Физика твердого тела, 1999, том 41, вып. 5 Influence of the Disorder in Doped Germanium Changed by Compensation on the Critical Indices of the Metal-Insulator Transition © Rolf Rentzsch, A.N. Ionov, Ch. Reich, V. Ginodman, I. Shlimak, P. Fozooni, M.J. Lea Freie Universitt Berlin, FB Physik, WE I, Institut fr Ex-physik, 14195 Berlin, Germany A.F. Ioffe Physico-technical institute of Russian Academy of Sciences, 194021 St.Petersburg, Russia The Resnick and Pearl Inst. Bar-Ilan University, Dept. of Physics, Ramat-Gan, Israel Department of Physics, Royal Holloway University of London, Egham, Surrey, UK E-mail: firstname.lastname@example.org We present a critical review of the present status of the critical exponent puzzle of the metal-insulator transition of doped semiconductors with the emphasis on the role of mesos- and macroscopically inhomogeneity caused by the disorder of intended or unintended acceptors and donors in the crystals. By using both the isotopic engineering and the neutron transmutation doping (NTD) of Germanium we found for low compensations (at K = 1.4 and 12%) that the critical exponents of the localization length and the dielectric constant are nearly = 1/2 and =1, which double for medium compensations (at K = 38 and 54%) to = 1 and =2, respectively.|
Till now there is an intensive debate in the literature isotopically engineered and neutron-transmutation doped.
whether the Metal-Insulator Transition (MIT) is a phase The crystals have in this case a well controlled disorder 74 transition of first or second order and what are the ex- via the compensation by the isotopic enrichment of Ge perimental conditions to obtain it at finite temperatures (K = 0.014, 0.12, 0.38 and 0.54 of n-type conductivity) and and in real (disordered) systesm [1,2]. If the MIT is as of a mesoscopically as well as macroscopically homogeneous second order phase transition a further challenge is the distribution of the impurities with N near Nc. In the case solution of the so called puzzle of the critical index, µ of low compensations we got smples on both sides of for the scaling behaviour of the metallic conductivity near the MIT.
the MIT, i. e. just above the critical impurity concentration Nc and as small compensation, K [2–14]. In particular, in several uncompensated material (Si : P [2–4], Si : As [5,6], 1. Sample Preparation Ge : As [7,8]), some experimental groups obtained µ 1/2, other µ 1 (Si : P , Ge: As and Ge: Ga ), which also Isotopically engineered bulk Ge-crystals were grown from 74 has been found in different compensated material [9–12]. pure Ge, enriched up to 94%, or by a mixture of Ge On the other hand the value of µ 1/2 is significantly with Ge of natural isotopic content. The isotopes Ge smaller than µ = 1 - 1.3 predicted theoretically for an and Ge transmute after irradiation with thermal neutrons 75 Anderson transition driven only by disorder [15–20] and to As-donors and Ge-acceptors. The four series of also greater than Chayes et al.  inequality µ > 2/3 n-type Ge with different isotopic abundance (in %) and for a MIT due to both disorder and electron-electron different K after NTD are shown in the table. The interaction. values of K are proportional to the product of the isotopic abundance and the thermal neutron cross-sections of all The main uncertainty in all previous experimental work isotopes producing impurities, K NGa/NAs whereas the = is whether the impurities, for instance, the donors at n-type impurity concentration is proportional to the irradiation dose.
conductivity during doping are distributed macroscopically homogeneous or not and whether during any chemical doping an unintended disorder via compensation by (backIsotopic abundance of the four series of NTD–Ge after massground) acceptors or defects is present or not. The spectroscopic analysis, compensation degree, conduction type and disorder in doped semiconductors arises mainly from the critical impurity concentration (see text) intended or unintended compensation K which is, for n70 72 73 74 Isotope Ge Ge Ge Ge Ge K (%) Type Nc (cm-3) type material, K = Na/Nd, as well as from correlated incorporation of donors and acceptors from melt grown Series 1 0.2 0.7 3.2 93.8 2.1 1.4 n 3.5·crystals and macroscopic inhomogeneity in the impurity Series 2 1.7 2.4 1.0 93.9 1.0 12 n 4.0·distribution. To avoid these uncertainties we have pre- Series 3 5.0 6.5 2.4 82.8 3.3 38 n 7.1·Series 4 8.1 11.2 ca.4 72.3 ca.4 54 n 1.5·pared four sets of germanium samples which were both 838 Rolf Rentzsch, A.N. Ionov, Ch. Reich, V. Ginodman, I. Shlimak, P. Fozooni, M.J. Lea 2. Results and Discussion All samples with N < Nc at T < 1 K exhibited a temperature dependence of resistance according to (T ) =0 exp(T0/T )1/2. (1) Eq. (1) corresponds to variable-range-hopping conductivity with a Coulomb gap at the Fermi level  and with T0 = 2.8e2/ak, (2) where a is the localization length and k is the dielectric constant. Fig. 1 and 2 show typical dependencies of the resistivity on temperature for low (K = 1.4 and 12%) and medium (K = 38%) disorder. One can see that Eq. 1 is fulfilled at low temperatures for all impurity concentrations where variable rangle hopping is obtained. According to the scaling theory of the MIT both a and k diverge at the MIT with power laws [2,15] a = C1aB (N/Nc) - 1 -, (3a) k = C2k0 (N/Nc) - 1 -, (3b) Figure 2. Temperature dependence of resistivity at medium compensation, K = 0.38.
with / = 2, In Eq. (3a, b) aB 4 nm is the Bohr radius of = the Arsenic donor, k0 15.2 is the static dielectric constant = and C1 and C2 are constants. As the result, the slope of the curves in figs. 1 and 2, i. e. T0 must decrease with a power p = + approaching, T0 = 0 at N Nc. Fig. 3 shows = T0 as function of Nd = n/(1 - K) for different K, where n is the free carrier concentration. The linear extrapolation Figure 3. Determination of Nc at T0 0.
of the curves in Fig. 3. gives us the value of Nc which rapidly increases with K. The scaling relation of T0 versus (N/Nc) - 1 at different K are shown in Fig. 4. At low disorder (K = 1.4 and 12%) the power p is close to the value of 3/2 and doubles at medium disorder (K = and 54%) to a value of about 3. Taking into account / = 2, one obtains 1/2 and 1, and 1 and 2, = = Figure 1. Temperature dependence of resistivity at low compensation, K = 0.014, 0.12. at low and medium K respectively. The values of a(K) Физика твердого тела, 1999, том 41, вып. Influence of the Disorder in Doped Germanium Changed by Compensation on the Critical Indices... combination of T0 (ak)-1 of the temperature dependence of resistivity without magnetic field by using Eqs. (1 and 2) we also estimated k(K). Both dependencies as functions of |(N/Nc)-1| are shown in Fig. 7 and 8, confirming the above estimates for the slopes 1/2 and 1, and 1 and = = for the scaling behavior of a and k, at low and medium K respectively.
From the experimental point of view the puzzle of the critical indices has been solved by well controlled disorder via the compensation degree and homogeneously doping by the combination of artificially changed isotopic content (isotopic engineering) and NTD which give rise to mesoscopic and macroscopic homogeneity. The determination of Nc from Figure 4. T0 vs. |(N/Nc) - 1| at different K.
Figure 6. Temperature dependence of the positive hopping magnetoresistance. The upper straight line shows the range of validity of Eq. (4).
Figure 5. Typical dependence of the hopping megnetoresistance of B2.
can be independently determined from measurements of the positive magnetoresistance. In all samples at T < 0.5 and at B=0.5-2 T the positive magnetoresistance was found  to fulfil the theory in :
3/ln (B)/(0) =+(e2/h2) a4(B2/T ) =+B2/B0(a, T )2, (4) where = 660 is a numerical coefficient. Fig. 5 shows the typical dependence of the hopping magnetoresistance, which confirms the quadratic dependence on a low magnetic field appearing in all samples. The analysis of the temperature dependence according to Eq. (4) is shown in Fig. 6 indicating the range of valid at about T = 0.2-1.5 K for this sample.
From Eq. (4) we calculated a(K) of all samples. By a Figure 7. a and k vs. |(N/Nc) - 1| at low K.
Физика твердого тела, 1999, том 41, вып. 840 Rolf Rentzsch, A.N. Ionov, Ch. Reich, V. Ginodman, I. Shlimak, P. Fozooni, M.J. Lea References  See for a review A. Moebius et al. Phys. Rev. B (1998), to be published.
 R.F. Milligan, T.F. Rosenbaum, R.N. Bhatt, G.A. Thomas. In:
”Electron-Electron Interactions in Disordered Systems” / Ed.
by A.L. Efros and M. Pollak. Modern Problems in Condensed Matter Sciences. 10. Elsevier Science Publi., North Holland (1985). P. 231.
 T.F. Rosenbaum, K. Andres, G.A. Thomas, R.N. Bhatt. Phys.
Rev. Lett. 45, 1723 (1982); M.A. Paalanen, T.F. Rosenbaum, G.A. Thomas, R.N.Bhatt. Phys. Rev. Lett. 48, 1284 (1982).
 P.F. Newman, D.F. Holcomb. Phys. Rev. B28, 638 (1983).
 W.N. Shafarman, D.W. Koon, T.G. Castner. Phys. Rev. B40, 1216 (1989).
 A.N. Ionov, M.J. Lea, R. Rentzsch. Pis’ma Zh. Eksp. Teor. Fiz.
54, 470 (1989). [JETP Lett. 54, 473 (1991)].
 P. Dai, Y. Zhang, M.P. Sarachik. Phys. Rev. Lett. 66, (1991).
 K.M. Itoh, E.E. Haller, J.W. Hansen, J. Emes, L.A. Reichertz, E. Kreysa, T. Shutt, A. Cummings, W. Stockwell, B. Sadoulet, Figure 8. a and k vs. |(N/Nc) - 1| at medium K.
J. Muto, J.W. Farmer, V.I. Ozhogin. Phys. Rev. Lett. 77, (1996).
 G. Hertel, D.J. Bishop, E.G. Spencer, J.M. Rowell, R.C. Dynes.
Phys. Rev. Lett. 50, 743 (1983).
the extrapolation of T0(N) 0 at Efros–Shklovskii variable  M. Yamaguchi, N. Nishida, T. Furubayashi, K. Morigaki, H. Ishimoto, K. Ono. Physica B118, 694 (1983).
range hopping agrees well with Nc from the metallic side.
 R. Rentzsch, K.J. Friedland, A.N. Ionov, M.N. Matveev, For low disorder (at K = 1.4 and 12%) that the I.S. Shlimak, C. Gladun, H. Vinzelberg. Phys. Stat. Sol. (b) critical exponents of the localization length and the dielectric 137, 691 (1986); R. Rentzsch, K.J. Friedland, A.N. Ionov.
constant are nearly = 1/2 and = 1, respectively.
Phys. Stat. Sol. (b) 146, 199 (1988).
The value of = µ at low disorder agree well with  M. Rohde, H. Micklitz. Phys. Rev. B36, 7572 (1987).
early Si : P  and Ge : As  as well as with recent results  H. Strupp, M. Hornung, M. lackner, O. Madel, H.V. Loehneyon uncompensated NTD Ge : Ga  results. At medium sen. Phys. Rev. Lett. 71, 2634 (1993).
desorder (at K = 38 and 54%) the critical indices double  I. Shlimak, M. Kaveh, R. Ussyshkin, V. Ginodman, L. Resnick.
Phys. Rev. Lett. 77, 1103 (1996).
to = 1 and = 2, respectively. These results are  E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramain accordance with results on different chemically doped krishnan. Phys. Rev. Lett. 42, 673 (1979).
material Si : P  and Ge : As/Ga  where the crystal  A. MacKinnon, B. Kramer. Phys. Rev. Lett. 47, 1546 (1981).
homogeneity could be less. Additionally some disorder by  T. Ohtsuki, B. Kramer, Y. Ono. Solid State Commun. 81, unintended compensation or correlated impurity distribution (1992).
is possible . However, the puzzle of the critical indices  M. Henneke, B. Kramer, T. Ohtsuki. Europhys. Lett. 27, between theory and experiment remains unsolved because (1994).
to our knowledge till now there is no unique theory taking  E. Hofstetter, M. Schreiber. Phys. Rev. Lett. 73, 3137 (1994).
 T. Kawabayashi, T. Ohtsuki, K. Slevin, Y. Ono. Phys. Rev. Lett.
into accounts both disorder and strong electron-electron 77, 3593 (1996).
 J. Chayes, L. Chayes, D.S. Fisher, T. Spencer. Phys. Rev. Lett.
54, 2375 (1986).
 A.L. Efros, B.I. Shklovskii. In: ”Electronic Properties of 3. Acknowledgements doped semiconductors”. Springer–Verlag, Berlin (1984).
P.P. 240, 211.
We thank Dr. M Lyubalin for the crystal growth of  R. Rentzsch, A.N. Ionov, Ch. Reich, M. Mller, B. Sandow, P. Fozooni, M.J. Lea, V. Ginodman, I. Shlimak. Phys. Stat.
isotopic engineered undoped material, Drs. I.M. Lazebnik, Sol. (b) 205, 269 (1998); R. Rentzsch, Ch. Reich, A.N. Ionov, W. Gatschke, and D. Gawlik for the neutron irradiation of P. Fozooni, M.J. Lea. Proc. of the 24th International Conferthe samples, Dr. V. Karataev for the mass-spectroscopic ence on the Physics of Semiconductors ICPS–24. Jerusalem, analysis of the isotopic abundance, M. Mueller, R. Ullrich (1998).
and V. Zarygina for the crystal preparation.
This work was only possible by financial support of the RFFI (grant N 97-02-18280), INTAS grant (93-1555/Ext.), German–Israeli Foundation (grant N I-0319-199.07) and the British–German Foundation (grant N 313/ARC 93-97).