Upper critical field as a probe for multiband superconductivity in bulk and interfacial STO

0.68 MB·

by  Jonathan M. Edge

Upper critical field as a probe for multiband
superconductivity in bulk and interfacial STO
4
1
0
2 J. M. Edge and A. V. Balatsky

t Nordita, KTH Royal Institute of Technology and Stockholm University,
c
O Roslagstullsbacken 23 106 91 Stockholm, Sweden,
Institute for Materials Science, Los Alamos National Laboratory, Los Alamos, NM,
6
87545, USA
1

E-mail: [email protected]
]
n
o
Abstract. WeinvestigatethetemperaturedependenceoftheuppercriticalfieldH
c c2
- asatooltoprobethepossiblepresenceofmultibandsuperconductivityattheinterface
r
p of LAO/STO. The behaviour of Hc2 can clearly indicate two-band superconductivity
u throughitsnontrivialtemperaturedependence. Forthedisorderscatteringdominated
s
. two-dimensionalLAO/STOinterfacewefindacharacteristicnon-monotoniccurvature
t
a of the H (T). We also analyse the H for multiband bulk STO and find similar
c2 c2
m
behaviour.
-
d
n
o
c
[

3
v
8
1
3
5
.
1
0
4
1
:
v
i
X
r
a
UppercriticalfieldasaprobeformultibandsuperconductivityinbulkandinterfacialSTO2
1. Introduction
Multiband superconductivity provides an intrinsically interesting extension of
superconductivity. Shortly after the publication of BCS theory [1], an earliest idea
of multiband superconductivity was proposed [2, 3]. It is characterised by having more
than one band in which Cooper pairs form. Thus two different superconducting gaps
may appear. Apart from the theoretical interest, multiband superconductivity also has
practical consequences. For example, some of the highest temperature superconductors
aremultibandsuperconductors. Thesearemagnesiumdiboride(MgB )withatransition
2
temperature of 39 Kelvin [4], and the iron-based superconductors [5, 6], with a maximal
critical temperature of about 56 Kelvin [7]. Additionally, multiband superconductivity
may lead to a higher upper critical magnetic field H that is also attributable to the
c2
interplay between the two gaps [8]. Indeed, in the realm of technology applications,
it has been speculated that due to these properties many future high magnetic field
superconducting magnets, such as those found in MRI scanners, will be made of
multiband superconductors [9].
Unambiguous detection of multiband superconductors requires advanced tech-
niques. Currently the main probes available are scanning tunnelling spectroscopy [10],
heat transport [11, 12], specific heat [13] and the superfluid density [14, 15]. Multiband
superconductivitymanifestsitselfthroughtheoccurrenceofmorethanonequasiparticle
coherence peak in tunneling spectroscopies [16]. However, short quasiparticle lifetimes
may smear these peaks and thus make them unobservable. Heat transport may also
be used to probe multiband superconductivity through its anomalous magnetic field
dependence. A single band superconductor shows a strong suppression of heat trans-
port all the way up to temperatures very close to the critical temperature. In contrast,
in multiband superconductors one of the gaps may be disproportionately suppressed
by a magnetic field, thus allowing that band to transport heat effectively [13]. These
techniques helped determine that e.g. MgB [11] and PrOs Sb [12] are multiband
2 4 12
superconductors.
The recent discovery of superconductivity at the LaAlO /SrTiO (LAO/STO)
3 3
interface [17] has made the discussion of the nature of the superconducting state and
possible multiband effects relevant [18]. In this paper we wish to put forward the
temperature dependence of the upper critical field as a probe for whether SrTiO (STO)
3
and particularly the interface between LaAlO (LAO) and STO are single or multiband
3
superconductors. The temperature dependence of the upper critical field may show
characteristic behaviour inherent to multiband superconductivity and has been used
previouslytodeterminethatiron-basedsuperconductorsaremultibandsuperconductors
[19].
STO has long been a material of interest. It was the first oxide which was found
to be superconducting [20]. Moreover, it was also the first material to show two-band
superconductivity, through the presence of two quasiparticle coherence peaks [16]. STO
can be tuned between single band and multiband superconductivity by changing the
UppercriticalfieldasaprobeformultibandsuperconductivityinbulkandinterfacialSTO3
levelofdoping[16]andrecentlytherehaveevenbeenindicationsthat, forcertaindoping
levels, the material may be a three-band superconductor[21, 22]. However, despite this
evidence STO is still not unanimously accepted as a multiband superconductor [23].
Since 2004 attention has shifted to a metallic interface between LAO and STO [24].
The system is remarkable since both LAO and undoped STO are insulators. Interest
grew even further in 2007 when superconductivity was discovered at the interface [17].
One of the most pertinent questions now concerns the origin of the superconducting
state at the interface.
One suggestion is that the metallic layer and thus the superconductivity is simply a
consequence of surface doping at the interface [17]. However, in addition to the doping
effectsit wassuggestedthat multiorbital effects[25] andmultibandeffectsare important
[18] and in fact enable multiband superconductivity [26]. The latter proposal, that the
superconductivity is a direct descendant of superconductivity from the bulk STO, is
supported by the fact that other interface layers apart from LAO also give rise to a
metallic and superconducting surface state of STO [27, 28]. Apart from the proposal of
”descendant” superconductivity at the LAO/STO interface, the alternative suggestions
were made that the superconductivity at the surface is of an entirely different origin,
resulting from a polar catastrophe and possibly spin orbit coupling that is a unique
property of the interface and has no analog in bulk STO [29]. The ongoing debate
underscores the importance of unambiguous tests that would clarify the nature of the
superconducting state. The investigation of H (T) is one of these tests.
c2
In this paper we propose a direct test of the hypothesis of two-band
superconductivity in bulk STO and the LAO/STO interface. We consider the
perpendicular upper critical magnetic field in order to see if its behaviour can indicate
whether the material is a single band or multiband superconductor. We concentrate on
the upper critical magnetic field since it is a quantity readily accessible to experiments.
Some other probes, like specific heat and heat transport, are not practical for LAO/STO
interfaces, thus making the temperature dependence of H one of the few available tools
c2
to further investigate superconducting states in these materials. In doing so, we also aim
to clarify the relationship between the superconductivity in the bulk and the interface
system.
The paper addresses both the case of bulk STO and LAO/STO interfaces. The
possible regimes include four cases: clean and disordered (in the sense of the ratio of the
coherence length to the mean free path) in bulk and interface STO. We first investigate
the dirty limit behaviour of the system. This is appropriate if the mean free path is
shorter than the superconducting coherence length ξ ≈ 70nm [17]; for interface systems
this is likely to be the relevant situation. Depending on doping, it is also a realistic
scenario for bulk STO, particularly at optimal doping [21]. Subsequently we address
what is expected in a clean system. However, if there are two superconducting gaps,
two coherence lengths and mean free paths are possible. In principle one could be in
a regime where one band is dirty and the other is clean. This regime would require a
complicated analysis and is outside the scope of this paper. Our work expands and adds
UppercriticalfieldasaprobeformultibandsuperconductivityinbulkandinterfacialSTO4
to earlier work that focused on H , but only considered the clean limit [25].
c2
In section 2 we consider the band structure for STO and motivate our treatment of
multiband superconductivity based on this band structure. In section 3 we show how
H (T) may be calculated for disordered multiband superconductors. Section 4 presents
c2
the results for multiband superconductivity for coupling constants relevant to STO and
also includes a more detailed investigation into the conditions under which H (T) may
c2
be used to detect multiband superconductivity. In section 5 we show the results for
H (T)inacleansystem. Section6firstpresentshowthegeneralcalculationofsection3
c2
needstobemodifiedinordertoconsiderthefinitethicknessofthesuperconductinglayer
at the LAO/STO interface. Subsequently the results for H (T) are presented.
c2
2. The band structure of STO
Undoped STO has filled oxygen p bands which are separated from the titanium d bands
by a large bandgap of 3eV [30]. Of these, the lowest result from the t orbitals, d ,d
2g xy yz
and d , which get filled once the system is doped. The t orbitals are split by the spin-
xz 2g
orbitinteractionandthecrystalfield. Thehighestenergybandissituatedapproximately
30meV above a doublet of bands split by an amount of the order of 2meV [31].
While this band structure may indicate that STO could form a one-, two- or three-
band superconductor, we will investigate the distinction between single and two-band
superconductivity only, as we wish to contrast single with multiband superconductivity.
Furthermore, two of the bands are very close in energy and can thus easily couple
together tightly and appear as a single band.
Twoimportantquestionswhichwillconcernusarethecouplingsbetweenthebands
and the degree of anisotropy within each band. We will be primarily interested in the
disorderedlimit, asdescribedinsection3. Disorderscatteringhastheeffectofaveraging
out Fermi surface anisotropies, such that one can effectively consider isotropic Fermi
surfaces. In the clean limit the Fermi surfaces of STO are not perfectly isotropic, but
for low degrees of doping we do not expect the anisotropies to be too great [21].
Additionally, disorder scattering will introduce a coupling between the bands. We
takethisintoaccountviatheinterbandcouplingconstantintheselfconsistencyequation
(see section 3). However, we do not expect this coupling to be very large and in
particular, we expect it to be much smaller than any coupling within the bands. This
is because the different t orbitals are orthogonal and have little spatial overlap. A
2g
coupling of the bands has to be able to effect an annihilation of a Cooper pair in one
band and create it again in another. This process will be suppressed if the bands do
not show great spatial overlap and is thus the justification for having a small interband
coupling parameter, as described in section 4.
UppercriticalfieldasaprobeformultibandsuperconductivityinbulkandinterfacialSTO5
3. Calculation of the upper critical field in the presence of disorder
At a quasi classical level the physics of a dirty superconductor can be described by the
Usaldel equations [32]. These give an accurate description of the physics when disorder
scattering is strong, such that anisotropies of the Fermi surface are averaged out. We
solve the multiband Usadel equations [8, 33] in the limit where the gaps ∆ are very
small. This describes the region very close to the transition from superconductor to
normal metal and the Usadel equations may be linearised, simplifying their solution.
By solving the equations as a function of an applied magnetic field and temperature we
thus obtain the temperature dependence of the upper critical magnetic field.
In our approach we closely follow the approach developed in Ref.[8]. We start with
the linearised Usadel equations.
2ωf −DαβΠ Π f = 2∆ (1)
1 1 α β 1 1
2ωf −DαβΠ Π f = 2∆ (2)
2 2 α β 2 2
f , i = 1,2, is the Green’s function of the system and in general depends on the
i
momenta,position,andtheMatsubarafrequencyω = 2πT(2n+1). Dαβ isthediffusivity
i
tensor within a band. Π is defined as Π = ∇+2πiA/φ , φ is the flux quantum. By
0 0
assuming the diffusivity tensor to be given by D = δ D and the vector potential to
m αβ m
be given by A = Hxyˆ, we can write these equations as
(cid:18)
4πiHx
2ωf −D ∇2 +∇2 +∇2 + ∇
m m x y z φ y
0
4π2H2x2(cid:19)
− f = 2∆ . (3)
φ2 m m
0
Since this equation only depends on x, we now assume that f is independent of y
m
and z (m ∈ {1,2}). Equation (3) can now be solved for ∆ and f using the ansatz
m m
f = h ∆ (x) and one obtains the solution
m m m
∆
m
f (x,ω) = (4)
m
ω +πHD /φ
m 0
∆ (x) = ∆(cid:48) e−πHx2/φ0 (5)
m m
with ∆(cid:48) being a constant. The solutions for f and ∆ can be inserted into the gap
m
equation for the two-band superconductor. This gives
(cid:88)ωD (cid:88)
∆ = 2πT λ f (x,ω) (6)
m mm(cid:48) m(cid:48)
ω>0 m(cid:48)
(cid:88) (cid:88)ωD ∆m(cid:48)
= λ 2πT (7)
mm(cid:48)
ω +πHD /φ
m(cid:48) 0
m(cid:48) ω>0
(cid:20) (cid:18) (cid:19)(cid:21)
(cid:88) 2γωD HDm(cid:48)
= λ ∆ ln −U . (8)
mm(cid:48) m(cid:48)
πT 2φ T
0
m(cid:48)
UppercriticalfieldasaprobeformultibandsuperconductivityinbulkandinterfacialSTO6
Figure 1. Temperature dependence of the upper critical field in the disordered limit
for the set of coupling constants (A) [26]. Different values of η = D /D correspond
2 1
to different ratios of the diffusivities.
ω is the Debye frequency and λ the superconducting coupling constants for the
D mm(cid:48)
different bands. In the last line we have used the equality
(cid:88)ωD 1 2γω (cid:18) X (cid:19)
D
2πT = ln −U (9)
ω +X πT 2πT
ω>0
with U(x) = ψ(x+1/2)−ψ(1/2) and where ψ is the di-gamma function. lnγ ≈ 0.577
is the Euler constant. We can convert this into a 2x2 system of equations for ∆ and
divide out the factor e−πHx2/φ0 and thereby replace ∆ with ∆(cid:48).
(cid:32) (cid:33)(cid:32) (cid:33)
(l−U(h))λ −1 (l−U(ηh))λ ∆(cid:48)
11 12 1 = 0 (10)
(l−U(h))λ (l−U(ηh))λ −1 ∆(cid:48)
21 22 2
(cid:124) (cid:123)(cid:122) (cid:125)
M0
Here l = ln 2ωDγ, h = HD1 and η = D2.
πT 2φ0T D1
Since these equations resulted from a linear expansion of the Usadel equations, they
are valid for small, or infinitesimal ∆(cid:48). Since ∆(cid:48), and thus ∆, is infinitesimal at H = H ,
c
these equations have a nontrivial solution at H = H . We thus need to find the solution
c
to the equation detM = 0. After some manipulation one arrives at the expression
0
a (lnt+U(h))(lnt+U(ηh))+a (lnt+U(h))
0 1
+a (lnt+U(ηh)) = 0 (11)
2
with t = T . Here the equation for T in a two-band superconductor has also been used
Tc c
in order to replace ω with T (equation (22) in ref.[8]). The coefficients a depend of
D c i
UppercriticalfieldasaprobeformultibandsuperconductivityinbulkandinterfacialSTO7
Figure 2. Same as figure 1, but for the set of coupling constants (B) [34].
the coupling constants as follows
2(λ λ −λ λ )
11 22 12 21
a = (12)
0
λ
0
λ −λ
11 22
a = 1+ (13)
1
λ
0
λ −λ
22 11
a = 1+ (14)
2
λ
0
(cid:113)
λ = λ2 +λ2 +4λ λ −2λ λ . (15)
0 11 22 12 21 11 22
Itisnowrelativelystraightforwardtosolvenumericallyfortherootsofequation(11)
as a function of H and t = T/T .
c2 c
4. Results for H in the presence of disorder
c2
4.1. Results for STO
We now address the behaviour of H (T) as a function of the coupling constants and
c2
the diffusivity parameters. Our aim is to clarify under which circumstances H (T) may
c2
be used as a probe for multiband superconductivity. We first investigate the coupling
constants applicable to STO.
There is no consensus for what the precise coupling constants for STO are. Two
such sets are found in the literature:
(A) λ = 0.14,λ = 0.13,λ = 0.02 [26]
11 22 12
(B) λ = 0.3,λ = 0.1,λ = 0.015 [34]
11 22 12
In figures 1 and 2 we have plotted H (T) for the two sets of coupling constants (A)
c2
and (B). Each plot contains the results for different ratios of the diffusivities in the two
bands η. If the diffusivities are the same in the two bands (D = D ), the H (T) curves
1 2 c2
are identical to those in single-band superconductors. Only once the diffusivities start to
UppercriticalfieldasaprobeformultibandsuperconductivityinbulkandinterfacialSTO8
differ appreciably, do the H (T) curves show a departure from single band behaviour.
c2
The characteristic two-band property of the H (T) curves, and thus the indicator for
c2
the presence of two-band superconductivity, is a change in the curvature of the H (T)
c2
curve, as can be seen most clearly in the blue dotted curve of figure 1. In the vicinity
of T , H initially grows very slowly, but at some temperature (here at T ≈ 0.5T ) it
c c2 c
starts growing dramatically until it saturates at T = 0. In contrast, for single band
superconductors or for equal diffusivities, H (T) starts growing rapidly at T and as
c2 c
T → 0 the growth rate monotonically decreases (see red curve in figure 1).
While we have no precise calculation for the ratio of the diffusivity, at constant
mean free time τ the diffusivities should be proportional to the square of the Fermi
velocity, since D ∼ l2 /τ = τv2 (l is the mean free path). The Fermi velocities in
mfp F mfp
STO differ by about a factor of 3 or 4 between the two bands [31]. Assuming the mean
free time to be the same, we thus obtain a ratio of diffusivities of about 10, which is
sufficient to observe the non-monotonic behaviour of the H (T) curvature.
c2
As we can clearly see from figures 1 and 2, the shape of the H (T) curves
c2
depends strongly on the values of the coupling constants chosen. The set of coupling
constants (A) is much more favourable for the detection of multiband superconductivity
than the set (B).
As we cannot be sure which set of coupling constants are precisely applicable for
STO, we now turn to a broader investigation of the upper critical field for more general
coupling constants.
4.2. More general parameter values
Here we explore in greater detail under which more general conditions two-band
superconductivitycanleadtoadiscerniblemodificationoftheH (T)curvewithrespect
c2
to the single band behaviour. In exploring this behaviour we explicitly go beyond the
values of the coupling constants expected for STO. We concentrate on the physics of
the bulk, as the physics of the interface is similar, as described in section 6.
We first investigate the possibly simplest situation in which one of the coupling
constants is zero, see figure 3. We choose λ = 0. In this case superconductivity only
22
exists in the second band as a result of the induced superconductivity due to λ . For
12
λ = 0 one obtains the single band H (T) behaviour. Although there is a dependence
12 c2
ofthecurvesonλ ,itisnotverystrong. Withoutaccesstotheentiretemperaturerange
12
0 < T < 1 it would be difficult to conclude whether or not multiband superconductivity
Tc
is present. The strongest departure from the single band behaviour of H (T) occurs at
c2
λ ≈ λ . If λ (cid:29) λ the two bands are strongly locked to each other and thus the
12 11 12 11
behaviour is similar to that for a single band system again.
In figure 4 we fix λ = 0.02, λ = 0.14 and vary λ . We observe that the
12 11 22
departure of the H curve from single band behaviour is strongest when the coupling
c2
constants within the bands are roughly equal. If their difference is too great, one of the
bands always dominates and the interplay of the two bands, which ultimately causes
UppercriticalfieldasaprobeformultibandsuperconductivityinbulkandinterfacialSTO9
Figure 3. Upper critical field in the case where one of the intra-band coupling
constants is zero. The parameters are given by λ =0.14,λ =0,η =0.1.
11 22
Figure 4. Upper critical field for the case in which a small interband coupling is
chosen, and one of the intraband coupling constants is varied. The parameters are
given by λ = 0.14,λ = 0.02,η = 0.1. The strongest departure from single-band
11 12
behaviour is observed when λ ≈λ .
11 22
the non-monotonic curvature of H (T), cannot be observed.
c2
In figure 5 we explore the behaviour of H (T) for different values of λ in the case
c2 12
when λ = λ , the case most favourable for the detection of the signature of multiband
11 22
superconductivityinH (T). Ifthecouplingbetweenthebandsisabsent, eachbandjust
c2
shows single band behaviour and there is no signature of multiband superconductivity
in the upper critical field. This is due to the fact that it is only the most dominant
band, the one with the larger coupling constant, which determines H . As can be seen
c2
from figure 5, the signature in the upper critical field can be best detected when λ is
12
significantly smaller than λ = λ , but non-zero. For λ ≈ λ a signature remains
11 22 12 11
but requires access to a very large range of T for it to be detected. Once λ (cid:29) λ
Tc 12 11
the bands are so strongly coupled that the system effectively behaves like a single band
system.
UppercriticalfieldasaprobeformultibandsuperconductivityinbulkandinterfacialSTO10
Figure 5. Upper critical field for the case λ = λ = 0.14 and η = 0.1, in which
11 22
now the inter-band coupling λ is varied. The circumstances most favourable for the
12
detection of multiband superconductivity ar when λ is much smaller than λ , but
12 11
non-zero.
From the above we may conclude that multiband superconductivity can be most
easily detected through measurements of the upper critical field when the coupling
constants within the two bands are approximately the same, the inter-band coupling
constant is significantly smaller than the intra-band coupling constants, and the
diffusivities in the two bands differ by at least a factor of 5.
We thus find that depending on what set of diffusivities and which of the two
coupling constants are realised in real STO, two band superconductivity might be
inferred from the shape of the H (T) curve. This observation can provide guidance
c2
for the search of multiband superconductivity in STO. On the other hand, a seemingly
trivial behaviour of the H (T) curve does not imply that STO is a single band
c2
superconductor. It has been argued that unconventional H behaviour could be
c2
expected even for a single band systems, as long as the single band is highly anisotropic
[13]. However,forSTOthisisnotexpectedtobethecase[21,31],andanunconventional
behaviour of H can be taken to be good evidence for multiband superconductivity.
c2
5. H for clean doped bulk STO
c2
For completeness we also present the case of clean bulk superconducting STO. Away
from optimal doping, bulk STO may enter a regime in which the mean free path is
larger than the superconducting coherence length [21]. In this regime a calculation for
the clean system is more appropriate. We therefore briefly present the results obtained
from the quasi-classical Eilenberger equations. The critical field for a three-dimensional