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Electron-electron interactions and the metal-insulator transition in heavily doped silicon.

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Ann. Phys. (Berlin) 523, No. 8 – 9, 599 – 611 (2011) / DOI 10.1002/andp.201100034
Electron-electron interactions and the metal-insulator
transition in heavily doped silicon
Hilbert v. Löhneysen∗
Physikalisches Institut and Institut für Festkörperphysik, Karlsruher Institut für Technologie,
Wolfgang-Gaede-Str. 1, 76131 Karlsruhe, Germany
Received 14 February 2011, accepted 6 March 2011
Published online 13 September 2011
Key words Metal-insulator transition, heavily doped silicon, electron-electron interaction, Hubbard splitting, thermoelectric power, specific heat, electrical conductivity, critical behavior, dynamic scaling.
This article is dedicated to Dieter Vollhardt on the occasion of his 60th birthday.
The metal-insulator (MI) transition in Si:P can be tuned by varying the P concentration or – for barely
insulating samples – by application of uniaxial stress S. On-site Coulomb interactions lead to the formation
of localized magnetic moments and the Kondo effect on the metallic side, and to a Hubbard splitting of
the donor band on the insulating side. Continuous stress tuning allows the observation of finite-temperature
dynamic scaling of σ(T, S) and hence a reliable determination of the critical exponent μ of the extrapolated
zero-temperature conductivity σ(0) ∼ |S − Sc |µ , i.e., μ = 1, and of the dynamical exponent z = 3.
The issue of half-filling vs. away from half-filling of the donor band (i.e., uncompensated vs. compensated
semiconductors) is discussed in detail.
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
Heavily doped semiconductors are prototype systems for the study of metal-insulator (MI) transitions
driven by the combined effects of disorder (Anderson transition) and electron-electron interactions (MottHubbard transition). In a strict sense, the MI transition occurs at T = 0 only because at finite temperature
T thermally activated carrier transport may occur. A metal is defined by a finite dc conductivity σ(T ) for
T → 0, while for an insulator σ(0) = 0. Zero-point quantum fluctuations become important at a quantum phase transition between these two groundstates [1]. The experimental determination of the critical
exponent μ describing how σ(0) vanishes at T = 0 as a function of a control parameter δ at a critical
value δc , i.e. σ(0) ∼ |δ − δc |μ , has been a subject of considerable controversy, as will be discussed below.
A self-consistant theory of the Anderson transition for noninteracting electrons was developed by Vollhardt and Wölfle [2, 3]. However, using both carrier concentration N and uniaxial stress S to tune the MI
transition, identifying the critical region of δ where critical behavior is expected to occur, and employing a dynamic finite-temperature scaling analysis, a consistent picture has emerged for Si:P which is the
prototype material for a localization transition in three dimensions.
For a disorder driven MI transition, the question of whether or not dopant atoms are distributed at random is of course crucial. Clustering or short-range ordering could drastically affect the properties near the
MI transition. For instance, extreme clustering, viz. phase separation, could drive the transition towards a
classical percolation transition. The distribution of dopants in semiconductors has been investigated by
scanning tunneling microscopy (STM) of freshly cleaved surfaces, where the spectral features can be
clearly attributed to dopant atoms by using scanning tunneling spectroscopy (STS). This was demonstrated
∗
E-mail: hilbert.loehneysen@kit.edu
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H. v. Löhneysen: Metal-insulator transition in doped silicon
for Si:P [4–6], Si:B [7], GaAs:SiGa [8], and n- and p-doped InAs [9, 10]. Recently, these studies were
extended for Fe atoms incorporated in a two-dimensional electron gas at an InSb surface [11]. Concerning
Si:P and Si:B, the spectroscopic identification of dopants at the (111) surface cleaved in situ in UHV allowed to investigate their spatial distribution. For both cases a random arrangement according to a Poisson
2
distribution, f (r) = 2πρr · e−πρr of the nearest-neighbor (nn) distances r was found even well above the
critical concentration Nc , with ρ denoting the surface dopant density. Of course, at a macroscopic level,
segregation and striations during crystal growth lead to concentration fluctuations at the 1 % level.
The long-range electron interactions around the MI transition have been analyzed in detail in the past.
On the metallic side, the progressive loss of screening upon approaching the MI transition and the concomitant enhancement of the electron-electron interactions, due to the slow diffusive electron motion arising
from disorder, leads to the well-known Altshuler-Aronov reduction of the density of states N (E) around
the Fermi energy EF and corresponding corrections to the Drude conductivity [12]. On the insulating side,
the dip at EF develops into a soft Coulomb gap, N (E) ∼ |E − EF |2 , giving rise to Efros-Shklovskii
variable-range hopping [13].
Deep in the insulating side, double occupancy of the donor impurity level – more exactly: of the 1s
donor groundstate whose sixfold degeneracy is split by the crystalline electric field of the Si host (valleyorbit splitting), with the (spin degenerate) orbital singlet 1s (A1 ) state as the lowest level – is energetically
unfavorable, with an on-site repulsion close to the donor ionization energy of 45 meV. Upon increasing the
donor concentration, the singly and doubly occupied states develop into lower and upper Hubbard bands,
respectively. Corresponding transport properties suggest a scenario of the MI transition involving states at
the edges of the lower and upper Hubbard band. Anomalies occurring in the transport data for Si:P well
below the critical concentration Nc ≈ 3.52 · 1018 cm−3 can be consistently attributed to a Hubbard gap,
as supported by the absence of such features in compensated Si:(P,B) away from half-filling of the 1s(A1 )
band, as will be discussed below.
In this review, we will discuss a few aspects of electron-electron interactions and the critical behavior at the MI transition focusing on three-dimensional n-doped Si and illustrating the advance that has
been reached over the last years [14]. The role of the on-site Hubbard interaction can be investigated by
comparative studies of uncompensated vs. compensated semiconductors, where the impurity band is at or
away from half-filling, respectively. We will also discuss the critical behavior of the conductivity. For a
discussion of the MI transition in two-dimensional systems (at zero magnetic field) which has received a
lot of attention in recent years, the reader is referred to [15–17]. Recently, a novel type of insulator with
topologically protected metallic surface states (topological insulator) has attracted much attention [18].
2 Metallic Si:P and Si:B – formation of local magnetic moments
and Kondo effect
It has been known for a long time that a small fraction of localized moments in doped semiconductors,
derived from the spin-degenerate groundstate of the hydrogen-like donor or acceptor wave function on
the insulating side, survive on the metallic side of the MI transition. These localized magnetic moments
have been mostly detected with magnetization [19, 20], magnetic resonance [22, 23], and specific-heat
measurements [24, 25]. The dependence of the concentration of localized moments on the P concentration
N has been mapped out systematically from specific-heat measurements for Si:P [26, 27] and Si:(P,B)
[28]. The concentration of local moments has been determined from the zero-field excess specific heat
and phonon contributions subtracted from the measured
ΔC = C −γT −βT 3 , i.e., the conduction-electron
specific heat C, via the entropy S = (ΔC/T )dT = NS ln2.
The random distribution of dopant atoms, as discussed in the introduction, leads to a wide distribution
of nn distances and, hence, nn exchange couplings Jnn between dopant pairs. This wide distribution of
exchange couplings led Bhatt and Lee [29] to a description of the magnetic properties of doped semiconductors in terms of a hierarchical coupling of antiferromagnetic pairs of localized moments. In this model,
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only free spins (of density Nfree (T )), i.e., those with Jnn < kB T , that are not coupled to pairs at a given
T , contribute to the specific heat at that temperature with Cloc = kB T · ln2 · dNfree (T )/dT , while the
magnetic susceptibility is given by χloc = μ0 μ2B Nfree (T )/kB T . In order to test the validity of the BhattLee model one can calculate Cloc from χloc with the above equations and compare it to ΔC as measured
directly [26, 27]. The overall good agreement found on the insulating side [20, 21] indicates the validity of
the Bhatt-Lee model. However, with increasing N ≥ 3.3 · 1018 cm−3 , i.e., on the metallic side of the MI
transition, the agreement quickly deteriorates [21].
The origin of localized moments in the metallic state has been investigated in a number of approaches
[27, 30–33], most of them starting with an Anderson-Hubbard model which is the simplest model Hamiltonian featuring the essential elements of disorder, through random hopping integrals tij , and on-site
Coulomb repulsion U :
H=
(εi − μ)niσ +
tij c+
ni↑ ni↓ .
(1)
iσ cjσ + U
iσ
ijσ
i
ciσ (c+
iσ ) is the annihilation (creation) operator for an electron with spin projection σ in the 1s(A1 ) groundstate of the dopant atom at site i, and niσ = c+
iσ ciσ . It is usually assumed that the five excited states
generated by valley-orbit splitting are sufficiently far away (experimentally ∼ 10 meV [34, 35]) that a
single-band model is applicable. Indeed, it has been inferred from infrared reflection measurements that
the MI transition in Si:P occurs for electronic states in the 1s(A1) impurity band which even for P concentrations N ≈ 2Nc is still energetically separated from the conduction band [34, 35]. Again, overall
similar behavior is found for the infrared reflection of compensated Si:(P,B) [35]. The host semiconductor
properties enter only through the fact that the positions i are random sites of the Si lattice and that the
effective mass and interaction are renormalized. The various approaches differ in the type of mean-field
model employed. In the approach taken by Langenfeld and Wölfle [33] the starting point is the observation
that the fraction of dopant atoms carrying a localized moment in the metallic state is relatively small, of the
order of several percent at Nc [26]. The donor sites i are assumed to be on a regular simple-cubic lattice
and the disorder is introduced through a variation of tij . Therefore an isolated moment (“impurity”) forming in an effective homogeneous medium is considered. The effective Hamiltonian is that of the Anderson
magnetic-impurity model with the local spin occupation number of an impurity atom, which is proportional
to the magnetization, to be determined self-consistently. The resulting concentration NM of local moments
(obtained applying the usual Anderson criterion for the stability of a local moment) is in good agreement
with the experimentally determined values NS on the metallic side. The coupling of the localized moments
to the itinerant electrons by an effective exchange interaction Jeff gives rise to the Kondo effect. Because
of disorder, both the density of states at the Fermi level N (EF ) and the effective exchange interaction
Jeff ≈ t̄2 /U where t̄ is the average hopping amplitude from the local-moment site to the neighboring P
sites, will fluctuate. This leads to a distribution P (TK ) of Kondo temperatures TK ∼ exp(−1/N (EF )Jeff ).
−αK
is found with αK ≈ 0.9 [27], which translates to a
An approximate power-law behavior P (TK ) ∼ TK
1−αK
specific-heat contribution CK ∼ T
because the Kondo specific heat is a universal function of T /TK .
Attributing ΔC in the metallic state of Si:P to the Kondo effect, αK = 1 − αc ≈ 0.8 is inferred experimentally [26], in rough agreement with the theoretical calculation. However, the Kondo susceptibility for
the same distribution P (TK ) should vary roughly as χK ∼ T −αK which is not observed. Rather, the experimental T dependence of χ can be approximated by an exponent of 0.5 [21]. Further work is necessary
to resolve this discrepancy. In the theory, interactions among localized moments (RKKY interaction) or a
possible quenching of the Kondo effect by disorder have not been considered.
As an interesting aside, we note that the specific-heat curves of the metallic samples all intersect at a
single crossing point, also called “isosbestic” point (Fig. 1a). As pointed out by Vollhardt, the existence
of an isosbestic point is a general feature of correlated systems [36, 37]. At high temperatures, the specific
heat first increases monotonically with N , i.e., dC/dN > 0, because of an increasing free-electron-like
contribution γT . At low T , the specific heat increases with N for small N because of the spin entropy that
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H. v. Löhneysen: Metal-insulator transition in doped silicon
Fig. 1 Specific heat C divided by temperature T vs. T
on a log-log plot for Si:P with different doping concentrations N . (a) the metallic samples exhibit a crossing
point, (b) no crossing point is observed for the insulating samples. Data after [27].
>
is lifted for T → 0. However, for N ∼ Nc , more and more P-derived electronic states are absorbed in the
Fermi sea, hence dC/dN < 0. Hence a crossing point occurs only on the metallic side of the transition, as
indeed seen by comparison with Fig. 1b where C(T ) of insulating Si:P is shown.
The effect of localized magnetic moments on the electrical resistivity via the process of spin-flip scattering cannot be identified unambiguously because ρ(T, B) is dominated by weak localization, long-range
electron-electron interactions, and wave-function shrinkage induced by a magnetic field B as discussed in
detail in the literature [38–41]. The thermoelectric power S, on the other hand, gives clear evidence for
scattering by localized magnetic moments [42, 43] because it is particularly sensitive to the Kondo effect.
Figure 2 shows S(T ) for Si:P with N = 4.1 · 1018 cm−1 and for Si:(P,B) with N = 4.24 · 1018 cm−3 .
The maximum of S(T ) is attributed to magnetic scattering since it is observed only for N slightly above
Nc where an appreciable density of localized moments exists. Even more convincing is the suppression of
the S(T ) maximum in large magnetic fields, also shown in Fig. 2. In B = 5–6 T we recover the negative
diffusion thermoelectric power S ∼ −T observed for N Nc in zero field [42, 43]. Assuming that the
thermoelectric power arises from the Kondo effect one can compare the S(T ) maximum to a corresponding
single-ion expression derived by Maki [44], cf. Fig. 2. The deviations between data and best fit might be
due to the neglect of a TK distribution in the fit where a single-valued TK = 0.8 K is assumed. Moreover,
the low-T behavior (T TK ) is only approximately correct in this model. The change of the conductivity
due to Kondo scattering as calculated from the fit amounts to only a few percent of the total variation of
σ(T ), in line with the lack of a sizable Kondo effect observable in σ(T, B). The similar dependence of the
thermoelectric power on N in Si:P [42] and Si:(P,B) [43] reflects the similar dependence of the density of
local moments on N [26, 28] in both types of materials.
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Fig. 2 Thermoelectric power S as a function of temperature T for (a) for Si:P with N = 4.1·1018 cm−3 ≈
1.16Nc and (b) Si:(P,B) with N = 4.24 · 1018 cm−3 ≈ 1.20Nc in various applied magnetic fields B: (a)
closed circles B = 0, open circles B = 1.5 T, closed triangles B = 3 T, open triangles B = 6 T; (b) closed
circles B = 0, open circles B = 1.3 T, closed triangles B = 2.6 T, open triangles B = 5.2 T. Dashed lines
indicate fits according to the single-ion Kondo model for B = 0 [44] and solid lines illustrate recovery of
the NFE-like behavior S(T ) ∼ −T in high fields. Data after [42, 43].
3 Insulating Si:P – evidence for Hubbard splitting
from transport properties
Marked differences between Si:P and Si:(P,B) in the transport properties occur well in the insulating
regime [46]. Figure 3 shows the thermoelectric power plotted as −S · N vs. T on a log-log plot. (Here
the T range is above 1 K so that the Kondo anomaly in metallic samples, i.e., the positive S(T ) maximum
superimposed on an overall negative diffusion thermopower with a concomitant sign change of S(T ), is
not seen and should not be confused with the behavior above 1 K.) For sufficiently high T ( 15 K), S is
always negative. While the −S · N curves for compensated Si:(P,B) fall onto an almost universal curve for
doping concentration N between 1.5 and 2.95·1018 cm−3 , such a scaling is seen for uncompensated Si:P
only above N0 = 2.78 ·1018 cm−3 . Below that concentration, S(T ) exhibits a sign change at a temperature
TS=0 which rapidly shifts to higher values with decreasing N . This sign change is visible in Fig. 3 as a
precipitous drop of log(−S · N ) with decreasing T . A further strong difference is seen in the electrical
resistivity ρ(T ) (Fig. 4). For N < N0 , ρ rises much faster with decreasing T for Si:P than for Si:(P,B),
while for N > N0 the behavior is similar for both types of material. The strong qualitative difference in
S(T ) and ρ(T ) emerging in Si:P upon crossing N0 points to different dominant transport processes above
and below N0 . Interpreting the steep ρ(T ) increase of Si:P below N0 as an activated process, we can extract an activation energy E2 . Figures 5a and b show a comparison of E2 and TS=0 . Although there is an
order-of-magnitude difference in absolute values of E2 /kB and TS=0 , the similarity of the concentration
dependence is striking. Hence we interpret the sign change of S(T ) as the onset of an activated process
over a finite gap. If one were forced to assume an E2 process also for Si:(P,B) – although the data actually
suggest Efros-Shklovskii variable-range hopping with an exponent p = 1/2, see inset of Fig. 4b – one
would obtain the open circles in Fig. 5b, i.e., no feature appears at N0 .
The sudden appearence of a hard gap at N0 well below Nc only in Si:P suggests that we are observing
the Hubbard gap due to the on-site Coulomb repulsion U . It has been speculated already many years ago
that the E2 process is indeed due to the Hubbard gap [47]. Some time ago the Hubbard gap was inferred
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H. v. Löhneysen: Metal-insulator transition in doped silicon
Fig. 3 Negative thermoelectric power times carrier concentration −SN vs. temperature T on a log-log
plot for insulating samples of (a) Si:P and (b) Si:(P,B). Data after [46].
Fig. 4 Electrical resistivity ρ plotted vs. inverse temperature T −1 for (a) Si:P and (b) Si:(P,B). Straight
lines in (a) indicate fits to obtain the activation energy E2 . Inset in (b) shows the Si:(P,B) data plotted as log
ρ vs T −1/2 . Straight lines indicate Efros-Shklovskii hopping. Data after [46].
for very dilute samples (N < 1017 cm−3 ) from optical measurements [48] while more recent infrared
transmission [49] and reflection [34, 35] experiments did not see any signature of a Hubbard gap closer to
the transition. A negatively charged cluster of four P sites was theoretically found to be stable, indicating
an apparent lack of electronic repulsion [50]. On the other hand, the negatively charged isolated P donor
in Si is barely stable (binding energy 1.7 meV), i.e., U is of the order of the ionization energy of 45 meV.
Therefore it is very likely that the on-site electron repulsion weakens progressively as the P concentration
(and carrier concentration) increases, until at N0 ≈ 0.8 Nc the two Hubbard bands start to overlap. Because
of disorder, however, metallic behavior does not incur immediately since the tail states of the Hubbard
bands are localized. It is only at the critical concentration Nc that the Hubbard bands are so close that the
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Ann. Phys. (Berlin) 523, No. 8 – 9 (2011)
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N << NO
Si:P
Si:(P,B)
N ~< NO
μ
(c)
μ
μ
NO < N < N C
μ
N0
N~> NC
occupied states
localized states
Fig. 5 (online colour at: www.ann-phys.org) (a) Temperature TS=0 of the thermoelectric-power zero and
(b) activation energy E2 vs. carrier concentration N for uncompensated Si:P (dark circles) and Si:(P,B)
(light circles). Data after [46]. (c) Qualitative sketch of the density of states of the impurity band of uncompensated Si:P for several P concentrations N , indicating the splitting into lower and upper Hubbard band
around the chemical potential μ. See text for details.
chemical potential is within the range of extended states. This scenario is schematically depicted in Fig. 5c.
It gives a physical picture of how disorder and electron-electron interactions drive the MI transition in Si:P.
Of course, the Hubbard features are absent in Si:(P,B) as they should because compensated semiconductors
are away from half-filling.
Interestingly, the density of localized magnetic moments in Si:P is not maximal at Nc but peaks rather
precisely at N0 where the two Hubbard bands in our scenario are suggested to merge [26–28]. Second, in
the dynamical mean-field theory (DMFT) treatment of the correlation-induced MI transition featured by
the Hubbard model which is mapped onto a generalized Anderson magnetic-impurity model, a peak in the
density of states occurs at the chemical potential right at the transition [50]. Experimentally, there is no
tendency that C/T for T → 0, i.e., below 100 mK, exhibits a pronounced maximum as a function of N ,
neither at Nc nor at N0 . Between N = 3.6 and 1.6 · 1018 cm−3 , C/T at 100 mK varies by roughly 30%
only. Figure 6 shows the specific heat at 100 mK as a function of N for Si:P and Si:(P,B). Here C is equal
to the total itinerant and localized electron contribution to C because the phonon contribution Cph can be
Fig. 6 Specific heat C measured at 100 mK for Si:P (Nc =
3.52 · 1018 cm−3 ) and Si:(P,B) (Nc = 3.54–4.97 · 1018 cm−3
for the samples studied)as a function of carrier concentration
N . Data after [27] and [28], respectively.
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H. v. Löhneysen: Metal-insulator transition in doped silicon
neglected at these low temperatures. Apart from a small shift, the overall form of C (100 mK) vs. N is very
similar although the density of localized moments is reduced by a factor of two in the latter [28].
4 Critical behavior of the conductivity at the metal-insulator transition
While early suggestions considered the MI transition to be discontinuous [45], scaling approaches for
noninteracting electrons suggested the existence of a continuous second-order phase transition for threedimensional systems [52] although this has been debated [53]. For a continuous MI transition at a critical
value δc of the nonthermal control parameter δ driving the transition, occurring in the strict sense only at
zero temperature, the dc conductivity at T = 0 is expected to vary as σ(0) ∼ |δ − δc |μ . Theoretically, μ
is usually inferred from the correlation-length critical exponent ν via Wegner scaling μ = ν(d − 2) where
d is the spatial dimension of the MI systems [54]. In a self-consistent theory of Anderson localization
(neglecting interactions), an exponent μ = 1 for d = 3 is suggested [2, 3]. Field-theoretical approaches are
discussed in [55]. Values of ν derived from numerical studies of noninteracting systems range between 1.3
and 1.6 [57–60]. Several investigations, notably on uncompensated semiconductors, have reported values
of the critical exponent μ = 0.5, in contrast to μ = 1 generally found for compensated semiconductors
and most amorphous metals [61]. However, there seems to be no clear-cut physical distinction between
these materials that would justify grouping them in different universality classes. The exponent μ = 0.5
was largely based on the stress-tuning experiments by Paalanen and coworkers [38, 62, 63] where uniaxial
T (K)
0.1 0.2
0.01
0.5
N = 3.21.1018 cm-3
16
(b)
σ (Ω -1cm-1)
12
8
4
0
0
0.2
0.4
0.6
T1/2 (K1/2)
0.8
1
Fig. 7 (a) Conductivity σ measured at 4.2 K vs. unaxial stress S for two Si:P samples close to the MI
transition, for different directions of S. The data are normalized to the value σ(S = 0). Lines are guides to
√
the eye. (b) Conductivity σ of a Si:P sample with P concentration N = 3.21 · 1018 cm−3 versus T for
several values of uniaxial stress applied along the [100] direction. From top to bottom: S = 3.05, 2.78, 2.57,
2.34, 2.17, 2.00, 1.94, 1.87, 1.82, 1.77, 1.72, 1.66, 1.61, 1.56, 1.50, 1.41, 1.33, 1.26, 1.18, 1.00 kbar. Solid
lines are connecting the very finely spaced individual data points. Data after [65].
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Fig. 8 (a) Low-temperature data of σ of Fig. 7b for S||[100] in the immediate vicinity of the metalinsulator transition plotted against T 1/3 . Dashed line indicates the conductivity at the critical stress (see
text). (b) Extrapolated conductivity σ(0) for T → 0 vs. uniaxial stress S for two P concentrations N = 3.21
and 3.43 · 1018 cm−3 (open and closed circles, respectively). The inset shows earlier σ(0) vs. S − Sc data
(triangles) with μ = 0.5 fit (dashed line) from [63] in comparison to our data for sample 1 (circles) with
μ = 1 fit (solid line) [65]. c) Scaling plot of σ/σc vs. |S − Sc |/Sc T y for Si:P with N = 3.21 · 1018 cm−3 .
Data after [65].
stress was used to drive an initially insulating (uncompensated) Si:P sample metallic. We suggested already
a number of years ago, for concentration-tuned Si:P, to limit the critical region of the MI transition on
the metallic side to concentrations where the conductivity decreases with decreasing T , i.e. to the range
dσ/dT > 0 [64]. The value of the conductivity at this crossover is σcr ≈ 40 Ω−1 cm−1 .
The situation concerning the T → 0 extrapolation to obtain σ(0) as a function of N or S being controversial, one may ressort to exploiting the scaling properties of the critical behavior at the MI transition [55].
However, the earlier stress-tuning data were shown to fail scaling behavior at finite T (often called dynamic
scaling) [55]. Therefore, we have systematically investigated the conductivity of Si:P under uniaxial stress
applied along different directions. Figure 7a shows the conductivity σ(S) measured at 4.2 K normalized to
the value for S = 0 for two barely insulating samples [65]. For S||[110], σ(S) increases monotonically,
while for S||[100] we found first a decrease and then an increase of σ. Finally, for S||[111] an overall weak
decrease of σ(S) is found. A nonmonotonic σ(S) dependence was reported previously at low T for P-,
Sb-, and As-doped Ge [56].
Figure 7b shows the electrical conductivity
σ(T ) for N = 3.21 · 1018 cm−3 for S||[100] between 1
√
and 3.05 kbar. The data are
√plotted vs. T which is the T dependence expected due to electron-electron
interactions, σ = σ0 + m T , and indeed observed well above the MI transition [38, 40]. Under uniaxial
stress between 1 and 2.57 kbar, the σ(T ) curves evolve smoothly from insulating to metallic behavior with
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H. v. Löhneysen: Metal-insulator transition in doped silicon
m > 0, and σ(T ) becomes nearly independent of T with a value σcr ≈ 12 Ω−1 cm−1 at ≈ 2.7 kbar. For
larger stress σ(T ) passes over a shallow maximum signaling the crossover to m < 0, as observed with
concentration tuning. It is interesting to note that σcr (S) ≈ 0.3 σcr (N ), thus severely limiting the critical
region. Closer inspection shows that the data near the MI transition are actually better described by a T 1/3
dependence for low T , see Fig. 8a. σ(0) obtained from the T 1/3 extrapolation to T = 0 is shown in
Fig. 8b, together with data for a sample closer to the critical concentration, yielding Sc = 1.75 kbar and
1.54 kbar for the two samples with 3.21 and 3.43 · 1018 cm−3 , respectively. Note that the critical stress Sc
is quite well defined, as σ(0) breaks away roughly linearly from zero within less than 0.1 kbar. In critical
region σ < σcr , μ is 0.96 and 1.09 for the two samples, respectively. μ ≈ 1 is found also when the
√
T extrapolation is employed. This behavior contrasts the earlier stress-tuning data [63] reproduced in
the inset of Fig. 8b, where appreciable rounding of σ(0) close to Nc is visible when plotted against S as
compared to our samples. However, those σ(0) data between 4 and 16 Ω−1 cm−1 are compatible with a
linear dependence on uniaxial stress. We note that the same value of μ was found for S [110] on the
sample with N = 3.21 · 1018 cm−3 .
The increase of σ with uniaxial stress is attributed to the admixture of the more extended 1s(E) and
1s(T2 ) excited states to the 1s(A1 ) groundstate of the valley-orbit split sixfold donor 1s multiplet [66].
However, the nonmonotonic σ(S) dependence for S||[100] is not readily explained, while S||[111] causes
an almost complete cancellation because all wavefunctions are affected in a similar fashion. The fact that
stress was applied to different directions in the previous and present studies, i.e., [123̄] and [100], respectively, may well be one reason for the different behavior of σ(T ).
In order to analyze the scaling behavior of σ at finite temperatures using the data of the sample with
N = 3.21 · 1018 cm−3 , we employ the scaling relation with a scaling function F ((δ − δc )/T y ) [67]
σ(δ, T ) = σc (T )F ((δ − δc )/T y )
(2)
where σc (T ) = σ(δc , T ) is the conductivity at the critical value δc of the parameter δ = S driving the MI
transition. If the leading term to σc (T ) is proportional to T x , one obtains x = μ/νz and y = 1/νz from a
scaling plot. Figure 7b shows that σ for S close to Sc does not exhibit a simple power-law T dependence
over the whole T range investigated. We therefore describe σc (T ) by the function σc (T ) = aT x(1 + dT w )
[65]. All σ(S, T ) curves with 1.00 kbar < S < 2.34 kbar up to 800 mK are then used for the scaling
analysis according to Eq. (4.1). The same procedure was repeated for other choices of σc (T ) between the
two measured σ(T ) curves embracing the critical-stress-curve with clearly less satisfactory results.
Figure 8c shows the scaling plot of σ(S, T )/σc (T ) vs. |S − Sc |/Sc T y . The data are seen to collapse
on a single branch each for the metallic and insulating side, respectively. The best scaling, as shown, is
achieved for y = 1/zν = 0.34. Together with μ = 1.0 as obtained from Fig. 8b and assuming Wegner
scaling ν = μ for d = 3, we find z = 2.94, consistent with σc ∼ T 1/z ∼ T 1/3 for T → 0 used to obtain
σ(0), see Fig. 8a.
From an analysis of σ(T ) of Si:P for different P concentrations N we had previously inferred μ = 1.3
from σ(0) vs. N [64] and z = 2.4 from a dynamic scaling analysis of metallic samples only [68]. Using
dynamic scaling of the form of Eq. (2) with δ = N we obtain y = 0.33, or zν = 3.0 [14]. Given the
uncertainty in the concentration determination, these values are broadly consistent with those obtained
from the present stress-tuning study for which we estimate an error of ±10% for μ and z. Bogdanovich
et al. [67] demonstrated that conductivity data for Si:B under uniaxial stress obey very nicely the dynamic
scaling on both metallic and insulating sides, yielding μ = 1.6 and z = 2, while concentration tuning of
σ(0) on the same system had suggested μ = 0.63 [70]. This large difference is not understood at present.
For Si:P, on the other hand, the exponents μ and z are in broad agreement with the expectation for a
noninteracting system. This is in line with our schematic scenario for the MI transition (cf. Fig. 5c) where
the transition proper is driven by disorder although interactions are vital in inducing the Hubbard splitting.
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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Ann. Phys. (Berlin) 523, No. 8 – 9 (2011)
609
The issue of uncompensated vs. compensated semiconductors at the MI transition has been addressed
with a series of experiments using neutron transmutation doping (NTD) [71–73]. Here, Ge crystals are irradiated with thermal neutrons leading to Ge:Ga via neutron capture with the reaction 70 Ge +n →71 Ge followed by electron capture with a half lifetime of 11.2 days, thus forming a 71 Ga acceptor. Rapid annealing
then removes most of the irradiation-induced defects. Using isotopically enriched Ge (70 Ge and 72 Ge with
96.2 and 3.8 at %, respectively) yields samples with small compensation ratio K = Nmin /Nmaj = 0.001
where Nmin and Nmaj denote the concentration of electrically active minority and majority impurities. Exposing nat Ge of natural isotopic composition to NTD leads to strong compensation with K ≈ 0.32 [73].
For nearly uncompensated 70 Ge:Ga, critical exponents μ = 1.2 and z = 31 were inferred from dynamic
scaling for eight samples in the narrow concentration range |δ̃| = |N − Nc |/Nc < 0.01. For larger |δ̃|,
μ = 0.5 was found. For deliberately compensated nat Ge:(Ga,As) obtained by NTD of Ge with natural
isotope abundance where K = 0.32, μ = 0.97 was found. Based on these results Itoh et al. [73] attribute
the exponent μ ≈ 1 to the compensated case and μ ≈ 0.5 to the uncompensated case. Clearly, more work is
necessary on this issue, in particular because for the range where μ ≈ 0.5 is observed, Wegner scaling [54]
μ = ν is not fulfilled.
5 Outlook
Considerable progress has been achieved in disentangling the roles of disorder and electronic correlation at
the metal-insulator transition in heavily doped semiconductors, at least on a qualitative level. However, we
are still far away from a complete understanding. This is true for the MI transition itself, which appears to be
a genuine quantum phase transition occurring strictly at T = 0 only, at least from the convergence of critical
exponents μ = 1 and z = 3 in Si:P, although the disparate exponents found in Si:B under concentration
and stress tuning constitute a puzzle. In the present case of the MI transition, critical fluctuations have not
yet been identified. However, noise measurements on Si:P exhibit a strong enhancement of non-Gaussian
conductance fluctuations [74]. In this respect, frequency-dependent experiments investigating dynamic
scaling would be very valuable. Dynamic scaling experiments have been performed only in a few cases,
notably amorphous Nb-Si alloys where ω/T scaling of the conductivity was observed [75]. Here recent
microwave experiments on insulating Si:P covering the range 30 GHz to 1.2 THz [76] should be extended
to metallic samples and to lower temperatures to check for ω/T scaling.
We need to distinguish between universal critical behavior and nonuniversal features such as different
prefactors of the temperature dependences of stress and concentration tuning, the width of the critical
region, and the crossover to an apparent conductivity exponent μ ≈ 0.5 sufficiently far away from the
critical point, i.e., beyond the critical region. The understanding of the latter features must probably by
sought in investigating the materials-related aspects in more detail, such as the relative importance of
disorder vs. interaction effects which might vary for different systems or even for different types of samples
in the same system. More work is needed to decipher the role of compensation on the critical behavior.
The generic phase diagram of electron systems featuring both correlations and disorder – calculated with
dynamical mean field theory – exhibits correlated-metal, Anderson-insulator and Mott-insulator phases as
well as coexistence regions [77] and might also be a guideline for future experiments.
Acknowledgements The work reviewed here grew out of a very fruitful collaboration with students, post-docs and
colleagues. Their contributions can be identified from the references cited. I thank them all for their important contributions, enthusiasm in carrying out difficult experiments, and useful discussions. I am particularly grateful to D. Vollhardt
and P. Wölfle for numerous enlightening discussions on the theoretical aspects of the metal-insulator transition. I thank
W. Zulehner, Wacker Siltronic, for the Si:P samples and D. F. Holcomb, Cornell University, for the Si:(P,B) samples.
www.ann-phys.org
c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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H. v. Löhneysen: Metal-insulator transition in doped silicon
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