Modeling novel effects in transient current measurements of single-crystal CVD diamond with carrier excitation by MeV α-particles
Graphical abstract
Introduction
In transient current measurements – often referred to as time-of-flight measurements – time resolved currents are recorded due to electrons or holes which are pulse generated in excess of their thermal equilibrium-densities and drift in an external electric field. When the drift distance is known, drift velocities and mobilities can be extracted from the transit times of the measured current pulses.
There is a great body of experimental data from single-crystal chemical vapor deposition (CVD) diamond obtained by the transient current technique (TCT) since knowledge of such basic electrical data is important for characterizing the material for research and applications.
This paper aims at explaining novel effects in transient current measurements performed in the group of H. Pernegger at CERN (Geneva, Switzerland) with excitation of free carriers by 4.6 MeV α-particles. The data published in three papers (Refs. [1–3]) are unique with regard to three aspects:
- (1)
The measurements were performed in a wide temperature range reaching from 295 K to 60 K [1] extended later to the range from 295 K to 2 K [2,3]. To the knowledge of the author no other investigations have been made at temperature as low as 2 K. These measurements showed remarkable features of both electron and hole current pulse shapes (cf. Fig. 1) hinting to a storage and releasing mechanism at around 80 K to 140 K related by the authors vaguely with excitons without further discussion.
- (2)
It is also remarkable that electron and hole current pulses have generically the same pulse shapes for all temperatures and electric fields being at very low and high temperatures rectangular-like and distinguished only quantitatively by virtue of different drift velocities. Naively, this seems not to be surprising but actually the pulse shapes are often very different for the two species deviating from the present rectangular shapes. Examples are found in [4] where at room temperature electron and hole spectra differ widely. Strongly fading amplitudes for holes across the pulse length have in this case been attributed to trapping of the holes by a negative space charge in the bulk related with a non-uniform effective electric field. Reversely, again at room temperature, generically identical pulse spectra were observed for electrons and holes at widely varied electric field strengths in [5]. Also, virtually rectangular-like hole pulse shapes have been reported in [6] at electric bias values of 70 V and 40 V though only at room temperature. In contrast, largely differing pulse shapes were reported in [7] changing in character from low to high excitation. In some TCT investigations mobilities of carriers are shown versus temperature but no pulse spectra are reported at all. Summarizing these few examples, the experimental manifestation of current pulses is manifold and the present pulses are singular in their straight regularity.
- (3)
The new data were obtained from three samples prepared from nominally identical commercially grown CVD single-crystal diamonds but processed in different laboratories. Values for electron drift velocities and mobilities are well comparable to those of published data from other samples. The data look very reliable and sound having been obtained in a laboratory with great pertinent expertise.
These statements conspire to suggest that in spite of the small experimental basis of only three samples the novel effects are real and the samples are - for the purpose of TCT - of outstanding quality. Hence, it appears worth to provide here a comprehensive model of the observed effects which has not been given so far. The original model was developed by the author in 2012 aiming at explaining the data in [[1], [2], [3]]. This turned out to be successful for several observations and the model was widely used in [3]. Other observations required amendments and led to the present final version which differs from the primary version in many decisive parts not yet published and mentioned in detail below.
Section snippets
Samples and excitation
For the sake of easy comprehensibility of the present paper sample data and excitation conditions are briefly resumed following the existing publications, particularly Ref. [3]. Three diamond samples were investigated grown commercially by the chemical vapor deposition (CVD) technique by Element Six Ltd. They are essentially identical in shape and material properties. Quoted was an impurity concentration below 2·1014 cm−3 and a nitrogen contents of <1 ppb. The samples have an area of 4.7 by
Measurements and modeling
Fig. 1 shows current pulse profiles for electrons and holes, respectively, at temperatures from 2 K to 295 K at bias voltages of |UBias| = 100 V and 500 V. At low temperatures the pulse profiles are rectangular-like with smeared edges and have constant intensity up to ≈70 K. They are nearly identical for T = 2…50 K and for T = 150 K…295 K. Unexpected profiles exhibiting rising and falling flanks seemingly of exponential characteristic develop in between. Independent of these effects the transit
Dependence of the activation energy Δ on the electric field
Most of the experiments were initially performed [1,3] at an electric field ξ = 1 V/μm (applied voltage ≈500 V) yielding an ample set of data points for Qtotal(T) as in Fig. 4. In the course of the experimental work Qtotal(T) was also measured for other field strengths extending from 0.2 V/μm (voltage ≈100 V) to 1.8 V/μm (voltage ≈900 V). Fits of Eq. (10) to these cases showed that the thermal activation energy Δ depends on ξ decreasing weakly for ξ ≥ 1 V/μm and increasing in succession more
Low-temperature case
At low temperature, T = 2…75 K, the collected charge is exclusively provided by the outer region. The data points lie approximately on a straight line up to the highest field strengths of ξ = 1.8 V/μm (Ubias = 900 V). The results for holes are again identical within experimental uncertainties (see Ref. [3], Fig. 5.22B). In the model, the number of electrons available for the drift current at t = 0 in the outer region is Na = N0/10μm·(xe + xh) depending via xe and xh on the electric field which
Valley re-population effects
In Refs [2,3]. the transit time t′ of electrons and holes was measured versus temperature for electric fields from ζ = 0.08 V/μm (applied voltage ≈40 V) to 1.8 V/μm (applied voltage ≈900 V) (Fig. 8). Holes exhibit at all bias voltages from 30 V to 900 V a monotonic increase, according to [3] partially compatible with a T3/2 temperature-dependence as typical for acoustic phonon deformation potential scattering.
Electrons, however, show non-monotonic behavior of t′(T) with maxima which become
Critical discussion of the model
A critical issue of the model is the total neglect of diffusion or other physical processes expanding the excitation volume during carrier relaxation. Diffusion of the carriers during relaxation will certainly happen. Also carrier-phonon scattering will cause an additional smearing and expansion of the carrier distribution. Moreover, the relaxing carriers generate a high density of phonons in situ and these may drive the carriers out of the excitation volume into the diamond bulk. This is the
Conclusion
We have interpreted published transient current measurements [[1], [2], [3]] performed under 4.6MeV α-particle excitation at temperatures from 2 K to 295 K and for electric fields from 0.08 V/μm to 1.8 V/μm by a physical model quantified in terms of rate equations: Electrons and holes are driven in opposite directions during relaxation from the high excitation states down to their respective band edges thus creating three regions in the direction of the electric field, two outer regions
Author statement
A statement of the author referring to the exceedance of the deadline and the changes made in the revised manuscript is given in the Letter to the Editor.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The author acknowledges a great many helpful annotations of Dr. Hendrik Jansen on details of his experimental work. I am deeply obliged to M.Sc. Angelika Kaiser of the Institut für Quantenmaterie/Quantum Matter, Group of Semiconductor Physics, Universität Ulm, for drawing many figures needed for comparison with the model in the present article using the published experimental data in [3].
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