Nonequilibrium transport and Electron-Glass effects in thin Ge{}_{\text{x}}Te films

Nonequilibrium transport and Electron-Glass effects in thin GeTe films

Abstract

We report on results of nonequilibrium transport measurements made on thin films of germanium-telluride (GeTe) at cryogenic temperatures. Owing to a rather large deviation from stoichiometry (10% of Ge vacancies), these films exhibit p-type conductivity with carrier-concentration N10cm and can be made either in the diffusive or strongly-localized regime by a judicious choice of preparation and post-treatment conditions. In both regimes the system shows persistent photoconductivity following excitation by a brief exposure to infrared radiation. Persistent photoconductivity is also observed in GeTe samples alloyed with Mn. However, in both GeTe and GeMnTe the effect is much weaker than that observable in GeSbTe alloys suggesting that antimony plays an important role in the phenomenon. Structural studies of these films reveal an unusual degree of texture that is rarely realized in strongly-disordered systems with high carrier-concentrations. Anderson-localized samples of GeTe exhibit non-ergodic transport which are characteristic of intrinsic electron-glasses, including a well developed memory-dip and slow relaxation of the excess conductance created in the excited state. These results support the conjecture that electron-glass effects with inherently long relaxation times is a generic property of all Anderson-localized systems with large carrier-concentration.

pacs:
72.80.Ng 78.47.da 72.15.Rn 72.20.Jv
1

I Introduction

The non-interacting Anderson-insulating phase has been called a Fermi-glass (1), presumably inspired by the spatial arrangement of the localized electronic wavefunctions resembling an amorphous structure. Further considerations, admitting for the long-range Coulomb interaction, inevitably present in a medium devoid of metallic screening, led several authors to suggest that a glassy phase, so called electron-glass (EG), should be observable in real systems (2); (3); (4); (5); (6); (7); (8); (9); (10); (11).

Experimental observations consistent with the anticipated glassy behavior were reported in several Anderson-localized systems (12); (13). On the other hand, these effects were not seen in either Si or GaAs, systems that are readily made insulating and exhibit strong-localization transport properties. To some researchers, this shed doubts on the notion that the electron-glass is a generic phenomenon peculiar to the Anderson insulating regime.

It has been conjectured that the absence of electron-glass features in Si and GaAs is related to their relative low carrier-concentration N (14). This was based on the observation that the dynamics in amorphous indium-oxide films (15) becomes much faster once N10cm. To date, a common feature in all Anderson-insulators that exhibit intrinsic EG effects, in addition to being strongly-localized, is their high carrier-concentrations, typically with N10cm. By “intrinsic” we mean that the non-ergodic effects appear in a given substance independently of the way the sample was prepared to achieve the required parameters (resistance at the measuring temperature, carrier-concentration, and dimensionality determined by the hopping-length to thickness ratio). Most importantly, the system has to exhibit a memory-dip with a width that is commensurate with the carrier-concentration of the material (14). This distinction is important; slow conductance relaxation by itself is not necessarily a sign for EG, slow relaxation (and 1/f noise) may occur in lightly-doped semiconductors, presumably due to extrinsic effects (13).

The correlation between high carrier-concentration and sluggish relaxation rates may suggest the relevance of many-body effects. However, a case may also be made for the difference in disorder being the reason behind the correlation with carrier-concentration. Note that the requirement of strong localization means that a system with higher carrier-concentration has perforce more disorder (required to overcome the higher kinetic energy associated with higher carrier-concentration). One may then argue that the reason for the slow relaxation (and the various glassy features) exhibited by systems with higher Fermi energies is their considerably larger disorder rather than due to correlation effects. It may transpire that there is a peculiar type of disorder that exists in the high-n systems that lightly-doped semiconductors cannot sustain and it is this ”defect” which slows down the relaxation of the system from an out-of-equilibrium state. It is therefore of interest to experimentally test more systems with diversified structural properties.

This work describes transport measurements on GeTe samples, yet another system with carrier-concentration (16) N 10cm, somewhat above the empirical limit for observing electron-glass effects (when the system is strongly-localized). Comparison with films made with the alloy GeSbTe reveals a much weaker persistent-photoconductivity (17) (PPC) in the GeTe films. The relaxation law from the photo-excited state also differ from that observed in GeSbTe presumably associated with another kind of a defect. The microstructure of the GeTe films prepared by the method of this work show some unique features such as preferred orientation (texture) over a extremely large spatial scale, a single-crystal-like attribute. Yet, Anderson-localized films of this material exhibit nonequilibrium transport effects including a memory-dip characteristic of the electron-glass phase just like found in all previously studied systems. The implications of these finding to the origin of slow relaxation of electron-glasses are discussed.

i.1 Sample preparation and characterization

Samples used in this work were prepared by e-gun depositing GeTe onto room temperature substrates in a high-vacuum system (base pressure 110mbar) using rates of 1-2Å/second. The source material was 99.999% pure GeTe (Equipment Support Company, USA). Film thickness was in the range of 30-75Å. Lateral dimensions of the samples used for the low temperature studies were 0.3-0.5mm long and 0.5mm wide. Two types of substrates were used; 1mm-thick microscope glass-slides, and 0.5m SiO layer thermally grown on 100 silicon wafers. These were boron-doped and had bulk resistivity 210cm, deep into the degenerate regime. This makes this substrate suitable to perform as a gate-electrode even at low temperatures. Samples deposited on these wafers were configured as three-terminal devices for field-effect measurements. These were designed to probe n(), the thermodynamic density of states versus energy of the material as well as to test for electron-glass behavior. Samples prepared on microscope glass-slides were mainly used for optical characterization and for Hall-Effect measurements, both performed at room-temperatures.

Each deposition batch included samples for optical excitation measurements, samples for Hall-effect measurements, and samples for structural and chemical analysis using a transmission electron microscope (TEM). For the latter study, carbon-coated Cu grids were put close to the sample during its deposition and received the same post-treatment as the samples used for transport measurements.

The Philips Tecnai F20 G2) was used to characterize the films composition (using energy dispersive spectroscopy, EDS) and microstructure. The EDS typically showed Ge deficiency so, we refer to our deposited films as GeTe. The Cary-1 spectrophotometer was used for optical measurements.

Films deposited at room temperatures were amorphous. TEM and associated diffraction pattern of typical GeTe sample deposited an hour prior to being inserted to the TEM are shown in Fig.1.

Figure 1: Bright-field micrograph and associated diffraction-pattern of the as-deposited GeTe film. The lumpy appearance of the film morphology is characteristic of many amorphous structures and it includes the contribution of the amorphous carbon that is the substrate in this case.

Crystalline samples of GeTe were obtained from the amorphous GeTe deposits by subjecting them to temperatures in the range of 470-490K for 2-3 minutes. The amorphous-crystalline transformation is reflected in the optical properties of the films as a mild change in color tint. In this regard the result is very similar to corresponding situation in the GeSbTe compound studied previously (12) as can be seen in the comparison shown in Fig.2.

Figure 2: Optical transmission through a 50Å GeTe film deposited on a 1mm glass-slide compared with that of a 120Å GeSbTe film (13).

In terms of other properties however, there are significant differences between our crystalline versions of GeTe and the GeSbTe. In particular, their microstructure is different; while both systems exhibit mosaic film structure with a tight, space-filling packing of the crystallites, the GeTe film shows a much more pronounced preferred orientation extending over large scales. This may be seen in both transmission electron microscope (TEM) micrographs depicted in Fig.3 (same sample as shown in Fig.1 after crystallization at 485K) and Fig.4 (same sample after being ‘aged’ for a week). The diffraction patterns in these figures were taken in selected-area mode covering 0.8 micron circle diameter. Pronounced preferred orientation was still conspicuous using a selected-area of 4 microns, which is at least order of magnitude larger than the average size of the grains in the studied films. This extensive texture, extending over a scale much larger than a typical grain size, was uniformly observed across the 3 mm TEM grid by scanning it with a constant electron-beam.

Figure 3: Bright-field micrograph and associated diffraction-pattern of the sample in Fig.1 after crystallization. The extensive texture is clearly observed in both the image and the diffraction pattern despite the plethora of point defects. The diffraction pattern is consistent with the rhombohedral (R-3m) phase of GeTe.

We found it hard to get films with appreciable sheet resistance even in quite thin specimen. The reason for that is presumably the reduced grain-boundary scattering and better mobility relative to that observed in the GeSbTe alloys (assuming that impurity-contents, and carrier-concentration are the same). Special measures had to be taken in fabricating films with high values for R (which were required for observing electron-glass properties). These included reducing the film thickness (down to 30Å relative to the constant 120Å used in the study of GeSbTe (13)), and aging the films in the lab atmosphere. A micrograph and associated diffraction of an aged film is shown in Fig.4.

Figure 4: Bright-field micrograph and associated diffraction-pattern of the sample in Fig.3 after aging it at room-temperature for a week (see text). Note the increase in diffuse scattering relative to the fresh film. However, the preferred-orientation, extending across grain boundary is still intact.

The room-temperature resistance of our films span the range 600-29k which yielded 0.8k-5.6M at T4K. The upper limit of this range was obtained by reducing the film thickness to 30Å and exposing the sample to the lab atmosphere for several days. However, we were yet unable to produce films with R5.6M at 4.1K. By comparison, using the GeSbTe alloy it was rather easy to obtain samples with R as high as 50M even with films that were 3-4 thicker (12) than the GeTe used in this work.

The diffraction pattern of the aged film shown in Fig.4 exhibited an increase in the diffuse scattering (compare Fig.4 with Fig.3) suggestive of an increased disorder. This is possibly due to enhanced surface scattering or the creation of a disordered dead-layer by oxidation process at the film-air  boundary. Energy dispersive spectroscopy revealed a 10-15% increase in oxygen content in the aged film relative to the fresh sample without a noticeable change in the Ge/Te ratio (being 0.90.05). X-ray photoelectron spectroscopy on GeTe pellets exposed to lab atmosphere found adventitious layers of carbon on their surface, which may also be source of enhanced scattering in films exposed to air. Another difference is the relative intensity of some rings that became more noticeable in the aged sample. Selected-area diffraction, sampling a circle of 0.8m diameter was used to scan across the sample and the same features were observed throughout, so structural inhomogeneities in this material are probably limited to spatial scales 1m.

The Hall effect that was monitored for some of the films showed a rather small change during the aging process; for a 100% increase of the film R the Hall resistance has increased by 6%. Based on these Hall effect measurements the carrier-concentration n of the films was N=(1.5-2)x10cm, somewhat smaller than the N=(4-9)x10cm found in the GeSbTe compound (13). In both cases the Hall effect had the sign of p-type carrier consistent with theoretical prediction for the material (18). The latter, based on equilibrium concentration of Ge vacancies in the ideal crystal anticipated carrier-concentration of 10cm holes. The carrier-concentration in our films, larger by roughly order of magnitude is probably a result of the abundant structural defects (readily observable in the TEM micrographs, Fig.3 and Fig.4) that apparently allows more Ge vacancies than the ordered crystal can sustain in equilibrium.

i.2 Measurement techniques

Conductivity of the samples was measured using a two terminal ac technique employing a 1211-ITHACO current preamplifier and a PAR-124A lock-in amplifier. All measurements were performed with the samples immersed in liquid helium at T4.1K held by a 100 liters storage-dewar. This allowed up to two months measurements on a given sample while keeping it cold (and in the dark) which was utilized to extend the time-duration of relaxation processes as well as many cycles of excitation-relaxation experiments. Fuller measurement techniques related to electron-glass properties are described elsewhere (19).

The ac voltage bias in conductivity measurements was small enough to ensure near-ohmic conditions (except for the current-voltage plots and the ‘stress protocol’ described in the Results section below). Optical excitations in this work were accomplished by exposing the sample to an AlGaAs diode operating at 0.880.05m, mounted on the sample-stage typically 10-15mm from the sample. The diode was energized by a computer-controlled current-source (Keithley 220).

Ii Results and discussion

ii.1 Persistent photo-conductivity in GeTe

A main difference between the transport properties of GeTe and the GeSbTe alloy is their different sensitivity to optical excitation. The experimental protocol used for observing photoconductivity is illustrated in Fig.5 using a diffusive GeTe film with R=5k and, for comparison, a GeSbTe film with similar R and thickness measured under the same conditions. The experiment begins 24 hours after the sample is cooled-down to 4.1K by recording G(t) for 1-2 minutes to establish a baseline conductance G. The IR source is then turned on for 3 seconds then turned off while G(t) continues to be measured. The brief IR burst causes G to promptly increase by G which decays slowly with time once the source is turned off (Fig.5). Both samples exhibit excess conductance that persists for a long time after the optical excitation. In terms of magnitude, the persistent photoconductivity (PPC) signal is however much more conspicuous in the GeSbTe film at all values of R. A detailed comparison of the PPC magnitude versus R illustrating the difference between the two systems is given in Fig.6. This figure includes three GeMnTe samples. These were prepared by co-depositing Mn with the GeTe compound to test the effect of magnetic impurities. The Mn inclusion had only a small effect on the samples mobility, reducing it by 10-20% (for 20% Mn) relative to the pure compound. As shown in Fig.6 it also had a negligible effect on the PPC performance of the compound.

Figure 5: Comparing the persistent-photoconductivity effect typically observed in our GeTe samples with that of a GeSbTe film. The R of the GeTe film used here is 5k while R of the GeSbTe film is 3k. The same protocol was used for both samples; same infrared intensity, distance from film, and duration of exposure. Inset shows the field-effect for the GeTe film.
Figure 6: The relative magnitude of the infrared induced excess conductance in the PPC state for our GeTe films as function of their R. These data are compared with the respective data for the GeSbTe studied previously (13) and measured under the same conditions. Also shown are results for three Mn-doped GeTe samples. Dashed lines are guides to the eye.

Actually it appears that the PPC in pure GeTe and GeMnTe compound differs from that in the GeSbTe system not just by magnitude. The relaxation law that fits the time dependence of the excess conductance G in the GeSbTe compounds showed a rather good fit to a stretched exponential law: G(t){-(t/)} with =0.1 for all samples with R in the 10-10 range. A similar expression could be fitted to the PPC data of our most resistive GeTe films (G for the lower resistance samples was too small to allow a meaningful fit) but with =0.14-0.22. An example of a fit is shown in Fig.7.

Figure 7: The persistent photoconductivity of a GeTe film with R=750k. The inset depicts a fit (dashed line) to a stretched-exponent for the associated excess conductance as function of time.

This, and the much smaller G (all other things being equal) suggests that the presence of the Sb plays a similar role in enhancing the PPC performance in germanium-tellurides compounds as that of In impurities in lead-telluride alloys (19). It would be of interest to find what other elements are effective in enhancing PPC in these systems. That is important for understanding the basics of the PPC phenomenon but also as a tool for elucidating the physics of the electron-glass; the PPC, when prominent enough, may be an effective way to increase the carrier-concentration in a system which is an important parameter in controlling the dynamics of the electron-glass (21).

ii.2 Strongly localized GeTe is an intrinsic electron-glass

Like in previously studied materials, a pre-requisite for observing electron-glass features is that the system must be strongly-localized. This applies in particular to the appearance of a memory-dip (MD) in the field-effect measurement, which is the identifying signature of the intrinsic (13) electron-glass. A memory-dip appeared in our films at T4K once their R/e. A well developed MD can be seen in Fig.8 for a GeTe film with R=195k.

The sign of G(V) (reflecting how thermodynamic density of states n changes with energy) is consistent with hole conduction (Fig.2 and Fig.8)

Figure 8: The field-effect for a GeTe sample with R=195k showing a a memory-dip with a relative magnitude of 0.75% magnitude (defined as by G/Geq where Geq is the equilibrium value of the conductance at the bath temperature). The dashed line is the thermodynamic part of the field-effect measurement (as in the weakly-disordered sample in Fig.5).

and the sign of Hall-effect measurements on these films. Both the slope of G(V) and the relative magnitude of the MD increases with disorder as shown in Fig.9. The disappearance of the MD as the system approaches the diffusive regime is common to all intrinsic electron-glasses, and has been seen in both two-dimensional and three-dimensional systems (22). This is a crucial attribute of the phenomenon and should be the starting point for any theoretical model.

Figure 9: (a) The relative change of conductance of the antisymmetric part of the field-effect as function of the GeTe films R. It is defined as: G(-30V)/G(+30V)-1. (b) The relative magnitude of the memory-dip for the GeTe films as a function of their R. Dashed lines are guides to the eye.

As may be expected, GeTe films that exhibit MD also show the other characteristic features of electron-glasses. Fig.10 shows the excitation and ensuing relaxation of the excess conductance due to a sudden change of the gate voltage.

Figure 10: Conductance versus time describing excitation of the electron-glass by a sudden change of the gate voltage. Sample has R=5.6M. The protocol involved a fast sweep of V(within 2s) from the equilibrium value V=0V, where the system spent 24 hours, to V=-10V where it was left for the duration of the time shown. The inset illustrates the logarithmic relaxation law of the excess conductance produced by the V change.

Another way to take the system away from equilibrium is the ‘stress-protocol’ (23). By applying sufficiently strong electric field across the sample (between the ‘source’ and ‘drain’), the system accumulates energy in excess of its thermal energy. This translates into excess conductance G building up over the time the field is on. Once the strong field is removed, and the conductance is monitored under Ohmic conditions, G decays with time and G approaches its equilibrium value. Both the buildup and decay of G involve a protracted process. Unlike the sudden increase of G when Visswitched (Fig.10), G grows continuously throughout the stress period without saturating. This is the analogue of the ‘time-dependent heat-capacity’ typical of glasses (24) which is due to the wide temporal spectrum of the system degrees of freedom. The stress protocol is illustrated in Fig.11a and Fig.11b for the conductance evolution G(t) during the relaxation and during the stress respectively. The conductance dependence on the applied voltage of this sample is shown in Fig.12 with the voltage values used during the stress and relaxation periods are marked on the G(V) curve.

Figure 11: The two parts of the stress protocol used on a GeTe sample with R=5.4M (a) Relaxation of the excess conductance after the stress was relieved. (b) The slow conductance buildup during the time stress was on.

The results of the stress-protocol (Fig.11) are essentially the same in all previously studied systems that exhibit electron-glass attributes (25). Qualitatively different behavior has been observed in granular systems (26). Granular systems differ from Anderson insulators in other aspects as well although they share some glassy features like a memory-dip (27).

It is also worth commenting on the mechanism by which the applied non-ohmic field takes the system out of equilibrium. The enhanced conductance that appears immediately after applying the field is associated with adiabatic modification of the hopping probabilities (28). This is the dominant effect when the resistance is large but Joule-heating is to some degree also responsible to the increase of G when a large voltage is applied (except when the applied field frequency exceeds the electron-phonon inelastic-rate (29)). Joule-heating is the reason for the slow buildup of excess conductance observed under large voltages. Qualitatively similar behavior is achieved by raising the bath temperature. However, comparing the behavior of G(t) under field F versus that of raising the bath temperature by T (to achieve the same initial G), demonstrated that under T the ensuing excess conductance increased with time at a faster rate (30). This is just a manifestation of the fact alluded to above; heating is only part of the reason for non-ohmicity in the hopping regime. The advantage of using voltage-swings over raising-lowering the bath-temperature is the higher speed and controllability of the procedure. The price is the uncertainty in assigning a value of ”effective-temperature” to the stress protocol; the value of G under non-ohmic fields is, in general, not a reliable thermometer; nonohmic measurements of G(V) are not simply related to the equilibrium values of G(T).

Figure 12: The conductance of the sample used for the stress protocol (Fig.11) as function of the applied voltage. Sample dimensions are L=W=0.5mm. Marked  by arrows are the voltages used during the stress and relaxation periods.

As mentioned above, the visibility of the memory-dip, over the antisymmetric G(V) (controlled by the energy dependence of the thermodynamic DOS), increases with R. The memory-dip for one of the most resistive samples we were able to manufacture in this study is shown in Fig.13 where it is compared with the MD of a GeSbTe film with the same R and measured under the same conditions (temperature, sweep-rate, and gate-voltage range).

Figure 13: Comparing the memory-dip of a GeTe sample with R=3M with the memory-dip of a GeSbTe sample with R=6.2M Both G(V) curves were taken with the same sweep rate. The arrows delineate the typical width of each memory-dip,

The main difference between the two G(V) curves in Fig.13 is the steeper antisymmetric contribution of the GeSbTe sample. Closer examination reveals that the typical width of the MD is also somewhat wider for the Sb-doped alloy. The narrower width of the MD in GeTe may be a result of the smaller carrier-concentration in this material which is in line with the general trend observed in previously studied electron-glasses. To date however, the only material where it was possible to change the carrier-concentration over a considerable range is amorphous indium-oxide (15). The carrier-concentration in GeTe is associated with Ge vacancies (17), and as demonstrated by Bahl and Chopra, the carrier-concentration in this system can be varied over a decade by heat-treatment during crystallization (16). GeTe may then be another system that allows testing the relation between the carrier-concentration, glass-dynamics, and the MD-width, by either controlling the sample stoichiometry during deposition, alloying with foreign elements, and thermal annealing. Future work will also focus on modifying the transport parameters of this system by various dopants.

With the addition of the currently studied GeTe, there are now seven different Anderson-localized systems that exhibit intrinsic electron-glass effects. The previously studied systems and their properties were discussed elsewhere (25). The only feature common to all these systems is having relatively large carrier-concentration; N5x10cm. These systems have quite different structural properties making it hard to conceive of a common defect that might be responsible for the long relaxation times observed in their nonequilibrium transport properties. Grain-boundaries for example, are not likely to be relevant as their contribution to transport must be very weak in GeTe relative to other electron-glasses while the electron-glass effects exhibited by all these systems are very similar; they all show slow relaxation and a memory-dip. It is therefore more likely that it is the magnitude of the disorder rather than its specific nature that is the important factor. This, in turn, suggests that quantifying the disorder in the Anderson-insulating phase may be a vital step in the quest to unravel the mechanism responsible for the electron-glass dynamics.

Acknowledgements.
This research has been supported by a grant administered by the 1126/12 grant administered by the Israel Academy for Sciences and Humanities.

Footnotes

  1. preprint:

References

  1. N. F. Mott, Phil. Mag., 26, 1015 (1972).
  2. J. H. Davies, P. A. Lee, and T. M. Rice, Phys. Rev. Letters, 49, 758 (1982); M. Grünewald, B. Pohlman, L. Schweitzer, and D. Würtz, J. Phys. C, 15, L1153 (1982); J. H. Davies, P. A. Lee, and T. M. Rice, Phys. Rev. B 29, 4260 (1984).
  3. M. Pollak and M. Ortuño, Sol. Energy Mater., 8, 81 (1982); M. Pollak, Phil. Mag. B 50, 265 (1984).
  4. G. Vignale, Phys. Rev. B 36, 8192 (1987).
  5. C. C. Yu, Phys. Rev. Lett., 82, 4074 (1999).
  6. M. Müller and L. B. Ioffe, Phys. Rev. Lett. 93, 256403 (2004).
  7. Vikas Malik and Deepak Kumar, Phys. Rev. B 69, 153103 (2004).
  8. R. Grempel, Europhys. Lett., 66, 854 (2004).
  9. Eran Lebanon, and Markus Müller, Phys. Rev. B 72, 174202 (2005); M. Müller and E. Lebanon, J. Phys. IV France, 131, 167 (2005).
  10. Ariel Amir, Yuval Oreg, and Yoseph Imry, Phys. Rev. B 77, 165207 (2008); Ariel Amir, Yuval Oreg, and Yoseph Imry, Annu. Rev. Condens. Matter Phys. 2, 235 (2011); Y. Meroz, Y. Oreg and Y. Imry, EPL, 105, 37010 (2014).
  11. M. Pollak, M. Ortuño and A. Frydman, ”The Electron Glass”, Cambridge University Press, England (2013).
  12. M. Ben-Chorin, D. Kowal and Z. Ovadyahu, Phys. Rev. B 44, 3420 (1991); M. Ben-Chorin, Z. Ovadyahu and M. Pollak, Phys. Rev. B 48, 15025 (1993).
  13. Z. Ovadyahu , Phys. Rev. B. 91, 094204 (2015).
  14. Z. Ovadyahu, Phys. Rev. B 78, 195120 (2008).
  15. A. Vaknin, Z. Ovadyhau, and M. Pollak, Phys. Rev. Lett., 81, 669 (1998).
  16. S. K. Bahl and K. L. Chopra, Journal of Applied Physics 41, 2196 (1970). More recent work was done on alloys based on this material mostly for study of the so called phase-change materials, for a review see: Simone Raoux, Annu. Rev. Mater. Res., 39, 25 (2009) and for a study of the metal-insulator transition in these compounds see: T. Siegrist, P. Jost, H. Volker, M. Woda, P. Merkelbach, C. Schlockermann and M.Wuttig, Nature Mater., 10, 202 (2011).
  17. M. G. Craford, G. E. Stillman, J.A. Rossi, and N. Holonyak, Phys. Rev., 168, 867 (1968); D. V. Lang and R. A. Logan, Phys. Rev. Lett., 39, 635 (1977); D. V. Lang, R. A. Logan, and M. Jaros, Phys. Rev. B 19, 1015 (1979); H. J. Queisser and D. E. Theodorou, Phys. Rev. Lett., 43, 401 (1979); Jennifer Misuraca, Jelena Trbovic, Jun Lu, Jianhua Zhao, Yuzo Ohno, Hideo Ohno, Peng Xiong, and Stephan von Molnár, Phys. Rev. B 82, 125202, (2010).
  18. A. H. Edwards A. C. Pineda, P. A. Schultz, M.s G. Martin, A. P. Thompson, and H.P. Hjalmarson, C. J. Umrigar, J. Phys.: C Condens. Matter 17,  L329  (2005); ibid Phys. Rev. B 73, 045210 (2006).
  19. A. Vaknin, Z. Ovadyahu, and M. Pollak, Phys. Rev. B 65, 134208 (2002).
  20. B. A. Akimova,. V. Dmitriev., R. Khokhlov, L. I. Ryabova, phys. stat. sol. (a) 137, 9 (1993).
  21. Z. Ovadyahu, Phys. Rev. Lett., 115, 046601 (2015).
  22. Z. Ovadyahu , Phys. Rev. B. 91, 054204 (2014).
  23. V. Orlyanchik, and Z. Ovadyahu, Phys. Rev. Lett., 92, 066801 (2004).
  24. Stephen M. Wolpert, Alexander Weitz, and Bernhard Wunderlich, Journal of Polymer Science, 9, 1887 (1971).
  25. Z. Ovadyahu, Phys. Rev. B 88, 085106 (2013).
  26. N. Kurzweil and A. Frydman, Phys. Rev. B 75, 020202 (2007).
  27. C. J. Adkins, J. D. Benjamin, J. M. D. Thomas, J. W. Gardner and A. J. McGeown, J. Phys. C: Solid State Phys. 17, 4633 (1984); G. Martinez-Arizala, D. E. Grupp, C. Christiansen, A. Mack, N. Markovic, Y. Seguchi, and A. M. Goldman, Phys. Rev. Lett., 78, 1130 (1997); G.         Martinez-Arizala, C. Christiansen, D. E. Grupp, N. Markovic, A. Mack, and A. M. Goldman, Phys. Rev. B 57, R670 (1998); T. Grenet, Eur. Phys. J, 32, 275 (2003); N. Kurzweil and A. Frydman, Phys. Rev. B 75, 020202 (2007); T. Grenet, J. Delahaye, M. Sabra, and F. Gay, Eur. Phys. J. B 56, 183 (2007); Tal Havdala, Ariel Eisenbach and Aviad Frydman, EPL, 98, 67006 (2012).
  28. R. M. Hill, Philos. Mag. 24, 1307 (1971); B. I. Shklovskii, Fiz. Tekh. Poluprovodn. 6, 2335 (1972) [Sov.Phys. Semicond. 6, 1964 (1973)]; N. Apsley and H. P. Hughes, Philos. Mag. 31, 1327 (1975); M. Pollak and I. Riess, J. Phys. C9, 2339 (1976).
  29. Z. Ovadyahu, Phys. Rev. Lett., 108, 156602 (2012).
  30. Z. Ovadyahu, Phys. Rev. B 73, 214208 (2006).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
157685
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description