PHYSICAL REVIEW B

VOLUME 53, NUMBER 6

1 FEBRUARY 1996-II

Superconductor-insulator transition in ultrathin Pb films:
Localization and superconducting coherence
K. Kagawa, K. Inagaki, and S. Tanda
Department of Applied Physics, Hokkaido University, Sapporo 060, Japan~Received 10 November 1995!
We have measured temperature-dependent resistivities of a set of Pb ultrathin films by repeating in situ
deposition at liquid-He temperature. Observed conductance in the fermion region can be described by asingle-parameter scaling function ~b function! with a self-consistent theory for localization in two dimensions.
Based on estimated values of electrical localization length ( j loc! and superconducting coherence length( j super!, the superconductor-insulator transition is confirmed to occur at the point where j loc;2jsuper . Moreover,
only samples exhibiting lnT dependence in the fermion region become global coherent superconductors at lowtemperature.

Recently, interplay between disorder and superconductivity in two dimensions ~2D! has gathered interest as a field ofstudy.1–10 In particular, superconductor-insulator ~SI! transition in 2D has attracted considerable interest in connection
with the quantum Hall liquid-insulator transition.11 Scaling
behavior for the SI transition has been the subject of numerous experiments4 – 6,12 in which a value of the critical sheet
resistance has been obtained near h/4e 2 . Fisher and coworkers argued the SI transition in a disordered superconducting thin film by scaling formulation.2,3 In this argument,
they did not directly address localization length of electrons~quasiparticles!. When one considers the formation of Cooper pairs, the relationship between electrical localization
length j loc and superconducting coherence length j superseems to be important. One can speculate that the SI transition occurs at the point where j loc5jsuper . If j loc is smaller
than j super , the system cannot form a Cooper pair but if
j loc is longer than j super , the system can form a Cooper pair.
However, there is no experimental evidence to support thistheory regarding the SI transition.
In this paper, we discuss experiments wherein we measured the temperature dependence of resistivities of ultrathin
Pb films by sequential repeating in situ deposition at lowtemperature. We indicate that all conductance data in thefermion region, where we can treat the carriers as fermion, fit
a single universal curve ~b function! of a self-consistenttheory for localization in 2D.13 We estimated j loc and j superfrom observed data, and show that the SI transition occurred
at a point where j loc;2jsuper . Furthermore, we discoveredthat films exhibiting lnT dependence in the fermion region
show global coherent superconducting behavior at low temperature; and that films exhibiting exp@ 2(T 0 /T) a # dependence in the fermion region show insulator ~Bose glass! behavior at low temperature.
The Pb films are grown by vapor deposition of the metal
onto liquid-He-cooled glass substrates. A sequence of films
can be fabricated and measured in situ. The film is 2.5 mm
34.0 mm in area. Temperature dependence of resistance ismeasured using a Keithley model 182 nanovoltmeter and aKeithley model 6512 electrometer in the range of 2;10 K.
Figure 1 shows R h ~T! curves of 39 sequential Pb films.
0163-1829/96/53~6!/2979~4!/$06.00

53

Film thickness is controlled by evaporation time and varies
from 13.8 to 42.8 Å. The thickness of the thickest film iscalculated as the average Pb grain height @Fig. 2 shows theatomic force microscope ~AFM! topography of the thickest
film#. The thickness of other films is evaluated from evaporation time. The film changes from an insulator to a superconductor with increasing film thickness. The sheet resistance of the thinnest film changes from 107 to 1013 V whiletemperature decreases from 10 to 2 K. In the thickest film,
superconducting transition takes place at 3.5 K and a residual
sheet resistance of about 20 V remains after the transitionoccurs. This residual resistance is observed in samples whichshow superconducting transition. Residual resistances decrease with temperature and reach a minimum near 2.3 K insome samples. A critical resistance which distinguishes insulators ~R increases as T→0! from superconductors ~R decreases as T→0! lies between 104 and 105 V. Considering
the fermion region and the boson region, where we can treat
carriers as bosons, we analyze the data below and above 4 K,separately. Our analyses include estimation of the criticalsheet resistance from the data observed below 4 K and esti-FIG. 1. Resistance of the samples vs temperature. The film isgrown sequentially in situ by evaporation of Pb. The film changes
to superconductor from insulator as film thickness increases. Filmthickness changes from 13.8 to 42.8 Å. The superconducting transition takes place at 3.5 K in the thickest film.
R2979© 1996 The American Physical SocietyR2980K. KAGAWA, K. INAGAKI, AND S. TANDA53FIG. 2. AFM topography of the thickest ~42.8 Å! film. This value represents the average Pb grain height. The thicknesses of other films
is calculated from the evaporation time.

mation of the electrical localization length j loc of each film
from the data observed above 4 K. The validity of the analyses will be discussed later in this paper.
In Fig. 3, we have plotted the temperature coefficient ofresistance ~TCR! below 2.3 K versus sheet resistance at 2.3
K for films near the SI transition. Films with positive valuesof TCR become superconductors at low temperature andfilms with negative values of TCR become insulators at lowtemperature. The critical sheet resistance which distinguishes
insulators from superconductors is regarded as the point
where TCR is zero. From Fig. 3 the critical resistance isfound to be between 20;23 kV. This value is close to h/e 2.
In a high-temperature region above 4 K, we measure characteristics of electrical transport from a strongly disorderedregime to a weakly disordered regime in a sequence of films.

The temperature dependence of resistivities of strongly disordered films takes the form R h ~T!5R 0 exp@(T0 /T)a#, whereT 0 is general activation energy. In hopping electrical transport, the value of a is unity and in variable range hopping~VRH! transport, the value of a is 1/3 in 2D.14 The conductance of films in a weak-localized regime shows logarithmic
temperature dependence. In localization theory,15 the conductance of a weak-localized regime is given bys 5 s 01

e2pln T,2 p 2\

where p is a constant determined by the scattering mechanism of electrons. The observed coefficient of lnT is 2.593105 and the value of p is 2.1, which is consistent with thetheory. Next, we analyze the observed data described so far
in terms of the conventional b function,16 b~g![d~lng!/d~lnL!, where L is the sample size and g is the dimensionlessconductance @ g5(\/e 2 ) s # . The sample size L is regarded asa cutoff length due to inelastic scattering: L 2 5DT 2 p , whereD is a diffusion constant of electrons. The b function is expressed asb ~ g ! 52

FIG. 3. Temperature coefficient of resistance ~TCR! vs sheet
resistance at 2.3 K. The sheet resistance is a parameter of the degreeof disorder in the films. A positive TCR indicates that a film becomes a superconductor ~T→0! and a negative TCR indicates that afilm becomes an insulator ~T→0!. The SI transition takes place atthe point where TCR is zero.

~1!

2 d ~ lng !
.
p d ~ lnT !

~2!

Using this form, we have plotted all the conductance dataabove 4 K in Fig. 4. We can see that all the data above 4 K
fit a single universal curve. We have also tried to plot the bfunction of all the conductance data below 4 K, but thesedata cannot be fitted to a single universal curve. Thus, fromFig. 4 we prove that all the films are in the disorder fermion
region above 4 K but are not in the disorder fermion region
below 4 K. The result that all the data are scaled by the bfunction above 4 K confirms our analyses.
Furthermore, to verify the criterion that the SI transition
occurs at the point where j loc5jsuper , we estimate the elec-SUPERCONDUCTOR-INSULATOR TRANSITION IN ULTRATHIN . . .

53

R2981

FIG. 4. The b function vs dimensionless conductance for all
films above 4 K. This figure shows that the conductance of all films
studied fit a single universal curve. This confirms that all films arein the fermion region above 4 K. Below 4 K, data cannot be fitted toa single universal curve.

trical localized length j loc from the high-temperature data.
Wölfe and Volhardt have produced a self-consistent theory13of the transition from weak to strong localization in a thinfilm, neglecting interaction. They found that\15@ ln~ 11y 2 !#~ 11y ! exp~ 2y ! ,e Rh 2p2
2~3!

where y5L/ j loc , j loc is an electrical localization length related to the elastic mean free path l by j loc5l exp(pkFl/2),
and k F is the Fermi wave number. At a finite temperature,sample size L is related to T by L;T 2 p/2. The temperaturedependence of the sheet resistance above 4 K can be written
in the form of Eq. ~3! because all data are fitted to the bfunction above 4 K. By fitting the data to Eq. ~3!, we estimate the electrical localization length j loc of each film. We
then estimate the superconducting coherence length ( j super!
of each film. j super is related to the elastic mean free path l by

FIG. 5. The superconducting coherence length ( j super! and theelectrical localized length ( j loc! vs the sheet resistance at 10 K. The
sheet resistance at 10 K is a parameter of the degree of disorder inthe films. The SI transition takes place at a point wherej loc.2jsuper , and the criterion that the SI transition takes place atthe point where j loc5jsuper is not observed.

FIG. 6. ~a! Conductance vs temperature for superconducting
films. The conductance shows logarithmic temperature dependence.
This indicates that the films are weak-localized regimes. Observed
coefficient of lnT is 2.593105 and the value of p is 2.1. ~b! Conductance vs temperature for insulator films near the SI transition.
The conductance shows VHR transport 1/s a }exp@2(T0 /T)1/3# . T 0
is the general activation energy.

j 2super5jBCSl, 17 where j BCS is a superconducting coherence
length in a clean superconductor.
We have plotted j super and j loc vs the sheet resistance at
10 K in Fig. 5. j loc and j super of the thickest film is about
23106 and 70 Å, respectively. If the SI transition is inducedby breaking the Cooper pair, the SI transition takes placewhen j loc becomes the same length as j super . In other words,
if j loc is larger than j super the film behaves as a superconductor, and if j loc is smaller than j super the film behaves as aninsulator. But in practice, the transition takes place at a point
where j loc;2jsuper . From the above result, it is found that
relationship of j super and j loc does not determine the SI transition immediately. Therefore, the SI transition is not inducedby the pair-breaking mechanism.
Furthermore, we discovered a difference between the insulator and the superconducting films in terms of the temperature dependence of resistivities in the fermion region.
The conductance of superconducting films exhibits logarithmic temperature dependence above 4 K. On the other hand,the conductance of the insulator films above 4 K near the SI
transition exhibits exp@2(T0 /T)1/3# dependence which indicates VRH transport. In Fig. 6 we have plotted conductancevs temperature near the SI transition for the superconducting
films and the insulator films that show VRH transport. We

R2982

K. KAGAWA, K. INAGAKI, AND S. TANDA

53

found that the films that exhibit VRH transport in the Fermi
region become insulators and that the logarithmic
temperature-dependent films become global coherent superconductors at low temperature.
We believe that our observed SI transition is a granular
one mediated by competition between grain charging andJosephson coupling. Granular systems are susceptible tocharging due to the fact that placing an electron on an isolated grain of capacitance C costs an electrostatic energy
E C . Thus, the SI transition is considered as competition between E C and Josephson-coupling energy E J , i.e., the system becomes an insulator when E C .E J and the system becomes a superconductor when E C ,E J . 18 We roughly
calculated E C and E J of the films near the SI transition. First,
we calculated E J using the Ambegaokar-Baratoff equation,
E J 5 p \D/4e 2 R N , 19 where D is the BCS gap energy for Pband R N is the normal resistance of the system. E J is about;5.6310223 J ~;4 K! using D/k B 512 K, R N 510 kV.
Next, we calculated E C using E C 5q 2 /4p e 0 k d @wherek5e~11d/2s! and d, s, and e are the size of grains#, thespacing of neighboring grains, and the dielectric constant,respectively.20,21 Our calculated value of E C is about ;1.2310221 J ~;80 K!, where q52e, e510 for the glass substrate, d530 nm, and s510 nm using the value measured by
AFM. The value of E C ~;20 K! for q5e is close to the valueof generalized activation energy (T 0 ) from 2D VRH conduction exp@2(T0 /T)1/3# . We claim that the films exhibiting
VRH conduction have a gap structure on the order of E C ,that phase fluctuation occurs at low temperature; and that thefilms exhibiting lnT dependence have no gap structure andbecome global coherent superconductors at low temperature.
We conclude that the relationship between j loc and j superdoes not determine the SI transition, but that the relationshipbetween E C and E J determines the SI transition. We suggestthat films in the region of j super,jloc,;2jsuper become Bose
glass phase2 at low temperature since the system can form a
Cooper pair in the region j super,jloc and these films have agap structure which induces VRH conduction in the fermion
region.
In our next experiment, we will observe the SI transition
at lower temperatures than we did during the present experiment and we will investigate the scaling feature of the Bose
glass-vortex glass transition.
In summary, we induced the SI transition by tuned disorder and we estimated the critical resistance which distinguishes superconductors from insulators ~20;23 kV!. We
proved that the relationship between electrical localized
length j loc and superconducting coherence length j super does
not determine the SI transition. The SI transition takes placeat the point where j loc.2jsuper . We observed characteristics
of electrical transport from a strongly disordered regime to a
weakly disordered regime in the fermion region ~T>4 K!,and found that all the conductance data above 4 K fit a single
universal curve ~b function! of a self-consistent theory. Below 4 K, all the conductance data do not fit a single universal
curve. An important feature of our findings is the differenceof temperature dependence of conductance in the fermion
region between insulating and superconducting films. We
claim that films in the region of j super,jloc,;2jsuper becomebose glass phase at low temperature; and that these films
have a gap structure which induces VRH conduction in thefermion region. We also claim that films exhibiting lnT dependence have no gap structure and become global coherent
superconductors at low temperature.

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This work was supported in part by a Grant-in-Aid of theJapan Ministry of Education, Science and Culture. We thankthe Kurata Science Foundation for its support. We also thankProfessor T. Nakayama for his valuable discussion.

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