Intrinsic Carrier Parameters and Optical Carrier Injection Method in High-Purity Diamonds

3.1 Time-resolved cyclotron method

Figure 3a shows a colored contour map of a real part of TRCR signal measured at 7.3 K excited by laser pulses at photon energy of 5.50 eV. Temporal profiles at the magnetic fields of 0.089, 0.122, 0.162, and 0.230 T are shown in Figure 3b. CR spectra at the delay times of 60, 200, and 600 ns are shown in Figure 3c, by slicing the data set at the delay times. The magnetic field was applied to an angle of 40° from the crystal axis of [001] in the (1–10) plane. In this orientation, four carrier species, light hole, heavy hole, and two electrons in inequivalent conduction valleys, indicated by lh, hh, e1, and e2, respectively, were distinguishable as shown in Figure 3a and c.

3.2 Optical carrier injection

Optical carrier injection is a key technique in our nanosecond TRCR method. As a diamond has an indirect band structure as shown in Figure 1a, the optical carrier injection at the lowest photon energy is established with the assistance of phonon emission/absorption to satisfy the energy and momentum conservations. The lowest excited state is an exciton band located below the indirect band edge by a binding energy larger than 80 meV [32, 33], whose fine structures were recently clarified [34]. To clarify the spectroscopic way of carrier injection, an excitation spectrum of TRCR signal at the fixed resonant magnetic field was measured with a thin CVD crystal of 70-μm in thickness to suppress the saturation by exciton absorption.

Figure 4 shows the TRCR excitation spectra obtained at 10, 80, and 300 K. The signals were averaged at the time windows, (a) 80–280 ns at 10 K, (b) 352–552 ns at 80 K, and (c) 156–356 ns at 300 K, after the signal decayed to the 1/e of the peak intensity [26]. For such late times, we observed that carriers were dominantly generated by dissociation of excitons [27]. The onset energy of the excitation spectra (a, b) at 5.493 eV coincides with the exciton generation edge assisted by emission of a transverse acoustic (TA) phonon (Eex+ℏωTA), where Eex = 5.406 eV is the exciton energy and ℏωTA = 87 meV is the TA phonon energy [35] (see the inset). The second onset at 5.547 eV is assigned to exciton generation assisted by emission of a transverse optical (TO) phonon (Eex+ℏωTO), where ℏωTO = 141 meV is the TO phonon energy [35]. Therefore, in the range of the excitation photon energy above 5.493 eV, free carriers can be generated via excitons that are generated by the assistance of phonon emission. At the lower temperatures as 10 K, free carriers were generated by two-body collision of excitons, as the CR signal intensity was proportional to the square of excitation intensity [22].

On the other hand, the signal at 300 K arose at the lower energy side with the onset at 5.265 eV. The onset energy coincides with the threshold for exciton generation assisted by absorption of a TO phonon (Eex−ℏωTO) (see the inset). The onset energy for TA phonon absorption (Eex−ℏωTA) is higher as indicated by the vertical broken line (the second one from the left) in Figure 4c. In the range of the excitation photon energy from 5.265 to 5.493 eV, free carriers can be generated only at higher temperatures via excitons that are generated by the assistance of phonon absorption. As the CR signal intensity measured at 300 K was proportional to the excitation intensity, the carriers are generated dominantly via a one-body process, such as thermal ionization of excitons [26].

Under the excitation in the range from 5.265 to 5.493 eV, where only the phonon absorption assists the process, the carrier number should increase with rising of the temperature according to the activation of phonons. The lower panel of Figure 5 shows temperature dependence of the temporal response intensity excited by laser pulse at 5.335 eV. Solid curves in the upper panel are the temperature dependence of the quantum statistical numbers n=1/expℏωi/kBT−1of TA and TO phonons, where ℏωi, kB, and Tare the phonon energies for i = TA or TO, Boltzmann constant, and temperature, respectively. The observed signal intensity rose around 150 K in coincidence to the appearance of TA phonon. This suggests that the phonon-assisted optical carrier injection is effective at device-operating temperatures.

In the subsequent Sections 3.3–3.5, we focus on the carrier properties at temperatures below 50 K. This temperature range is uniquely reached by our method owing to the optical carrier injection without the need of thermal activation of carriers from deep levels. For an efficient carrier generation at these temperatures, the excitation wavelength was chosen in the range of 219.4–226.4 nm. Furthermore, we discuss the CR spectra at the later delay times after 600 ns (see Figure 3) by eliminating the plasma shift effect at the earlier delay times depending on experimental conditions [16, 31].

3.3 Determination of effective masses

Both CR spectra of real and imaginary parts were well fitted by the formula for the complex conductivity [15]:

where Γjis a half width and Bjis a central magnetic field at the resonance taken to be common for both real and imaginary parts, Bis an external magnetic field, and jrepresents each resonant component (j=lh,hh,e1,e2). We can obtain the effective mass mj∗and momentum relaxation time τjfrom these fitting parameters by using the relations, m∗=eB0/ωand 1/τj=ωΓj/Bj. Figure 6 plots the extracted effective masses mj∗at the magnetic field orientations: the positive angles indicate the magnetic field in the (1–10) plane for the rotation axis along [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], whereas the negative angles indicate the magnetic field in the (100) plane for the rotation axis along [100].

The effective masses of electrons were simulated according to the following equation [15]:

where g1θ=±sinθ/2,g2θ=g3θ=±cosθon (1–10) plane and g1θ=±sinθ,g2θ=±cosθ,g3θ=0on (100) plane. The curves calculated with parameters mt/m0=0.280and ml/m0=1.560[21] trace the data points very well. The curves of e1 and e2 in the (1–10) plane originate from two and four equivalent valleys, respectively. The curve of e2 splits into two curves in the (100) plane, where electron of a constant effective mass is referred as e3.

On the other hand, the effective masses of holes were simulated according to the equation for light (−) and heavy (+) holes [15]:

where fθ=1−3cos2θ2/4in the (1–10) plane and fθ=1−3cos2θsin2θin the (100) plane. The thick lines for lh and hh were calculated with parameters A=−2.670,B′=1.245,C=1.898[21]. The thin lines are calculated with the upper and lower limit of errors, ΔA=±0.020,ΔB′=±0.025,ΔC=±0.048. Moreover, the split-off hole mass calculated from the parameter as mso/m0≈ℏ2/2m0Ais also plotted by a broken red line. These parameters give band dispersion at the Γ point,

with a transformation by B′=B2+C2/4. The finite value of Cgives rise to the warping of the valence band.

As we report in detail in Ref. [21], it is experimentally figured out that the electrons are in highly asymmetric valleys along the <001> directions, that is, at the △ points, with the transverse effective mass (mt=0.280m0) and longitudinal effective mass (ml=1.560m0). And, the doubly degenerate valence-band maxima are located at the Γ point and warping. The values of effective masses were listed in Table 1 in comparison to those of silicon and germanium [15]. The calculated values of the density of mass for electron med=mt2ml1/3and hole mhd=mlh3/2+mhh3/2+mSO3/22/3were also listed with the value of direct bandgap energy at Γ point.

Figure 7 compares the angular dependence of effective masses of diamond with those of silicon and germanium [15] with the same angular definition as in Figure 6. The conduction-band minimum in silicon is located at the △ points as in a diamond, while that in germanium is located at the L points. It is easily recognized that the effective masses in diamond are largest among group-IV semiconductors, reflecting the largest direct bandgap energy at the Γ point which causes relatively larger contribution of the first perturbation term in the k·p theory to the effective mass of an energy band.

3.4 Sample dependence of carrier lifetime

A well-resolved spectrum of TRCR at the lower temperatures allows extracting the effective masses in good accuracy as described in Section 3.3. We compared the TRCR signals in different samples as reported in Refs. [24, 25]. The sample showed the narrower spectral width as presented in Figure 3 possesses the smaller concentration of donor and acceptor, that is, nitrogen and boron. Figure 8 shows temporal profiles of five different samples. The sample displayed a slow rise and decay in a couple of hundred nanoseconds (sample A) which is identical to that in Figure 3. The narrow spectral width is caused by the long carrier scattering time. From the comparison of CR spectra of CVD diamonds to those of dislocation-free HPHT diamonds, we found the fact that the TRCR detection is rather insensitive to crystalline dislocations [24]. It is known that a typical dislocation density in CVD diamond is lower than 10⁴ cm⁻² [36], corresponding to dislocation periods larger than 100 μm. On the other hand, the cyclotron radii in the measurement with X band microwave were 86 and 55 nm for light and heavy holes, respectively. As the carriers rotate in the much smaller spatial extension than the typical dislocation period in CVD samples, the CR detection is rather insensitive to dislocations. Instead, impurity scattering by neutral nitrogen atoms is found to be dominant at low temperatures (as described in Section 3.5), because their average separations are comparable to the cyclotron radii in the present case.

From these facts, we emphasize that the accurate determination of effective masses as described in Section 3.3 became possible, since we could use a highly pure diamond produced by the CVD method under the optical carrier injection.

The rise time and decay time in the temporal profile of TRCR in Figures 3b and 8 reveal carrier generation and trapping mechanism. The finite rise time reflects the time required for carrier creation by exciton collision. A detailed formula giving an approximate rise time in connection with the lifetime is described in Refs. [22, 29]. The shorter decay time is probably caused by the higher density of impurity concentrations (in comparison among samples A–C) and by the higher density of stacking faults and substitutional impurities (in comparison samples E–D). It has been known that incorporation of defects occurs more easily in a (111)-oriented diamond than in an (001)-oriented diamond.

3.5 Carrier scattering time and drift mobility

The temperature dependence of TRCR spectrum provides the aspect of carrier scattering mechanisms. Figure 9a shows the normalized temporal curves measured at 0.16 mT at various temperatures. The rise time and the decay time of the signal increased as the temperature is rising. The longer rise and decay times at higher temperatures indicate elongating of the carrier lifetime [22]. This is probably caused by trapping of carriers into impurity states at lower temperatures. Similar shortening of the exciton lifetime at low temperatures was clarified in Ref. [37] by comparing exciton lifetimes in samples containing different concentrations of impurities.

Figure 9b shows the CR spectra at 7.3, 10, 20, 30, and 40 K taken at a delay time of 1 μs after the laser pulse with averaging window for ±40 ns [23]. The four peaks were separately observed up to 20 K and the width broadened with increasing temperature. These spectra were analyzed by the abovementioned spectrum fitting. The carrier scattering times τj’s, in other words, momentum relaxation times, were taken from the width Γjas a function of temperature. The fitting of the structureless spectra at the higher temperatures was possible by fixing the resonance field of the four components. The fitting curve of each spectrum is shown by the thin red lines.

The spectral width Γjat low temperature sharply depends on the impurity concentration as mentioned in Section 3.4. The carrier scattering at low temperature is mainly governed by the neutral impurity scattering instead of the ionized impurity, because the optically injected carriers neutralize ionized impurity centers. Figure 10a shows the temperature dependence of the inverse carrier scattering times 1/τjextracted from the spectra measured with a CVD diamond shown in Figure 9b. The data points are well reproduced by the sum of the longitudinal acoustic (LA) phonon scattering rate 1/τacjand a constant term bj(solid lines). The LA phonon scattering rate 1/τacj(broken lines) was calculated for electrons and holes by using Eq. (6) in Ref. [38] with a deformation potential of 8.7 eV [35] (6.8 eV) for e1 (e2) electrons or 10.0 eV for both types of holes in diamond. Deviation from the straight line of the T3/2law which was known for acoustic phonon scattering appeared below 40 K due to inhibition of LA phonon emission at low temperatures. The constant terms were attributed to neutral impurity scattering rates. The best fit values were be1=1.4×109s−1, be2=1.3×109s−1, blh=2×109s−1, and bhh=8×109s−1. As discussed in Ref. [23], these values were larger than the values calculated by modified Erginsoy’s formulas, βe=3.4nAaA+20nDaDℏ/me∗for the electron [39] and βh=20nAaA+3.4nDaDℏ/mh∗for the hole, where aADis a Bohr radius and nADis the concentration of acceptors (donors). This fact indicates that the estimated nIaII=A,Dwere smaller than those in the actual situations probably owing to the too much simplified estimation of the Bohr radius by the hydrogenic model.

Now as the parameters of the effective mass m∗and carrier scattering time τwere individually obtained from the analysis of CR spectra, we can derive the drift mobility by the relation μ=eτ/m∗. Figure 10b shows the drift mobilities of electron and hole. We used a conductivity mass for the electron, me∗=3/1/ml+2/mt=0.386±0.026m0. For the valence bands, effective masses were averaged for the warped energy surfaces mlh∗/m0=0.261±0.005for the light hole and mhh∗/m0=0.663±0.032for the heavy hole. The total hole mobility was obtained by weighting the light and heavy hole mobilities as μ=plhμlh+phhμhh/plh+phh=μlh+rμhh/1+rwith the ratio r=phh/plh=mhh/mlh3/2=4.05derived from the density-of-state ratio for the degenerate energy bands [40]. The drift mobility at 10 K was obtained as μe=1.52±0.27×106cm2/Vsfor electrons and μlh=2.26±0.38×106and μhh=0.24±0.03×106cm2/Vsfor the light and heavy holes. The ratio of light to heavy hole mobilities μlh/μhh≅9.35is in better agreement with the ratio of mhh∗/mlh∗5/2=10.3than mhh∗/mlh∗=2.54. This fact indicates that light and heavy holes in ultrapure diamond relax inside each band. This is in contrast to the case in p-type germanium [40], where the intra-/inter-band scattering is dominant for the heavy-/light-hole and then μlh/μhhis proportional to mhh∗/mlh∗.

We evaluated the mobility up to 300 K by extrapolating the T3/2relation as shown by the solid lines, because the optical phonon process dominates carrier scattering only above 400 K in diamond [41]. The values of the mobility at 300 K were derived as μe=7.3×103cm2/Vsfor electron and 5.3×103cm2/Vsfor total hole. The values are higher than those evaluated by Reggiani et al. [8] and Nava et al. [9] for a natural n(p)-type diamond crystal or by Isberg et al. [10, 17] with the CVD diamond crystals. The values are summarized in Table 3 in comparison to those in silicon and germanium at 300 K [42]. The highest mobility of both electron and hole responsive to 10 GHz at 300 K indicates the outstanding possibility of a diamond in the field of power electronics and optoelectronics.