ESTIMATION OF THE CHARGED DEFECT DENSITY FROM HOT-ELECTRON TRANSPORT STUDIES IN EPITAXIAL ZnO

1. Introduction

Zinc oxide (ZnO) is a transparent semiconductor with a direct and relatively wide bandgap [1–3]. Its superiority in temperature resistance and high electron saturation velocity [4] is also promising to thin film transistors for high-temperature and high-power operations [5–7]. Many properties of ZnO are similar to those of GaN. Particular advantages of ZnO over GaN include higher electron drift velocity and breakdown field [8], bulk growth capability, amenability to conventional chemical etching, low growth temperature, high radiation hardness and large exciton binding energy [9–13]. The  epitaxial technology of ZnO is less expensive, albeit relatively not so well developed compared to that of GaN.

The electron low-field mobility in ZnO is mainly determined by scattering on charged defects. There is an experimental evidence that double-charged oxygen vacancies act as shallow donors and charged scattering centres [14]. The  oxygen vacancies were studied by an electrochemical method [15], optical absorption [16], optically detected electron paramagnetic resonance [17], positron annihilation spectroscopy [18], or after studying oxygen self-diffusion in oxygen-isotope ZnO [14]. The density-functional theory calculations were also applied to study oxygen vacancies [19]. The typical density of the vacancies exceeded 10¹⁷ cm^–3 [15, 16, 18]. The highest electron low-field mobility around 400 cm²/(Vs) was reported at room temperature in epitaxial ZnO layers with low residual impurity densities [20]. Polar scattering by longitudinal optical (LO) phonons was found important at low and moderate electron densities [21]. The room temperature electron mobility (430 cm²/(V s)) calculated for the electron and donor densities of 10¹⁷ cm^–3 [22] slightly exceeded the maximum measured value.

The results of Monte Carlo simulations [4, 8, 23, 24] were used for the interpretation of the experimental studies [25–27] of high-field electron transport in epitaxial ZnO channels. The dependence of electron drift velocity on the electron density was deduced at room temperature for doped ZnO [26]. High electric fields (430  kV/cm) and velocities (~2.7  ×  10⁷  cm/s at 320  kV/cm) were reported for nominally undoped ZnO epilayers at room temperature [27].

The goal of this paper is to examine the electron scattering by charged defects through the study of hot-electron effects in nominally undoped ZnO epilayers at room temperature. The current–voltage dependence is measured up to high electric fields, and the experimental results are fitted with the  solution of the  Boltzmann kinetic equation. The  results are used for the  estimation of electron density, drift velocity and low-field mobility. The electron scattering by the double-charged oxygen vacancies and other charged defects is discussed.

2. Theoretical background

The Boltzmann kinetic equation is used for the investigation of electron scattering and hot-electron transport in nominally undoped ZnO. The highly-efficient spherical harmonics approach [28] is employed for the  calculation of the  differential mobility deduced from a small-signal response at 1 GHz at a constant applied electric field. The calculations are performed for the  nonparabolic Γ₁ valley with the electron effective mass of 0.21 m_e and the non-parabolicity coefficient of 0.4 eV [23]. More details on the model are available elsewhere [22, 27]. Eight spherical harmonics are found sufficient for an accurate enough approximation of the electron distribution function. The change in calculation results is negligible if higher spherical harmonics are included.

The electron scattering on acoustic phonons, longitudinal optical phonons and screened point defects is taken into account. Deformation-potential and piezoelectric mechanisms are considered for the acoustic phonon scattering. The FrÖhlich approximation is used for interaction with the optical phonons. Charged defects are treated in the screened Coulomb potential approach within the Brooks and Herring model for point scattering centres. The  electron gas degeneracy is not included in the present calculations. The electron density is considered lower than the total ionized centre density because of the  known compensation of the shallow donors by other charged defects acting as acceptors. The total effective density of the charged defects is introduced as follows:

Here the defect density N_i of an ith type is weighted by the  number Z_i of elementary charges on a single defect of the ith type.

The differential mobility $\mu \left(E\right)=\frac{d{v}_d}{dE}$ is deduced from the small-signal response at 1 GHz, and the results are illustrated in Fig. 1 as a function of the applied electric field. The curves stand for different values of the effective defect density N_eff. The density of the hot-electron gas is kept constant at 1.8  ×  10¹⁷  cm^–3. The  mobility decreases as the electric field increases. The curves merge at electric fields over 100  kV/cm, and the value becomes almost independent of the defect scattering. The calculations are terminated at E = 120 kV/cm because electron runaway develops at higher fields unless the upper valleys are taken into account. The values of the differential mobility at low and moderate electric fields are compared in Fig. 2.

An almost linear dependence of the mobility on N_eff^–1 at low fields (Fig. 2, triangles) shows that the defect scattering plays the dominant role. The deviation from the linear dependence (triangles and line) illustrates a weak contribution of the phonon scattering. This is a strong argument that the low-field mobility is controlled by the ionized defect scattering.

On the other hand, the longitudinal optical phonon scattering becomes important at high electric fields. According to calculations, the  hot-electron differential mobility is almost independent of the defect scattering (Fig. 2, squares). This conclusion is in contrast to the known behaviour of the hot-electron drift velocity: the ionized donor scattering reduces the drift velocity considerably at any fixed electric field (except for the extremely high ones) [22, 24].

Consequently, the value of the differential mobility at high fields can be used as a reference for the estimation of the electron density and the drift velocity if the  differential conductivity is available from the experiment. Indeed, the differential conductivity of hot electrons at moderate fields (E = E_m) is

where µ(E_m) is the differential mobility at E = E_m and $\sigma \left(E_m\right)=\frac{dj}{dE}$ is the  differential conductivity measured as the  slope of the  current density-field j(E) dependence at E = E_m. From this equation the electron density n(E_m) can be estimated. Once the electron density is known, the electron drift velocity can be estimated from the  measured current density. For demonstration of these possibilities on ZnO epilayers, the  hot-electron transport experiment is carried out.

3. Samples

The investigated ZnO epilayers were grown by plasma-assisted molecular beam epitaxy (PA-MBE) at Virginia Commonwealth University (USA). Highly-resistive (60 kΩ mm), carbon compensated 2.5 µm Ga-polar (0001) GaN layers, prepared by metal-organic chemical vapour deposition on c-plane sapphire, served as substrates. Two Zn-polar ZnO structures included into this study were grown at 670°C with two different Zn cell temperatures employed for the  growth (340 and 347°C for the growth of the #1153 and #1161 structures, respectively). Other parameters used for depositing a low-temperature (LT) nucleation layer at 300°C as well as the oxygen gas flow rate (1.6  sccm) and plasma power (400  W) used for the  growth at 670°C were identical. The  resultant growth rates were 115  nm/h for #1153 and 135  nm/h for #1161. More oxygen-rich conditions (#1153) as well as lower substrate temperature caused a  lower point defect density (either oxygen vacancies or Zn interstitials) and gave rise to lower electron density.

The transmission line measurement (TLM) patterns were processed with evaporated Ti/Au (25  nm/30  nm) stacks acting as ohmic contacts. The channel width w was 300 µm, the thicknesses were 350 and 375  nm for #1153 and #1161, respectively, and the  interelectrode distances L were 1.7, 3.9, and 6.9, 9.9 and 15.8 µm. The contact resistance was estimated at low electric fields from the dependence of the sample resistance on the interelectrode distance. The channel resistance, the length and the cross section area were used to estimate the low-field conductivity σ₀ (Table 1).

The electron mobility available from the  Hall effect measured in the van der Pauw configuration was substantially lower than that obtained from the  magnetoresistance measurements carried out on the TLM samples [27], and a new method based on the hot-electron effect was developed for the  electron density and the  low-field mobility. This enabled the estimation of the oxygen vacancy and other charged centre densities.

4. Experiment

Electron transport measurements at high electric fields were carried out on the two-electrode samples selected from the  TLM patterns. The  measurement procedure and the  determination of current–voltage (I–U) dependence is described in Refs. [29, 30]. The data for evaluating the current and the voltage drop along the channel were obtained from the oscilloscope amplitudes when either a reference cable or the sample under test was present in the circuit. The employment of nanosecond pulses allowed us to minimize the channel self-heating due to current [31]. The average electric field was estimated as E  =  (U  –  IR_c)/L. Figure 3 presents the dependence of the current on the  voltage pulse width for the  ZnO samples: the highest current values are attained with the shortest pulse duration (3 ns).

The transient studies are carried out with short pulses of voltage. The current I(t) and the voltage U(t) are obtained from the waveforms of the transmitted and reference signals. The current deduced during the  rise edge nearly coincides with that available from the falling edge. This indicates that the electron density remains constant in the electric field range up to 320 kV/cm [27].

The coincidence suggests that neither the sample temperature nor the  electron density change with time and field, i.e. n₀ = n(E_m), where n₀ is electron density at low electric fields. 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 domain of the short pulse of voltage. This is a good base for the  interpretation of the  experimental results on the current in terms of the electron scattering.

The electron drift velocity v_dr(E) was estimated from the current I measured in the direction perpendicular to the c axis of ZnO according to the expression

valid for the uniform electric field E and the field-independent electron density n₀ (here e is the elementary charge, S = wd is the channel cross-section area).

Some samples suffered from soft damage at high electric fields. The damage was characterized by the change of the zero-field channel resistance measured before and after the high-field experiment. We present results up to a damage level not exceeding 5%. The threshold field for soft damage can be introduced, as reported previously [30].

5. Experimental results

Figure  4 presents the  transmitted current (a) and electric field (b) waveforms for the  ZnO sample #1161. The highest current of 0.8 A corresponds to 320  kV/cm peak field at 26  dB attenuation. Further, the  voltage pulse length and the  interelectrode distance were varied in order to find out the  optimal conditions for resolving the  effect of scattering on the  hot-electron transport.

Figure 5(a) presents the relatively long voltage pulse (100  ns) measurements of the  current on the average electric field for the sample #1153. An essential deviation from the ohmic behaviour appears at average electric fields exceeding 10 kV/cm in long channels (L  =15.8  µm, stars). Moreover, the deviation is sensitive to the channel length – it is weaker in shorter channels. In particular, the results for 1.7 µm samples (squares) and those for 3.9 µm (bullets) nearly match (within experimental error) at electric fields below 50  kV/cm. The enhanced non-ohmic behaviour in the long-channel samples might be an indication of either enhanced scattering or reduced electron density due to sample self-heating and hot-electron trapping, respectively.

The use of short nanosecond voltage pulses (3 ns) enabled us to reach the highest electric field of 430±50 kV/cm (Fig. 5(b), squares; the error is due to different contact resistance) for the 1.7 µm long channel and attain the  maximum current density of 1.15 A for the L = 6.9 µm sample #1161 (Fig. 5(b), triangles). Both current values are close to each other (triangles and diamonds) up to 230 kV/cm indicating that other effects are ruled out except for the hot-electron effect on the mobility. No soft damage was found for the channels #1161, and the  highest applied field was limited by the  setup. When the  field exceeded a  certain value, ~350  kV/cm for the  6.9  µm channel (Fig. 5(b), triangles), the slope of I–E dependence started increasing. Moreover, the shape of the current pulse did not follow the  shape of the  voltage pulse. In particular, for the  3.9  µm channel (Fig. 5(b), diamonds), the current increased when the  voltage decreased at the  end of the  pulse. These distortions in the pulse shape might be attributed to the increased electron density caused either by the  impact ionization or de-trapping effects.

6. Discussion

The n-type conductivity of nominally undoped ZnO epilayers can be caused by oxygen vacancies [14, 17]. The redundant Zn atom gives its two electrons to the  conduction band, and the  oxygen vacancy becomes positively charged. One can estimate the  density of vacancies by measuring the density of electrons in the conduction band. However, part of free electrons is captured by other charged centres (e.g. dangling bonds, impurities, and other point centres) present in epitaxial ZnO layers. Thus, the real density of electrons is always lower than the double density of vacancies.

We estimated the  density of vacancies with the aid of electron scattering on them. Let us define the double charged density of vacancies as N₊ Also, N_– is the  density of other charged defects, each capturing an electron. At the equilibrium, we have

where Z₊ = 2 is the number of elementary charges per vacancy and Z_– = 1 defines the negative charge of a dangling bond with one electron. The charged defects scatter electrons. In the  point scattering centre approximation, the  scattering is proportional to the  density of point defects multiplied by the squared number of elementary charges of the  defect. Thus, after Eq. (1), the  total scattering is proportional to the  sum of contributions due to positively charged vacancies plus Z²_– times the negatively charged defects:

One can approximate the results of current density on electric field (Fig. 6 of this work or Fig. 5 in [27]) with lines. For the sample #1153, the Ohm’s law is valid at low electric field (solid line), while the dominant scattering mechanism is due to optical phonons at moderate fields (50 < E_m < 250 kV/cm), and the  so-called second Ohm’s law is valid (Fig. 5, dashed line [27]). The differential conductivity estimated at moderate fields in #1153 from the slope of the dashed line is σ(E_m) = 2.55 S/cm. The  ratio σ₀/σ(E_m)  =  1.84. When the  current is measured with the  aid of nanosecond voltage pulses, the density of electrons does not depend on the  electric field in the  investigated samples, and we have σ(E_m)/σ₀ = µ(E_m)/µ₀.

We estimated the  ratio of the  mobility at the  low field, µ₀, and the  differential mobility at the moderate field, µ(E_m), by solving of the Boltzmann kinetic equation through the  spherical harmonics expansion method. The dependence of the calculated ratio on the sum N_eff (see Eq. (5)) is illustrated in Fig. 7.

The electron degeneracy is insignificant in the  nominally undoped ZnO, and it is useful to consider the  results when the  electron density is constant with varying N_eff. Once the  ratio σ(E_m)/σ₀  =  µ(E_m)/µ₀ is valid, the  square and the arrow for the sample #1153 leads in Fig. 7 to N_eff  =  8.2  ×  10¹⁷  cm^–3. Knowing N_eff, we estimate the  low-field mobility µ₀  =  195  cm²/(V s) from Fig. 2 and find the electron density n(E_m) = σ(E_m)/ (eµ(E_m))  =  n₀  =  σ₀/(eµ₀)  =  1.5  ×  10¹⁷  cm^–3. This electron density value is consistent with the magneto-resistance measurement results. Then we put N_eff = 8.2 × 10¹⁷ cm^–3 and n₀ = 1.5 × 10¹⁷ cm^–3 values into Eqs.  (4) and (5) and find the  results for the  #1153 sample (Table  1). The  oxygen vacancy density is estimated as N₊ = 1.6 × 10¹⁷ cm^–3 and the density of other charged defects is found as N_– = 1.7 × 10¹⁷ cm^–3. These numbers are in fair agreement with the findings by other methods [18].

The same procedure is performed for the #1161 sample (Table  1) which was grown under higher Zn cell temperature (347°C). The  higher Zn cell temperature favours the formation of higher density oxygen vacancies (1.9 × 10¹⁷ cm^–3), as shown in Table 1.

Once the electron density is known, the electron drift velocity can be estimated from the measured current density j  =  I/S and the  estimated electron density according to Eq. (3). The highest experimental value for the electron drift velocity of ~2.9 × 10⁷ cm/s (~2.7 × 10⁷ cm/s) is found at 320 kV/ cm in the sample #1153 (#1161) with the electron density of 1.5 × 10¹⁷ cm^–3 (1.9 × 10¹⁷ cm^–3).

A correlation with the growth conditions is resolved: a slightly higher drift velocity is estimated for the sample #1153 with a lower density of oxygen vacancies. This wafer has been grown at lower Zn cell temperature (Table 1).

The contact resistance plays a role in the determination of the  applied electric field in the  ZnO channel. For example, the error of 14 Ω in the ohmic contact resistance for the  sample #1161 shifts the highest electric from 320 kV/cm (R_c = 15 Ω) to 300 kV/cm (R_c = 29 Ω).

7. Conclusions

The nanosecond pulsed technique is used to single out the hot-electron transport and scattering in the nominally undoped ZnO epilayers at room temperature. The  Boltzmann kinetic equation is solved within the  spherical harmonics expansion method for considering the  effect of electron scattering. The  approach is proposed for estimating the  density of oxygen vacancies (1.6– 1.9  ×  10¹⁷  cm^–3) and that of other single-charged defects (1.7–2.0 × 10¹⁷ cm^–3). The estimated density of the oxygen vacancies correlates with the Zn cell temperature.

The epilayers with the  shortest inter-electrode distances withstand average electric fields up to 430±50  kV/cm. Two ranges of electric field are resolved. The results on the non-ohmic transport are interpreted in terms of scattering by charged point defects and phonons: the electron drift velocity reaches the highest value of ~2.9 × 10⁷ cm/s at 320 kV/cm electric field when the electron density is 1.5 × 10¹⁷ cm^–3.

Acknowledgements

This research was funded by the Research Council of Lithuania (Grant No. APP-5/2016). The authors are also grateful to the VCU Microelectronics Materials and Device Laboratory for the ZnO samples.

References