Charge Transport Phenomena Unique to Diamond

Mater. Res. Soc. Symp. Proc. Vol. 1591 © 2014 Materials Research Society DOI: 10.1557/opl.2014.295

Charge Transport Phenomena Unique to Diamond Kiran K. Kovi1, Nattakarn Suntornwipat1, Saman Majdi1, Markus Gabrysch1, Johan Hammersberg1 and Jan Isberg1 1

Division of Electricity, Department of Engineering Sciences, Box 534, Ångström laboratory, Uppsala University, SE-75121,Uppsala, Sweden.

ABSTRACT Diamond is a unique material in many respects. One of the most well-known extreme properties of diamond is its ultrahardness. This property of diamond actually turns out to have interesting consequences for charge transport, in particular at low temperatures. In fact, the strong covalent bonds that give rise to the ultrahardness results in a lack of short wavelength lattice vibrations which has a strong impact on both electron and hole scattering. In some sense diamond behaves more like a vacuum than other semiconductor materials. In this paper we describe some interesting charge transport properties of diamond and discuss possible novel electronic applications. INTRODUCTION There are a range of techniques to carry out charge transport measurements. Unfortunately, most of these techniques, e.g. Hall measurements, are not applicable to materials with high resistivities, but there are some useful techniques that are more appropriate. The Time-of-Flight (ToF) technique is ideally suited for drift velocity measurements at very low carrier concentrations and for highly resistive materials. The ToF technique is a powerful method that has been used to investigate the electrical properties of semiconductors. It is based on a direct time-resolved measurement of the current, arising from the drift of free charge carriers in an applied electric field. If the electric field distribution across the sample is known, the transit time of the charge carriers is directly related to the drift mobility. This method has been used to measure, e.g., drift velocity in amorphous silicon [1], silicon [2,3], cadmium telluride [4] and also in natural and single-crystalline CVD diamond [5–16]. The ToF technique can also be used to measure carrier lifetimes [6,17] and to extract the electric field distribution in particle detectors and device structures [18]. Our group has used the ToF technique with low-intensity pulsed UV excitation to study charge transport in single-crystalline CVD diamond over a wide range of temperatures (5-500 K) [9,13,19]. At temperatures above 150 K, the drift velocities of holes and electrons exhibit a similar behaviour as functions of the applied electric field [9]. For both carrier types the drift velocities show a linear behaviour at low fields; at higher fields the drift velocities saturate. Below 150 K, however, holes and electrons behave very differently. This can be traced to the completely different structure of the valence and conduction bands in diamond [20,21] and to the high energy of optical phonons in diamond [22]. The very high energy of optical phonons leads to an absence of such phonons at low temperatures and therefore diamond behaves, due to the lack of optical phonon scattering, more like a vacuum than other semiconductor materials do. In particular, for electron transport interesting phenomena can be

observed: at temperatures below 150 K negative differential electron mobility has been observed in ToF drift mobility measurements and below about 80 K valley polarized electron states with long relaxation times have been observed. These topics are the explained in more detail in the following sections.

Negative differential electron mobility means that in a certain range of applied electric fields the drift velocities of electrons decrease with increasing electric field. This phenomenon is normally associated with III-V or II-VI semiconductors with an energy difference between different conduction band valleys. The observation of negative mobility in diamond, an elemental group IV semiconductor, can be explained in terms of repopulation effects between different equivalent conduction band valleys. At temperatures well below 80 K intervalley scattering between different conduction band valleys becomes negligible and electron transport depends on the actual valley occupied. THEORY Negative differential electron mobility (NDM) for electronic transport is normally associated with semiconductor materials with non-equivalent valleys, i.e., materials where the conduction band has non-equivalent local minima separated by a small difference in energy. Such materials include III-V semiconductors GaAs, GaN, GaSb, InP, InAs and InSb as well as II-VI semiconductors ZnSe and CdTe. The NDM in these materials arises when electrons are scattered into satellite valleys where they have a higher effective mass. The mechanism is as follows: in an external electric field of sufficient magnitude, electrons in a valley with low effective mass are heated efficiently by high-field effects and scatter into valleys of slightly higher energies with higher effective masses. In these valleys, the heating is reduced and the probability of backscattering is lowered. This leads to a preferential population of high effective mass valleys. Since the magnitude of the population is field dependent, it gives rise to the NDM. For example, in GaAs – the prototype NDM material, NDM is observed at room temperature for electric fields above 3.5 kV/cm and the peak electron velocity is 4·107 cm/s. If the NDM effect is pronounced enough (as e.g. in GaAs), electric current instabilities build up and give rise to Gunn oscillations [23–25]. This effect is, utilized in Gunn diodes for microwave applications.

intervalley fscattering intravalley scattering

intervalley g-scattering

Figure 1. (Left) The six conduction band valleys in the first Brillouin zone in diamond. The minima are situated on the Cartesian axes at 76 % of the distance from the Γ point to the zone boundary. (Right) Different types of electron scattering mechanisms. Diamond has a very different conduction band structure (see fig. 1) than the III-V and II-VI materials mentioned in the previous paragraph. In diamond (as well as in silicon) the conduction band has six isolated conduction band minima in the first Brillouin zone along the [100], [010] and [001] crystallographic axes. These minima have, by symmetry, exactly the same energy and effective masses in (unstrained) diamond. Thus, naively, one may not expect diamond to exhibit NDM at all. However, the anisotropy of the valleys must be taken into account. With the electric field applied along, e.g., the [100] axis in diamond, electrons in the two valleys on the [100] axis respond to the field with the longitudinal effective mass (ml) and electrons in the other four, perpendicular, valleys respond with the transversal effective mass (mt). The difference between transversal and longitudinal effective masses mt < ml results in heating of the electrons in the four perpendicular valleys (hot valleys) which are then depleted, while the two parallel valleys (cool valleys) are enriched. This repopulation effect is counteracted by scattering processes that tend to equalize the valley populations. Such processes include phonon scattering, impurity intervalley scattering and electron-electron scattering. Thus, the observation of NDM resulting from equivalent valley repopulation should be possible to observe in an experiment only for very low impurity and carrier concentrations. This repopulation can be observed in Monte-Carlo (MC) carrier transport simulations. Figure 2 shows the result from a MC simulation of 3000 electrons with no applied field and with an applied field along the [100] direction, respectively. It is clear from this simulation that there is an almost complete repopulation from the hot to the cool valleys in the latter case.

Figure 2. Monte-Carlo transport simulation at 77K showing the electron population density in the conduction band. The left figure shows an equal population of the six valleys with no applied electric field. On the right an electric field, 860 V/cm, is applied along the [100] direction leading to almost complete repopulation into the two valleys aligned with the field.

EXPERIMENT In measurements of the electron drift velocity using the ToF technique in undoped electronic grade single-crystalline CVD samples from Element Six Ltd, we have observed NDM in diamond in the temperature interval 100-150 K and for electric fields in the range 300-700 V/cm [26]. This NDM is due to the previously described valley repopulation. Figure 3 shows the measured drift velocity for electrons (and holes, for comparison) at 120 K.

Figure 3. Measured electron and hole drift velocities vs. applied electric field in the [100] direction. The electron drift velocity decreases with increasing electric field in the interval 300700 V/cm.

It may also be noted from Figure 3 that hole transport behaves very differently from electron transport, and indeed no NDM can be observed for hole transport at any temperature for which we have performed measurements (5-500 K) [19]. This is to be expected. The valence band of diamond consists of three bands: the heavy- hole (hh), light-hole (lh) and split-off (so) bands. The hh and lh bands have their maxima (minimum hole energy) at the Γ point of the Brillouin zone. These two bands are degenerate at the Γ point. The so-band also has its maximum at the Γ point, but with a small energy difference, 13 meV. However MC transport simulations for holes have shown that repopulation between these different bands is relatively small and consequently repopulation has a relatively small influence on the drift mobility, in agreement with our observations. At temperatures well below 100K, intervalley phonon scattering rates become very small in diamond and consequently we have been able to observe nonequilibrium valley populations with relatively long relaxation times, e.g. 300 ns at 77K. This finding has recently been reported by our group in ref [27], where we also report techniques to generate such nonequilibrium populations. See also ref [28]. In addition we have shown that such nonequilibrium populations, "valley polarized electrons", can be transported across macroscopic distances using weak electric fields with negligible loss of valley polarization and that the valley polarization can be detected using Hall angle measurements. Valley relaxation times ~1ns have previously been reported in MoS2 [29–31]. Figure 4 is a plot of the measured (using the ToF technique) drift velocity vs. electric field applied in the [100] direction, for electrons in cool (on the [100] axis) and hot (on perpendicular axes) valleys, respectively, confirming the strong valley transport anisotropy.

Figure 4. Measured electron drift velocities vs. applied electric field in the [100] direction at 70 K. Two different velocities are observed depending on which valleys the electrons occupy. At electric fields >400 V/cm phonon emission results in almost complete repopulation into cool valleys and hot valley electrons can no longer be observed.

From the ratio hot/cool valley drift mobility at low fields the ratio of longitudinal to transversal effective mass can be calculated: we find ml/mt = 5.2 ± 0.2 [27] in agreement with recent ab-initio bandstructure calculations: ml/mt = 5.2-5.5 [32]. In recent cyclotron resonance measurements a slightly higher value was obtained: ml/mt = 5.5 [33]. The ratio is also in agreement with older band structure calculations of Baldereschi (found in ref. [17]) yielding 5.0 ± 0.5, but substantially higher than the result of Willatzen et.al. [20] who obtained 4.4 for the ratio. Previous experimental determinations were derived from measurements of drift velocity [34] and from the excitation spectrum of phosphorous in diamond [35] yielding 3.9 and 5.9 for the ratio, respectively. CONCLUSIONS Below 150 K diamond exhibits negative differential electron mobility, i.e., in a certain range of applied electric fields the drift velocities of electrons decrease with increasing electric field. Negative electron mobility is normally associated with III-V or II-VI semiconductors with an energy difference between different conduction band valleys. The observation of negative mobility in diamond, an elemental group IV semiconductor, can be explained in terms of repopulation effects between different equivalent conduction band valleys. Because of this effect it may be possible to detect Gunn oscillations in diamond, although this has not yet been achieved. At temperatures well below 80 K intervalley scattering between different conduction band valleys becomes small and electron transport depends on the actual valley occupied. This opens up the possibility to use diamond in "Valleytronics", i.e., electronic circuits where the valley polarization is used to carry information. However, this will require the development of crucial valleytronic circuits, such as "valley gates" and "valley amplifiers", which will be a challenging task. The possibility of using valley polarized states for quantum computing is another intriguing possibility for the future [27,36]. ACKNOWLEDGMENTS The authors wish to acknowledge the Swedish Research Council (VR) for financial support through Grants 2010-4011 and 2012-4819.

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