility at the low electric fields is in a reasonable agreement (within $\u223c15%\u201320$%) with the magnetoresistance measurements when the magnetoresistance factor is ignored (Table I).

## VI. SUMMARY

We applied the nanosecond pulsed technique for the hot-electron transport study in the nominally undoped ZnO epitaxial layers at room temperature. In order to minimize the self-heating effects, we employed short voltage pulses. The material withstood average electric fields up to $430\xb150$ kV/cm when the interelectrode distances were short. The transient measurements of the current demonstrated no significant change in the electron density at the applied electric fields up to 320 kV/cm. The hot-electron effects are interpreted in terms of the electron drift velocity. The highest electron drift velocity is determined as $2.7\xb10.3\xd7107$ cm/s at the electric field of $\u223c320$ kV/cm. This experimental result approaches the theoretic limit ($3.1\u22123.2\xd7107$ cm/s) predicted by the known Monte Carlo simulations. The solution of the Boltzmann kinetic equation is in satisfactory agreement with the experimental data at low and moderate electric fields.

## ACKNOWLEDGMENTS

This research was funded by the Research Council of Lithuania (Grant No. APP-5/2016). The work at VCU was supported by the Air Force Office of Scientific Research (AFOSR) under Grant No. FA9550-12-1-0094.

## REFERENCES

*GaN and ZnO-Based Materials and Devices*, edited by S. Pearton (Springer, Berlin, 2012).