Journal of the Physical Society of Japan
Vol. 73, No. 1, January, 2004, pp. 173–183#2004 The Physical Society of Japan

Complete Scaling Analysis of the Metal–Insulator Transition in Ge:Ga:Eﬀects of Doping-Compensation and Magnetic Field
Kohei M. I TOH1 , Michio W ATANABE2 , Youiti O OTUKA3 , Eugene E. HALLER4 and Tomi OHTSUKI5
1

Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522
2
Macroscopic Quantum Coherence Laboratory, FRS, RIKEN, Saitama 351-0198
3
Tsukuba Research Center for Interdisciplinary Materials Science and Institute of Physics, University of Tsukuba, Ibaraki 305-8571
4
UC Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S.A.
5
Department of Physics, Sophia University, Tokyo 102-8544(Received September 9, 2003)

We report on the complete scaling analysis of low temperature electron transport properties with andwithout magnetic ﬁeld in the critical regime for the metal–insulator transition in two series ofhomogeneously doped p-type Ge samples: i) nominally uncompensated neutron-transmutation-doped(NTD) 70 Ge:Ga samples with the technological compensation ratio K < 0:001, and ii) intentionallycompensated NTD nat Ge:Ga,As samples with K ¼ 0:32. For the case of the uncompensated series in zeromagnetic ﬁeld, the critical exponents , , and  determined for the electrical conductivity (),localization length (), and impurity dielectric susceptability (imp ), respectively, change at the very
vicinity of the critical Ga concentration (N  Nc ). Namely, the anomalous critical exponents, e.g.
  0:5, change to   1 only within the region 0:99Nc < N < 1:01Nc . On the other hand, the same
critical behavior,   1, was found for the K ¼ 0:32 series in much larger region 0:25Nc < N < 2:4Nc .
This ﬁnding suggests that the   1 critical behavior observed for the nominally uncompensated series
in the extremely narrow region is due to the presence of the self-compensation of acceptors by native
defects and/or technologically unavoidable very small amount of doping compensation (K < 0:001).
Therefore, the width of the concentration that can be ﬁtted with   1 around Nc is likely to scale withthe degree of compensation (K), and disappears in the limit K ! 0, i.e., only the region with theanomalous exponent   0:5 remains for the case of K ¼ 0. An externally applied magnetic ﬁeld tonominally uncompensated samples also broadens the width of   1 around Nc , but with a mechanism
clearly diﬀerent from that of compensation. The uniﬁed description of our experimental resultsunambiguously establishes the values of the critical exponents , , and  for doped semiconductors withand without compensation and magnetic ﬁeld.
KEYWORDS: doped semiconductor, metal–insulator transition, scaling theory, Mott–Anderson transition, hopping
conduction
DOI: 10.1143/JPSJ.73.173

1.

compensated Ge:Ga samples by tuning N and B.
Since the classic experiment of Rosenbaum et al. thatshowed   0:5 for stress (S)-tuned Si:P,8) a wide variety ofexperiments probing the value of  has been performed on
nominally uncompensated semiconductors as a function of N
or S in the absence of the externally applied magnetic ﬁeld
B. The results are truly puzzling as summarized in Table I.
One immediately sees in Table I that diﬀerent values of 
have been reported even for the same system.   0:5 ispuzzling from a theoretical point of view since it violates
Chayes et al.’s inequality   2=326) assuming Wegner’s
scaling law  ¼  for the three dimension27) where  is thecritical exponent for the localization length  and forcorrelation length 0 .

Introduction

The metal–insulator (MI) transition in the presence ofboth disorder and electron–electron interaction turns out tobe one of the most challenging subjects in condensed-matterphysics. Despite many decades of theoretical1–6) andexperimental eﬀorts,7–24) researchers are yet to agree uponan uniﬁed description of the phenomena.25) The dopinginduced MI transition in single crystalline semiconductors isthe best example of disorder and interaction induced
transition that has been studied extensively via measurements of physical quantities such as the electrical conductivity, dielectric constant, and heat capacity. In particular,
the critical behavior of the electrical conductivity at zerotemperature ð0Þ has been evaluated as a function of aparameter t that describes the degree of the disorder andinteraction;ð0Þ / jt=tc  1j

Table I. Conductivity critical exponents () reported for a wide variety of
nominally uncompensated (K  0) semiconductors in the absence of
externally applied magnetic ﬁeld.

ð1Þ

where  is the conductivity critical exponent and tc is thecritical value of t that separates the insulating and metallic
phases. The MI transition in semiconductors has been
investigated as a function of the doping concentration (N),externally applied magnetic ﬁeld (B), and externally applied
uniaxial stress (S), i.e., t  N; B; S, and tc  Nc ; Bc ; Sc ,respectively, in eq. (1). In this paper we probe the MI
transition in nominally uncompensated and intentionally173

Semiconductor system



Si:P

0.5,8Þ 1.0,9Þ 1.210Þ

Si:As

0.5,11Þ 1.012Þ

Si:B

0.65,13Þ 1.614Þ

Ge:As

0.5,15Þ 1.212Þ

Ge:Sb

0.916Þ

Ge:Ga

17;18Þ

0.5,

1.219Þ

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K. M. I TOH et al.

The following example of the Si:P situation illustrates
very well how various research groups have reacheddiﬀerent values of . Because the MI transition discussed
here is considered to be a quantum phase transition that
occurs at zero temperature, one has to rely on particulartheoretical models to extrapolate information, which represents the zero-temperature states of physical parameters suchas electrical conductivity , localization length , impurity
dielectric susceptability imp , etc. Rosenbaum et al. havemeasured the temperature dependence of  of nominallyuncompensated Si:P down to T ¼ 3 mK with ﬁnely-tuned
uniaxial stress S, and estimated the zero temperature
conductivity ð0Þ for a given S by linear extrapolation of to T ¼ 0 K assuming  / T 1=2 . They determined   0:5
by achieving a good ﬁt for a wide range on the metallic side
(0:7Sc < S < Sc ) of ð0Þ vs S by eq. (1).8) More recently, agroup from Universität Karlsruhe questioned the relatively
wide range of N and S that could be ﬁtted with   0:5, andproposed   1:3 on N-tuned Si:P by redeﬁning Nc at avalue 6% smaller than that of Rosenbaum et al., and bylimiting the critical region only to Nc < N < 1:07Nc .10)
Rosenbaum et al. immediately argued that the   1:3
region analyzed by the Karlsruhe group was an artifact due
to an inhomogeneous dopant distribution,28) but the Karlsruhe group responded that only the region scalable with theso-called ﬁnite temperature analysis should be considered as
the true critical region.29) The ﬁnite temperature scaling
takes the form;5)
ðt; TÞ / T x f ðjt=tc  1j=T y Þ:ð2ÞHere y ¼ 1=z where z is the dynamical scaling exponent.
The critical exponent is given by  ¼ x=y. Equation (2) has
two advantages over the conventional analysis involving eq.
(1). Firstly, eq. (2) allows one to use values of ðt; TÞrecorded at ﬁnite temperatures, i.e., the conventional
extrapolation to T ¼ 0 can be avoided. Secondly, eq. (2)allows one to evaluate ðt; TÞ recorded on the both sides of
the transition (t < tc and tc < t), while eq. (1) can be used
only on the metallic side.
The Karlsruhe group has shown that their N-tuned datascale according to eq. (2) only in the vicinity of Nc and not
for the wide range evaluated by Rosenbaum et al. In thesame spirit, the Karlsruhe group has further analyzed thedata of S-tuned Si:P and obtained   1 by limiting thecritical region only to the vicinity of Sc .9) In our view, thesituation similar to Si:P applies to all the other systems listed
in Table I. Researchers, including ourselves,19) have obtained   1 for nominally uncompensated systems only
when they have limited the critical region to a very small
range, typically within a few percent of Nc or Sc . It should
also be noted that the values of  reported for intentionallycompensated samples were always   1 for a wide rangeof N above Nc ; very often up to 50% and more.30–34) The
relation between   1 observed only in the vicinity of tc in
nominally uncompensated samples and the similar   1
observed for a wide range above tc in intentionallycompensated samples must be clariﬁed.
Moreover, it is important to point out that many of theprevious experiments performed on the nominally uncompensated samples were not supposed to be able to analyze
such a small region, within a few percent of Nc or Sc ,because a typical spatial ﬂuctuation of the doping concentration within a typical sample size of a few mm could easily
be a few percent when it was prepared by the standard meltdoping method.35) In this regards, Rosenbaum et al.’swarning against the analysis of the very vicinity of tc 28)
should be taken very seriously. Doping ﬂuctuations also
make it impossible to determine Nc and Sc precisely for the
reliable determination of . Only a few experimental groups,
including ourselves, have recognized this problem andemployed a method known as neutron-transmutation-doping(NTD) in order to realize completely homogeneous doping
down to the atomic level.15,17–24,32) The NTD preparation ofsamples is absolutely crucial for the successful scaling
analysis as we will show in this paper.
The present paper reports the experimental studies on theeﬀects of the doping compensation and externally appliedmagnetic ﬁeld on the MI transition of doped semiconductors.
In order to better illustrate the signiﬁcance of the present
results, we shall summarize important conclusions of ourearlier results17–24) at the beginning of the paper, and proceed
to the discussion of the compensation and applied magnetic
ﬁeld.
2.

Experiment

2.1 Sample preparation and characterization
A chemically very pure 70 Ge crystal of isotopic composition [70 Ge] = 96.2 at.% and [72 Ge] = 3.8 at.%, and a nat Ge
crystal with natural isotopic abundance [70 Ge] = 20.5 at.%,[72 Ge] = 27.4 at.%, [73 Ge] = 7.8 at.%, [74 Ge] = 36.5 at.%,and [76 Ge] = 7.8 at.% were grown using the Czochralski
method. The as-grown crystals are free of dislocations, ptype with an electrically active net-impurity concentration
less than 5  1011 cm3 .
A series of nominally uncompensated samples with thecompensation ratio K < 0:001 were prepared by the following procedure. The thermal neutron irradiation leading to
NTD was performed at the University of Missouri Research
Reactor with a thermal to fast neutron ratio of 30 : 1.
Wafers sliced from the 70 Ge crystal become p type due toneutron capture 70 Ge + n!71 Ge followed by electron
capture with a half-life of 11.2 days forming a 71 Ga acceptor.
The small fraction of 72 Ge capturing neutron becomes 73 Ge
which is stable, i.e., no further acceptors or donors are
introduced to the 70 Ge crystal. The rapid-thermal annealing
after NTD at 650 C for 10 s removed most of the irradiationinduced defects from the samples. A short annealing time isimportant in order to avoid the redistribution and/or
clustering of the uniformly dispersed 71 Ga acceptors. The
concentration of the electrically active radiation defects
measured with deep level transient spectrometry (DLTS)
after the annealing is less than 0.1% of the Ga concentration,36) i.e., technological K < 0:001. Note here that theso called self-compensation near N  Nc may lead to alarger K.37) The dimension of most samples used for
conductivity measurements was 6  0:9  0:7 mm3 . Four
strips of boron-ion-implanted regions on a 6  0:9 mm2 faceof each sample were coated with 200 nm Pd and 400 nm Au
pads using a sputtering technique. Annealing at 300 C for
one hour activated the implanted boron and removed thestress in the metal ﬁlms. The Ga concentration N in our 70 Ge
samples after NTD is given precisely by

J. Phys. Soc. Jpn., Vol. 73, No. 1, January, 2004

½71 Ga ðcm3 Þ ¼ 0:1155  n ðcm2 Þ;

K. M. I TOH et al.

175

ð3Þ

where n is the thermal neutron ﬂuence. Therefore, we havean ability to control N  ½71 Ga with the precision of 0.01%.
A series of intentionally compensated samples has been
prepared by NTD of the nat Ge crystal using the sameprocedure as the one employed for the nominally uncompensated 70 Ge samples. Upon thermal neutron irradiation,
the ﬁve stable isotopes of nat Ge undergo the following
nuclear reactions after neutron capture: 70 Ge + n!71 Ge!
71
Ga acceptor, 72 Ge + n!73 Ge, 73 Ge + n!74 Ge, 74 Ge +n!75 Ge!75 As donor, and 76 Ge + n!77 Ge!77 As !77 Se
double donor. The isotopic abundance in nat Ge and thethermal neutron capture cross sections yield a p type Ge
crystal after NTD with the ﬁxed compensation ratio K:38)
K

½Minority Impurity ½As þ 2½Se
¼
 0:32½Majority Impurity
½Ga

ð4Þ

It is therefore possible to ﬁnely tune the net-carrier
concentration [Ga]–[As]–2[Se] with precise control of theneutron ﬂuence n while maintaining the compensation ratio
constant at 0.32. One of advantages of working with NTD
nat
Ge is that we can reliably use other group’s data recordedwith NTD nat Ge because the compensation ratio is always
ﬁxed at 0.32 and dopants are homogeneously distributed
regardless of who prepares it. For the work discussed here
we have prepared ﬁve samples of NTD nat Ge. Theirtemperature dependence of the conductivity has been
analyzed together with data reported for eight NTD nat Ge
samples by Zabrodskii and Andreev.39)
2.2 Measurements
The electrical conductivity () of the nominally uncompensated samples has been measured using a standard lockin method at 21.0 Hz down to T  20 mK. The cryostat andall the analog instruments were placed inside a shieldedroom. The temperature of samples in a 3 He–4 He dilution
refrigerator has been calibrated very carefully as describedin ref. 18. Magnetic ﬁelds were applied in the direction
perpendicular to the current ﬂow by means of a superconducting solenoid. The inﬂuence of the magnetic ﬁeld on
the temperature calibration has been examined.40)
The conductivity measurements of the compensatedsamples were performed using a 3 He refrigerator down toT  300 mK. The two point measurements with the implanted contacts were employed for many of the highresistivity samples. Sample heating was avoided by using anelectrical power of less than 1014 W.
3.
3.1

Results I: Nominally Uncompensated NTD 70 Ge:Ga
SamplesZero-temperature scaling analysis of the metallic
samples
The temperature dependence of the electrical conductivityðTÞ for eight samples with positive d=dT in the scale ofT 1=3 is shown in Fig. 1.19) Nc ¼ 1:859  1017 cm3 for thisnominally uncompensated series of samples as it will beshown later by numerical ﬁtting with eq. (2). Based on thisﬁnding, the concentrations N of Ga of the eight samples lie
between 0:994Nc and 1:028Nc with the top three samples
being metallic at T ¼ 0 K. Mott’s minimum metallic con-Fig. 1. Electrical conductivity vs. T 1=3 for the eight NTD 70 Ge:Ga
samples in the vicinity of Nc . From bottom to top in units of 1017 cm3 ,the concentrations N are 1.848, 1.850, 1.853, 1.856, 1.858, 1.861, 1.863,
and 1.912, respectively. Samples with ﬁlled marks (top three curves) turnout to be metallic while the ones with other marks are insulating for
T ¼ 0 K.

ductivity min for Ge:Ga is estimated to be 7 S/cm using therelation min  CM ðe2 =h ÞNc1=3 with CM ¼ 1=20. The temperature dependence of ðTÞ is expected to be proportional toT 1=2 when it is governed mainly by electron–electroninteraction. Therefore, ð0Þ is obtained usually by extrapolation assuming ðTÞ / T 1=2 , and such an analysis was
performed in our earlier work.17) However, we havesubsequently found that the samples near the transition(<1% of Nc ) obeyed  / T 1=3 instead of T 1=2 as shown in
Fig. 1.18) This observation of the change in the temperaturedependence from T 1=2 to T 1=3 as N approaches Nc isinteresting because T 1=2 is predicted when interaction isimportant (Mott transition) while T 1=3 is expected41) whendisorder is dominant (Anderson transition).42) It indicatesthat the nature of the transition for the two concentration
regions, Nc < N < 1:01Nc and 1:01Nc < N, is diﬀerent forour nominally uncompensated samples.
In ref. 18 we have developed a method that allows for an
appropriate extrapolation of ðTÞ to T ¼ 0 when thetemperature dependence changes from T 1=2 to T 1=3 as N !
Nc . Figure 2 shows the zero-temperature conductivity ð0Þ
determined for a total of twelve metallic samples, including
the three metallic samples with ﬁlled marks in Fig. 1, that wemeasured previously18,19) as a function of N=Nc  1. The
best ﬁt to the twelve data points with eq. (1) yields  ¼
0:50 0:04 with Nc ¼ 1:860  1017 cm3 ,18) i.e., Nc ¼
1:860  1017 cm3 is employed for the horizontal axis ofFig. 2(a). On the other hand, as we will show later, the best
ﬁt with eq. (2) with a limited number of the samples within
1% of Nc is obtained with  ¼ 1:2 and Nc ¼ 1:859 
1017 cm3 , i.e., Nc ¼ 1:859  1017 cm3 is used for Fig.
2(b). Only the two metallic samples closest to Nc in Fig. 2(b)lie within 1% of Nc and they are the samples which exibit
 / T 1=3 . All the samples with N=Nc  1 > 102 exhibit
 / T 1=2 . The dashed curve is the best ﬁt for them with
ð0Þ / ðN=Nc  1Þ with  ¼ 1:2. What is interesting is the

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K. M. I TOH et al.

Fig. 3. Finite-temperature scaling analysis of ðN; TÞ using eq. (2) withx ¼ 1=3, y ¼ 2=3, and Nc ¼ 1:860  1017 cm3 , and (b) x ¼ 0:38, y ¼
0:32, and Nc ¼ 1:8590  1017 cm3 . The solid curve in (b) is the best ﬁt
to the data. The symbol for each sample is the same as the one in Fig. 1.

Fig. 2. Zero-temperature conductivity ð0Þ vs N=Nc  1 determined
experimentally ( ) by extrapolation of ðTÞ to T ¼ 0 for (a) Nc ¼
1:860  1017 cm3 and (b) Nc ¼ 1:859  1017 cm3 . They include ð0Þ
for a total of twelve samples as a summary of the previous measurements.18,19) The solid line in (a) represents the best power-law ﬁt byð0Þ / ðN=Nc  1Þ where Nc ¼ 1:860  1017 cm3 and  ¼ 0:50
0:04. The dotted line in (b) is a ﬁt for the two samples closest to Nc
with  ¼ 1:2 for it will be shown in Fig. 3(b) that the two samples obey
 ¼ 1:2 with Nc ¼ 1:859  1017 cm3 rather than  ¼ 0:5 with Nc ¼
1:860  1017 cm3 . The solid curve in (b) is the best ﬁt assuming
 ¼ 0:5.

fact that there is a clear kink in ð0Þ vs N=Nc  1 for the caseof Fig. 2(b), and the fact that  ¼ 1:2 applies only for thetwo samples that show T 1=3 dependence.
3.2 Finite temperature analysis19)
The dependence ðN; TÞ / T 1=3 at N  Nc immediately
implies x  1=3 in eq. (2) since the contribution to thetemperature dependence from the f ðjN=Nc  1j=T y Þ term
becomes small for the region N  Nc . It was also shown in
Fig. 2(a) for a wide range of the concentration Nc < N <1:4Nc that  ¼ 0:5 when the conventional zero temperature
analysis was performed with Nc ¼ 1:860  1017 cm3 .
Therefore, x ¼ 1=3 and y ¼ 2=3 in eq. (2) are expectedfrom the relation  ¼ x=y for the analysis of the wide
concentration region Nc < N < 1:4Nc . Figure 3(a) shows theﬁnite temperature scaling plot of ðN; TÞ using eq. (2) withx ¼ 1=3, y ¼ 2=3, and Nc ¼ 1:860  1017 cm3 . ðN; TÞrecorded between T ¼ 20 and 750 mK was used for the
analysis in Fig. 3(a). Fairly good scaling was obtained on themetallic side as expected, while scaling on the insulatingside is clearly unsatisfactory. In order to ﬁnd a better set of x,y, and Nc , numerical ﬁtting has been performed using thefollowing non-linear equation;"
ðN=Nc  1ÞðN=Nc  1Þ 2xðt; TÞ ¼ T a0 þ a1
þ a2
Ty
Ty
#


ðN=Nc  1Þ 3
þ a3
:
ð5Þ
Ty
It is important to stress here that the conductivity of boththe metallic and insulating samples must be described by asingle scaling function, e.g., eq. (5) with a single set ofparameters. When we impose this strict restriction, it is notpossible to achieve satisfactory ﬁtting with eq. (5) for the
samples with the wide range of N used in Fig. 3(a).
Figure 3(b) shows the result of the ﬁtting analysis when
the critical region was limited to N ¼ Nc 0:01Nc , i.e., only
the data from ﬁve insulating and two metallic samples
closest to the transition are ﬁtted. The solid curve in Fig. 3(b)is the best ﬁt obtained numerically with x ¼ 0:38, y ¼ 0:32,
Nc ¼ 1:8590  1017 cm3 , a0 ¼ 5:75, a1 ¼ 580, a2 ¼
1:97  104 , and a3 ¼ 3:15  106 in eq. (5), which agree
very well with the conductivity of both the metallic andinsulating samples. A similar ﬁt with a fourth-order equation
leads to practically the same set of parameters with theabsolute value of the fourth-order term being much smaller
than those of the lower-order terms, i.e., third-order ﬁtting is
suﬃcient. The major consequence of this analysis is  ¼
x=y ¼ 1:2 0:2 which satisﬁes the Chayes et al.’s inequality
 ¼   2=3.26) Although the range of N we chose forscaling may appear to be quite small, we emphasize againthat this range is the only region that can be scaled with eq.
(2) and that the seven samples included are the only ones
having ðN; TÞ / T 1=3 with approximately the same slope

J. Phys. Soc. Jpn., Vol. 73, No. 1, January, 2004

K. M. I TOH et al.

177d=dT 1=3 (see Fig. 1).
Following the success of scaling for the very narrow jN 
Nc j 0:01Nc region, we have attempted to scale thosesamples outside of the N ¼ Nc 0:01Nc region, i.e., theN  0:01Nc and N 0:99Nc regions with eq. (5). The
metallic side of this region is characterized by   0:5 as wehave found in Fig. 2. However, even with exclusion of thejN  Nc j 0:01Nc samples, we could not obtain satisfactoryﬁtting with eq. (5). Therefore, the so called   0:5 region is
not scalable with the ﬁnite temperature scaling eq. (2).
3.3

Scaling of the variable range hopping resistivity as
N ! Nc from the insulating side20,21)
Since it has become clear from the ﬁnite temperaturescaling and the zero-temperature scaling of the metallic
conductivity that the nature of the conduction within 1% of
Nc is diﬀerent from that of outside (N < 0:99Nc and
1:01Nc < N), it becomes important to evaluate the behaviorof the charge transport on the insulating side in detail.
Figure 4 shows the temperature dependence of theresistivity  of eighteen insulating samples in the temperature range T ¼ 20{250 mK. In general the temperaturedependence of the resistivity  for variable range hopping
conduction is given by ¼ 0 T q expðT0 =TÞp

ð6Þ

where p ¼ 1=4 and 1=2 have been predicted for hopping
across1) and within43) a parabolic-shaped Coulomb gap in thedensity of the states, respectively. q ¼ 0 is usually assumed
since the temperature dependence in the strongly localized
regime is determined mainly by the value of p in the
exponential term. Note that ln  is plotted against T 1=2 in
Fig. 4 because most of the curves in the low temperature
limit appear to form straight lines supporting the relationln  / T 1=2 . However, a closer inspection of the samples
close to Nc has revealed that q ¼ 1=3.20,21) Figure 5(a)shows ln  vs T 1=2 for the four samples close to Nc that do
not actually obey eq. (6) when q ¼ 0 is assumed. On theFig. 5. (a) ln  vs T 1=2 and (b) ln( multiplied by T 1=3 ) vs T 1=2 . From
top to bottom N=Nc ¼ 0:994; 0:995; 0:997, and 0.999.

other hand, it becomes consistent with eq. (6) when q ¼
1=3 is assumed as seen from the straightness of the data inthe plot of ln T 1=3 vs T 1=2 in Fig. 5(b) with respect to thatin (a).  / T 1=3 is consistent with the observation  ¼
1 / T 1=3 in Fig. 1.
Because the variable range hopping theory of Efros andShklovskii43) applies to our nominally uncompensatedsamples all the way to Nc from the insulating side, we shalluse the following relation to scale T0 in eq. (6) with p ¼ 1=2
and q ¼ 1=3:
kB T0 Fig. 4. ln  vs T 1=2 for 18 insulating samples. The concentration N from
top to bottom in the unit of Nc are 0.923, 0.943, 0.948, 0.951, 0.952,
0.957, 0.966, 0.971, 0.981, 0.986, 0.989, 0.990, 0.991, 0.992, 0.994,
0.995, 0.997, and 0.999. (Nc ¼ 1:859  1017 cm3 ).

1
2:8e2
4 0 ðNÞðNÞ

ð7Þ

in SI units, where ðNÞ ¼ h þ imp ðNÞ is the dielectric
constant and ðNÞ is the localization length. imp ðNÞ is theimpurity dielectric susceptability and h ¼ 15:4 is therelative dielectric constant of the host germanium. As N
approaches Nc from the insulating side one expects
imp ðNÞ ¼ 0 ð1  N=Nc Þ and ðNÞ ¼ 0 ð1  N=Nc Þ .
With these relations T0 becomes
kB T0 ¼

2:8e2
ð1  N=Nc Þ ;4 0 0 0

ð8Þ

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K. M. I TOH et al.

Fig. 6. T0 as a function of 1  N=Nc determined with p ¼ 1=2 and q ¼
1=3 in eq. (6). The solid and dashed curves are the best ﬁts to the data
for the regions 1  N=Nc > 102 and 1  N=Nc < 102 , respectively.

assuming imp ðNÞ
h in the critical regime. The criticalexponent ¼  þ  for T0 has been determined experimentally for our nominally uncompensated samples by evaluating the slope T0 of each sample shown in Fig. 4 as a functionof N. As seen in Fig. 6, there is a kink at 1% below Nc which
divides the regions into ¼ 0:95 0:08 for 0:9Nc < N <0:99Nc and ¼ 3:5 0:8 for 0:99Nc < N < 0:999Nc . The
solid curve in Fig. 6 represents calculated T0 for the 0:9Nc <N < 0:99Nc region using eq. (8) with ¼ 1, 0 ¼ 15:4
(dielectric constant of host Ge), and 0 ¼ 8 nm (Bohr radiusof Ga acceptors) as was done by Ionov et al.,44) which agrees
surprisingly well with the data in the¼ 1 region.
However, it is obvious that we expect 0 to be the dielectricsusceptability of a single Ga acceptor instead of h ¼ 15:4
for the critical region. The dashed curve in Fig. 6 is the best
ﬁt to T0 for the 0:99Nc < N < 0:999Nc region using eq. (8)with ¼ 3:5, 0 ¼ 2  104 , and 0 ¼ 8 nm. Note that thevalue 0 ¼ 2  104 is reasonable for a single Ga acceptor.
3.4

Scaling of the localization length  and the impurity
dielectric susceptability imp for N ! Nc from theinsulating side21)
Our next step is to separate T0 into  and imp based on themagnetoresistance measurement performed on our nominally uncompensated series of insulating samples that demonstrates Efros and Shklovskii’s variable range hopping
conduction all the way up to Nc . For =
1, themagnetoresistance of the variable range hopping is expressed by;43)ln½ðB; TÞ=ð0; TÞ  0:0015 ð= Þ4 ðT0 =TÞ3=2 ;ð9Þ
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
where h =eB is the magnetic length in SI units.
According to eq. (9), the magnetic-ﬁeld variation of ln  at
T ¼ const. is proportional to B2 , i.e., ln ðB; TÞ ¼ln ð0; TÞ þ CðTÞB2 , and the slope CðTÞ is proportional toT 3=2 . Since eq. (9) is equivalent to CðTÞ=T 3=2  0:0015ðe=h Þ2 4 T03=2 ;ð10Þ is given by  5:1ðh =eÞ1=2 1=4 T03=8 :ð11ÞTherefore, it is possible to determine  of each sample from

Fig. 7. Localization length  vs 1  N=Nc (lower data set) and thedielectric susceptibility imp arising from the impurities vs Nc =N  1(upper data set). Dashed curves are the ﬁts to the data with ðNÞ /ð1  N=Nc Þ and imp ðNÞ / ðNc =N  1Þ .

the magnetoresistance measurements and calculate imp
using eqs. (7) and (11). Details of such procedures have been
given in ref. 21.
Figure 7 shows  and imp ¼  h determined for our
nominally uncompensated samples.21) We should note thatboth  and imp are suﬃciently larger than the Bohr radius(8 nm for Ge) and static host dielectric constant of Ge( h ¼ 15:4), respectively. Just like for the case of T0 shownin Fig. 6, both  and imp show sharp kinks at N  0:99Nc .
The concentration dependence of  and imp below andabove 0:99Nc can be independently ﬁtted very well with thescaling formula ðNÞ / ð1  N=Nc Þ and imp ðNÞ /ðNc =N  1Þ , respectively, as shown in Fig. 7.
The most important outcome here is the experimental
conﬁrmation of Wegner’s scaling law27)    ( 1:2) andthe prediction of the scaling theory 2   ( 2:3) for theregion 0:99Nc < N < 0:999Nc . On the other hand,   0:5
determined from the zero temperature extrapolation, is notequal to   0:33 for the region N < 0:99Nc , though therelation 2   ð 0:62Þ is satisﬁed. It was already indicatedby the ﬁnite temperature scaling that only the region within
1% of Nc is scalable with eq. (2). The fact that Wegner’s
scaling law is valid only within 1% of Nc supports our earlier
ﬁnding that the scaling theory is applicable only to theregion within 1% of Nc and not to the region outside for thecase of the nominally compensated Ge.
Figure 8 summarizes the behavior of T0 and ð0Þ as afunction of normalized concentration N=Nc for the nominally uncompensated samples. The solid curves are thescaling curves with ¼ 0:95 for the insulating samples, and ¼ 0:5 for the metallic samples that are valid for the
samples outside of 1% of Nc . The dashed curves are thescaling curves with ¼ 3:5 for the insulating samples, and ¼ 1:2 for the metallic samples that are for inside of 1%
of Nc . The plot shows convincingly that the nature of theregion within 1% of Nc is diﬀerent from that of outside.

J. Phys. Soc. Jpn., Vol. 73, No. 1, January, 2004

Fig. 8. T0 ( ) and ð0Þ ( ) as a function of normalized cocentration
N=Nc for the nominally uncompensated samples. The solid curves are thescaling curves with ¼ 0:95 for the insulating samples, and  ¼ 0:5 forthe metallic samples that are outside of 1% of Nc . The dashed curves are
the scaling curves with ¼ 3:5 for the insulating samples, and  ¼ 1:2
for the metallic samples that are inside of 1% of Nc .

Fig. 9. ln  vs T 1=2 of thirteen NTD nat Ge:Ga,As with K ¼ 0:32
evaluated in this study. Filled circles represent the samples preparedand measured by the present authors while open circles are the data
provided by Professor Zabrodskii.39) The concentrations N from top tobottom in the unit of Nc are 0.25, 0.35, 0.45, 0.53, 0.56, 0.60, 0.71, 0.72,
0.93, 1.28, 1.36, 1.94, and 2.40.

4.

K. M. I TOH et al.

179

Fig. 10. T0 (and
) and ð0Þ ( ) as a function of normalized
concentration N=Nc for the deliberately compensated, K ¼ 0:32 samples.
The solid scaling curves are the best ﬁts obtained with ¼ 3:06 and ¼ 0:97. Filled circles represent the samples prepared and measured by
the present authors while open circles and squares represent the samples
prepared and measured by Professor Zabrodskii and co-workers.39)

concentration 0:25Nc < N < 2:4Nc yields ¼ 3:06 0:25
and  ¼ 0:97 0:07, and they are found to satisfy the
expected relation  3  3 so that Wegner’s relation
 ¼ . Our results, ¼ 3:06 0:25, is in excellent agreement with Rentzsch et al.’s results,
 3, previously
reported for two series of n-type NTD 74 Ge:As with K ¼
0:38 and 0.54 in the concentration range 0:2 < N=Nc <0:91.45) It was also shown by Katsumoto et al. that   2 forcompensated samples,34) i.e., the combination of our resultsfor deliberately compensated samples ¼  þ   3 with
Katsumoto’s   2 yields   1   that satisﬁes Wegner’s
scaling law. The signiﬁcance of ¼ 3:06 0:25 and  ¼
0:97 0:07 for a wide range 0:25Nc < N < 2:4Nc of the
K ¼ 0:32 series is the fact that they agree very well with
¼ 3:5 0:8 and  ¼ 1:2 0:2 for a very narrow range
0:99Nc < N < 1:01Nc of the nominally uncompensatedseries.
Figure 11 shows the ﬁnite temperature scaling of the K ¼
0:32 series. For this purpose we have performed a leastsquare ﬁtting with open circles only because the open circles
cover a wide range of concentrations spanning both theinsulating and the metallic phases, while ﬁlled circles cover

Results II: Deliberately Compensated NTD nat Ge:Ga
SamplesFigure 9 shows the temperature dependence of theresistivity for thirteen deliberately compensated (K ¼ 0:32)
NTD nat Ge:Ga,As samples analyzed in this study. Some ofthe data (open circles), which have been published in ref. 39,
were provided by Professor Zabrodskii. The ranges oftemperatures used for the measurements are T ¼ 0:25{1 K
and 0.65–2 K for ﬁlled and open circles in Fig. 9, respectively.
For our deliberately compensated series, the criticalexponent ¼  þ  for T0 has been determined experimentally by evaluating the slope T0 of each sample shown in
Fig. 9. Similarly, the conductivity exponent  has been
determined based on the extrapolation of ðN; TÞ to T ¼ 0 K.
The results are summarized in Fig. 10. The least-square
ﬁtting (solid curves in Fig. 10) for the wide range of

Fig. 11. Finite temperature scaling analysis of ðN; TÞ using eq. (12) withx ¼ 0:33 and y ¼ 0:32. The solid curve is the best ﬁt to the data
represented by open circles. Open and ﬁlled circles are the same as inFig. 9. One in every three data points is shown for open circles.

180

J. Phys. Soc. Jpn., Vol. 73, No. 1, January, 2004

K. M. I TOH et al.

only the limited regions of the insulating phase.
"ðt; TÞ
ðN=Nc  1ÞðN=Nc  1Þ 2
ln
¼
a
þ
a
þ
a
0
1
2
Tx
Ty
Ty
#


ðN=Nc  1Þ 3
þ a3
:
ð12Þ
Ty
The solid curve in Fig. 11 is the best ﬁt obtained numericallywith x ¼ 0:33, y ¼ 0:32, a0 ¼ 1:05, a1 ¼ 4:55, a2 ¼ 5:69,
and a3 ¼ 2:90 using eq. (12). It is shown convincingly thatthe conductivity of the K ¼ 0:32 series (especially open
circles for they are the data used for ﬁtting) collapse to form
one universal curve for a very wide range of concentration
0:25Nc < N < 2:4Nc . As expected, the value of the conductivity critical exponent  ¼ x=y ¼ 1:01 0:04 foundfrom this analysis is in excellent agreement with  ¼
0:97 0:07 determined by the zero-temperature extrapolation in Fig. 10. The deviation of open and ﬁlled circles inFig. 11 at lower temperatures is due to the fact that the data
recorded in the present study ( ) extend to much lower
temperatures. The open circles are expected to merge withﬁlled circles when the measurements are extended to lower
temperatures. However, one sees in Fig. 11 that each series( and ) collapses onto a separate single curve and thediﬀerence between the two is very small. The result of theanalysis with eq. (12) is practically unchanged even if we
include both open and ﬁlled circles.
5.

Fig. 12. Zero-temperature conductivity ðB; 0Þ vs normalized concentration N=Nc ð0Þ  1, where ðB; 0Þ is the zero-temperature conductivity for aconstant B and Nc ð0Þ ¼ 1:859  1017 cm3 is the critical concentration at
B ¼ 0. From top to bottom the magnetic ﬂux density increases from 1 T to
8 T in steps of 1 T. The dashed curve at the top is for B ¼ 0.   1 holds
within the shaded region where < 0 , while   0:5 is valid for outsidewhere> 0 .
and 0 are the magnetic and correlation lengths,
respectively.

Results III: Eﬀect of Magnetic Field

Similar to the eﬀect of compensation,   1 has beenfound experimentally for magnetic ﬂux density B of theorder of one tesla for nominally uncompensated semiconductors: Ge:Sb,16,46) Ge:Ga,18) Si:B,47,48) and Si:P.49) In
order to discuss the eﬀect of externally applied magnetic
ﬁeld on the nature of the MI transition, it is important toconsiderthe length scale known as the magnetic length
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 h =eB.50) When a variety of length scales such as thecorrelation length, thermal diﬀusion length, inelastic scattering length, spin scattering length, and spin–orbit scattering length are larger than , the system is considered to be inthe ‘‘magnetic-ﬁeld regime’’. The length scale of interest to
us is the correlation length 0 on the metallic side (N > Nc )that diverges at N ¼ Nc . When the ﬁeld strength is weak, the‘‘magnetic-ﬁeld regime’’ where we assume   1 to hold isrestricted to a narrow region of the concentration around Nc .
Outside this region, the system crosses over to the ‘‘zeroﬁeld regime’’ where  ¼ 0:5 is found. Such behavior is
depicted in Fig. 12 where ðB; 0Þ, the zero-temperature
conductivity at constant B, is plotted as a function of
N=Nc  1. The shaded region in Fig. 12 is the ‘‘magneticﬁeld regime’’ where < 0 , i.e.,   1, while outside of theregion is the ‘‘none-magnetic-ﬁeld regime’’ where > 0 ,i.e.,   0:5. The correlation length 0 as a function of
N=Nc ð0Þ  1 in Fig. 12 has been estimated assuming thatvalues of the correlation length 0 are equal to the localization length  given in Fig. 7 with the mirror symmetryaround Nc , i.e., ð1  N=Nc Þ ¼ 0 ðN=Nc  1Þ, since theexponent is the same and the amplitude ratio is usually ofthe order of unity. The exponent  obtained for the‘‘magnetic-ﬁeld regime’’ by eq. (1) with t  N and tc  Nc
are  ¼ 1:03 0:03 at B ¼ 4 T and  ¼ 1:09 0:05 at

Fig. 13. ðB; TÞ vs T 1=2 of the N ¼ 1:063Nc sample at the constantmagnetic ﬁeld B. From top to bottom: B ¼ 0; 2:0; 4:0; 4:8; 4:9; 5:0; 5:1;
5:2; 5:3; 5:4; 5:5; 5:6; 5:7; 5:8; 5:9; 6:0; 7:0 and 8.0 T. One in every thirty
data points is shown.

B ¼ 5 T.
Figure 13 shows the magnetic ﬁeld and temperaturedependence of the conductivity ðB; TÞ of a sample having
N ¼ 2:004  1017 cm3 . In contrast to Fig. 1 of the N-tuning
experiment with B ¼ 0, ðB; TÞ / T 1=2 is observed for awide range of constant B, i.e., ðB; TÞ is plotted against T 1=2 .
The zero temperature conductivity ðB; 0Þ obtained by thesimple extrapolation of ðB; TÞ to T ¼ 0 assuming ðB; TÞ /T 1=2 decreases with increasing B. This particular sample wasfound to undergo a MI transition at Bc ¼ 5:5T with  ¼

J. Phys. Soc. Jpn., Vol. 73, No. 1, January, 2004

K. M. I TOH et al.

Fig. 14. Finite temperature scaling analysis of ðB; TÞ using eq. (2) withx ¼ 0:50, y ¼ 0:54, and Bc ¼ 5:45 T, The solid curve is the best ﬁt to thedata. The symbol for each sample is same as the one in Fig. 13.

1:1 0:1 based on the analysis using eq. (1) with t  B andtc  Bc .
Figure 14 shows the ﬁnite temperature scaling of ðB; TÞ
for B ¼ 2{6 T and T ¼ 50{500 mK for the metallic samples
and B ¼ 2{6 T and T ¼ 100{250 mK for the insulating
samples. These ranges of B and T have been chosen in orderto ensure that ðB; TÞ of each sample is / T 1=2 withapproximately the same d=dT in Fig. 13. The solid curve in
Fig. 14 is the best ﬁt using eq. (2) with t  B and tc  Bc ,and x ¼ 0:50, y ¼ 0:54, Bc ¼ 5:45 T, a0 ¼ 7:43, a1 ¼ 8:30,
a2 ¼ 0:217, and a3 ¼ 0:063. The conductivity criticalexponent  ¼ x=y ¼ 0:93 0:10 obtained here for themagnetic-ﬁeld tuning of a Ge:Ga with N ¼ 1:063Nc is ingood agreement with the results of the conventional
extrapolation analysis  ¼ 1:1 0:1.22) We have shown
theoretically in ref. 22 that the critical exponent  obtainedby N-tuning with constant B is equal to  obtained by Btuning as long as they are in the ‘‘magnetic-ﬁeld regime’’.
6.

Discussion

The critical exponents obtained for Ga:Ge are summarized in Table II. For the case of nominally uncompensatedsamples (K  0) with B ¼ 0,   0:5 is obtained for thewide concentration range 1:01Nc < N < 1:4Nc by the conventional extrapolation of ðTÞ to T ¼ 0. However, the ﬁnite
temperature scaling is not applicable to this region.   0:5
obtained from extrapolation is not equal to   0:33, i.e.,
Wegner’s scaling law of  ¼  is not satisﬁed, though theprediction of the scaling theory 2   seems to hold for thewide range in the insulating side. It is important to noticethat all of   0:5 previously reported for a variety of doped

181

semiconductors (see Table I) have been obtained for thewide range of concentrations on the metallic side by the
extrapolation method, which means the behavior is universal. This situation changes dramatically if we limit theregion to 0:99Nc < N < 1:01Nc . Both the ﬁnite temperaturescaling and extrapolation to zero-temperature work, andWegner’s scaling law     1:2 and the prediction of thescaling theory 2   are satisﬁed. Together with z  3, allof the critical exponents we obtain for this narrow region of
nominally uncompensated Ge:Ga agree with what have beenobtained with nominally uncompensated Si:P by the Karlsruhe group.9,10) Again, this shows how universal thisbehavior for doped semiconductor is regardless of thematerial system (Si or Ge) and conduction type (n or p
types). In fact, all of   1 cases listed in Table I have beenfound by limiting the critical region very close to Nc , as wasdone in our analysis.
For the case of intentionally compensated samples(K ¼ 0:32) with B ¼ 0, both the ﬁnite temperature scaling
and extrapolation to zero-temperature work for a surprisingly large region 0:25Nc < N < 2:4Nc . Wenger’s scalinglaw     1:0 and the prediction of the scaling theory2   are satisﬁed. z ¼ 3 is found from the ﬁnite temperature scaling.
For the case of nominally uncompensated samples (K 0) with ﬁnite magnetic ﬁelds B > 0, the ﬁnite temperaturescaling with B and T applies as long as the system is in themagnetic ﬁeld regime, i.e., < 0 as shown in Figs. 12 and14.   1:0 and z  2 have been determined for themagnetic ﬁeld induced transition.
The most distinct feature in Table II is the fact that all of
the critical exponents found for K  0 with B ¼ 0 for
0:99Nc < N < 1:01Nc agree within their experimental errors
with those for K  0:32 with B ¼ 0. This is a strong
evidence for the fact that the critical behavior in theseregions is governed by the same physics, namely dopingcompensation, i.e., the width of the   1 region isdetermined by the degree of doping-compensation. It appear
to scale with the value of K and disappear for completely
uncompensated (K ¼ 0) semiconductors. The experimental
problem, of course, is that it is impossible to prepare K ¼ 0
samples both technologically and thermodynamically.37) The
reason for the wide scattering of critical exponents reported for nominally uncompensated systems in Table I
may be very simple; researchers have employed samples
prepared in diﬀerent manners (leading to diﬀerent values ofK) and they probed a wide variety of widths around Nc and/or Sc , i.e., some found   1 governed by the eﬀect of
unavoidable compensation and others found   0:5 meas-

Table II. Critical exponents of Ge:Ga. ½ð0Þ have been determined by the conventional extrapolation to zero temperature using eq. (1). x and y have been
determined by the ﬁnite temperature scaling [eq. (2)]. Columns with  cannot be determined because the ﬁnite temperature scaling is not applicable.
Missing critical exponents (open columns) are to be determined in the future. The exponent in parentheses has been taken from ref. 34. The error bars for
those not speciﬁed are typically 10%.
 [(0)]K  0, B ¼ 0, jN=Nc j > 1%

0:50

0:04

K  0, B ¼ 0, jN=Nc j < 1%xy

 (¼ x=y)





z






0:95

0:08

0.33

0.62

0.38

0.32

1:2

0:2

3:5

0:8

1.2

2.3

3

3:06

0:25

1

(2)

3



K ¼ 0:32, B ¼ 0

0:97

0:07

0.33

0.32

1:01

0:04

K ¼ 0, B 6¼ 0

1:1

0:1

0.50

0.54

0:93

0:10

2

182

J. Phys. Soc. Jpn., Vol. 73, No. 1, January, 2004

K. M. I TOH et al.

uring the region further away from Nc and Sc . A perfectly
uncompensated doped semiconductor is a half-ﬁlled Hubbard system where electron–electron interaction becomes
more important with respect to the eﬀect of disorder. On theother hand, compensation introduces more positively andnegatively charged ionized impurities, which act as sourcesof disorders. This very simple picture phenomenologicallydescribes why the critical exponents of the ‘‘compensatedregion’’, 0:99Nc < N < 1:01Nc of the K  0 series and0:25Nc < N < 2:4Nc of the K ¼ 0:32 series, agree withpredictions of non-interacting theories. In the future it will
be important to experimentally ﬁnd how the width of the  1 region scale with the value of K. Growth of single
crystalline Ge with controlled mixture of 70 Ge and 74 Ge
followed by NTD is the ideal way to prepare samples with acontrolled range of compensations.45,51) There have been one
experimental attempt to probe the critical exponents as afunction of compensation. Rentzsch et al. have reported 
1:5 for the series with K ¼ 0:014 and 0.12.44) However, they
could not resolve the two independent regions,  3:5 foraway from Nc and  1 for near Nc because the number ofsamples they evaluated was limited to six. It would be of agreat interest for Rentzsch et al. to prepare more samples,
identify the two regions, and determine how the width of the‘‘compensated region’’ changes as a function of thecompensation.
Now we turn our attention to the eﬀect of the magnetic
ﬁeld. The magnetic ﬁeld, which breaks the time reversal
symmetry, is expected to bring the system into a diﬀerent
universality class, leading to a diﬀerent value of .3,4)
However,   1 obtained from the ‘‘compensated region’’with B ¼ 0 agrees with  ¼ x=y  1 obtained from thetransition of the K  0 series in the constant magnetic ﬁeld
(Fig. 12) and by the magnetic ﬁeld (Fig. 14) within theirexperimental errors. One may naively conclude from thisobservation that the eﬀects of the compensation andmagnetic ﬁeld are the same. However, one should note thatthe dynamical exponent z  3 for the compensation is
diﬀerent from z  2 for the magnetic ﬁeld. Therefore, theuniversality classes of the ‘‘compensated region’’ and the‘‘magnetic ﬁeld region’’ are diﬀerent.

and  ¼ 1:43 0:04 for the unitary universality class,52,53)
while our experiment ﬁnds     1. Electron–electroninteraction may be responsible for this small diﬀerence eventhough the eﬀect of disorder is dominant for this region. The
second issue relates to the question why such a wide range ofconcentration (0:25Nc < N < 2:4Nc ) can be ﬁtted with theﬁnite temperature scaling for K ¼ 0:32 while theory is
expected to be valid only in the vicinity of Nc .
We believe that the   0:5 region situated outside of the  1 region is the intrinsic part of the uncompensatedsystem. Many aspects of it do not agree with scaling theory:
Wegner’s scaling law is broken ( 6¼ ), the ﬁnite temperature scaling is not applicable, and   0:33 severely
violates the inequality   2=3. Although there have been
a number of theoretical proposals to explain these anomalous phenomena,54–59) none of them have been accepted
widely. The present experiment has revealed the anomalous
aspects of the uncompensated system so unambiguously thatit remains to be a theoretical challenge to describe suchphenomena.

7.

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Concluding Remarks

We have performed a complete scaling analysis of the lowtemperature electrical conductivity of nominally uncompensated and deliberately compensated Ge:Ga samples. We
have determined the critical exponents for conductivity,
dielectric constant, and localization and correlation lengths,
and have suggested that the large scattering in the values of
the critical exponents shown in Table I is due to the eﬀect ofcompensation.
All aspects of the     1 region dominated bycompensation are in complete agreement with the predictions of the original scaling theory for non-interacting
system.2) The same results have been found for nominallyuncompensated Si:P and other systems, i.e., the phenomenon
is universal. While everything appears to be consistent withthe picture of purely disorder driven transitions, twoimportant issues remain. The ﬁrst one is the fact that moreadvanced calculations for purely disorder induced transitions
predict  ¼ 1:57 0:02 for the orthogonal universality classAcknowledgements
We acknowledge K. Slevin, T. Kawarabayashi, S.
Katsumoto, Y. Ono, A. Kawabata, R. N. Bhatt, M. Sarachik,and V. Dobrosavljevic for valuable discussions, J. W.
Farmer for the neutron irradiation, V. I. Ozhogin for the
supply of the Ge isotopes, J. W. Beeman for the help withsample preparation, and A. G. Zabrodskii for the conductivity data on NTD Ge. One of us (KMI) acknowledges the
Aspen Center for Physics for the wonderful discussion
opportunities on metal–insulator transition with physicists
from all over the world during his stay during the summersof 2001–2003. The work at Keio was supported in part by
the Grant-in-Aid from the Ministry of Education, Culture,
Sports, Science and Technology. The work at Berkeley was
supported in part by the Director, Oﬃce of Energy Research,Oﬃce of Basic Energy Science, Materials Science Divisionof the U.S. Department of Energy under Contract No. DEAC03-76SF00098 and in part by U.S. NSF Grant No. DMR0109844.

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