Table Of ContentAngular-dependent upper critical field of overdoped Ba(Fe Ni ) As
1−x x 2 2
J. Murphy,1,2 M. A. Tanatar,1,2 D. Graf,3 J. S. Brooks,3
S. L. Bud’ko,1,2 P. C. Canfield,1,2 V. G. Kogan,1 and R. Prozorov1,2,∗
1Ames Laboratory US DOE, Ames, Iowa 50011, USA
2Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA
3National High Magnetic Field Laboratory, Florida State university, Tallahassee, Florida 32310, USA
(Dated: 27 January 2013)
In-planeresistivitymeasurementsasafunctionoftemperature,magneticfieldanditsorientation
with respect to the crystallographic ab−plane were used to study the upper critical field, H , of
c2
two overdoped compositions of the iron-based superconductor Ba(Fe Ni ) As , x =0.054 and
3 1−x x 2 2
x=0.072. Measurements were performed using precise alignment (with accuracy less than 0.1o) of
1
magnetic field with respect to the Fe-As-plane. The dependence of the H on angle θ between the
0 c2
2 field and the ab-plane was measured in isothermal conditions in a broad temperature range. We
found that the shape of Hc2(θ), clearly deviates from Ginzburg-Landau functional form.
n
a
PACSnumbers: 74.70.Dd,72.15.-v,68.37.-d,61.05.cp
J
7
2 I. INTRODUCTION pling. Theoretical modeling explicitly included these ef-
fectsintoconsiderationandproducedtheangulardepen-
]
n The theory of the upper critical field in superconduc- dence of the Hc2(θ) which was virtually identical to the
o tors was essentially developed by the mid-60s,1–3 with dependence of Eq. 1.28
c the linear H (T) behavior close to T described by Recent developments in understanding of the upper
- c2 c
r the anisotropic Ginzburg-Landau theory,4,5 leading in critical field were greatly stimulated by the discovery
p tetragonalcrystalstodependenceofH onangleθ with in 2008 of the layered FeAs superconductors,29 which
u c2
s respect to the ab-plane: opened up a new avenue in high Tc research. The up-
. per critical fields of iron pnictides are very high,30 and
t
a besides the potential for high field applications,31 this
m H (θ)= Hc2,ab , γ = Hc2,ab . (1) brings the possibility of the paramagnetic effects at low
- c2 (cid:113)(γ2 −1)sin2θ+1 H Hc2,c temperatures.32
d H
Since Fe-As layers form a main building block of all
n
Renewed interest to studies of the upper critical field iron-based superconductors,33–36 these compounds show
o
c was brought by the discovery of materials with poten- anisotropy of the electronic structure, reflected in the
[ tiallyunconventionalpairingmechanisms,heavyfermion, anisotropy γH of the upper critical field. Unlike sig-
organic and especially cuprate superconductors, with nature layered materials, organic superconductors37 and
1
v the well documented d-symmetry of the superconduct- cuprates,38 the values of γH in iron arsedines are small,
6 ing order parameter in the latter case.6 It was quickly for most compounds in the range 2 to 4,39–48 with few
9 recognized that the angular dependence of the upper exceptions49–51 were values as high as 7 to 10 are found,
3 critical field, in particular on the angle φ in the basal see Ref. 52 for review. Furthermore γH in iron ar-
6 plane of the tetragonal crystals, can be used to probe senides shows notable temperature dependence, presum-
1. the anisotropy of the superconducting order parameter.7 ably reflecting multi-band effects.46 Low anisotropy val-
0 These ideas were further developed for unconventional uessuggestthatatleastclosetothecriticalinzero-field,
3 superconductors with various exotic order parameters Tc(H =0), we are dealing with orbital Hc2.
1 in a series of papers by K. Maki and co-workers.8–12 Thedetailedstudyoftheanisotropyoftheuppercrit-
:
v They stimulated a series of experimental studies, in par- ical field along principal a and c directions as a func-
i ticular in Sr RuO ,13–15 heavy fermion16 and organic tion of doping was undertaken in the series of electron-
X 2 4
superconductors.17–21 doped compounds Ba(Fe1−xCox)2As2 (BaCo122 in the
ar Although these developments were correctly catching following).42 It was found that γH changes significantly
the importance of the anisotropy of the superconducting between under-doped x < 0.08 and overdoped regions
gap structure for the angular-dependent H , the the- of the phase diagram, presumably reflecting change of
c2
oretical data analysis was oversimplified by assumption the electronic structure. Comparison with optimally
of simple cylindrical or spherical Fermi surface shapes. doped Ba(Fe1−xNix)2As2 (BaNi122 in the following),
The importance of the Fermi surface topology for the x=0.04647,48 suggested that the anisotropy is not par-
anisotropyoftheH wasbroughttofocusbythediscov- ticularly sensitive to the amount of disorder introduced
c2
ery of pronounced multi-band effects in superconductiv- by dopant x.
ity of MgB .22–24 The upper critical field of this layered Ontheotherhand,severalstudiespointedoutcomplex
2
material is anisotropic and can be varied significantly by superconducting gap structure of iron arsenides, both in
carbon and aluminum doping,25–27 controlling the mean BaCo12253–58 and analogous BaNi122.59 For both these
freepathofthecarriersandchangingtheinter-bandcou- materials,itwassuggestedthatnotablesuperconducting
2
3 0 0 7 0
x = 0 .0 7 2
6 0
x = 0 .0 5 4 )
)2 0 0 m
m c H ||a b
c
W mW 5 0
m(cid:1) ( 100 (cid:1) (
1 0 0 T
) s
(KT 50 T m 4 0
T
c
0
0.00 0.05 0.10
x
0 -4 -2 0 2 4
0 1 0 0 2 0 0 3 0 0 4 0 0 (cid:2) (cid:1)( d e g r e e s )
T ( K )
c -axis H -rotation
z plane
FIG. 1. (Color Online) Temperature-dependent resistivity
of two samples of BaFe Ni As used in this study, with
1−x x 2
x=0.054 (slightly overdoped) and x=0.072 (strongly over-
doped), with doping level indicated with arrows with re-
H
specttotemperature-dopingphasediagramofBaNi122after
Ref.47and48shownintheinset. Notepronouncedcurvature
θ
of the ρ(T) for T >T , typical of overdoped compositions.61 y
c
Sampleresistivityvalueisdefinedwithaccuracyofabout20%
due to uncertainty of geometric factors, see Refs. 46 and 62 Sample
for details.
x
gap modulations along c-axis should be characteristic of
the overdoped regime. Since this gap variation might be
reflectedintheangulardependenceoftheH (θ),wede-
c2
FIG. 2. (Color Online) During experiments in single axis ro-
cidedtoperformsystematicstudyofthesecompositions.
tationsystemof35Tmagnet,thedirectionofmagneticfield
To our knowledge, there was only one study addressing
was aligned parallel to the conducting plane by resistivity
angularvariationofH iniron-basedsuperconductors,60
c2 measurements in field H slightly lower than H , in which
c2(cid:107)
providing no high angular resolution data.
sampleresistanceshowsstrongangulardependence,blackline
In this paper we perform a detailed characterization in the top panel. The curve was measured in one-sided mo-
of the upper critical field in overdoped BaNi122 as a tion of the rotator to avoid backlash, with deep minimum
function of temperature and field direction. Our main correspondingtoH (cid:107)aborθ=0condition. Theredopensym-
experimental finding is a clear deviation of the angular bolsshowalignmentmeasurements,takeninasecondangular
dependence from Eq. 1, particularly strong for the di- sweepofthesamerotationdirection,andstoppedatθ=0. H
and T sweeps were used to determine the phase diagrams in
rections of magnetic field close to H (cid:107) c, where orbital
H (cid:107)abcondition,andthenmagneticfieldangleθwithrespect
effects should be the strongest.
to the plane was changed by continuing rotation of the sam-
ple in the same direction as during alignment. Because the
orientationofthesampleinthethirddirection,perpendicular
II. EXPERIMENTAL totherotationplane,wassetbyeyetheremayexistnon-zero
angleϕbetweenthefield-rotationplaneandtheplaneofthe
Single crystals of Ba(Fe Ni ) As were grown from normaltothesample. Inmostcasesthisangleshouldbeless
1−x x 2 2 than 5o.
FeAs flux using a starting load of metallic Ba, FeAs
and NiAs, as described in detail elsewhere.48 Crys-
tals were thick platelets with large faces correspond-
ing to the tetragonal (001) plane. Actual composition Samples for in-plane resistivity, ρ, measurements were
of the crystals was determined using WDS x-ray elec- cleaved with a razor blade into rectangular strips with
tron probe microanalysis.48 The two compositions stud- typical dimensions, 2×(0.1−0.3)×(0.03−0.1) mm3
ied were on the overdoped side of the phase diagram, andthelongsidecorrespondingtotetragonala-axis. All
slightlyoverdopedx=0.054(T =16K)andstronglyover- sample dimensions were measured with an accuracy of
c
doped x=0.072 (T =7.5 K), whereas maximum T =19 K about 10%. Contacts to the samples were made by at-
c c
is achieved at optimal doping, x =0.046,47,48 see dop- taching silver wires using ultrapure tin, resulting in an
opt
ing phase diagram in inset in Fig. 1. ultra low contact resistance (less than 10 µΩ).63 Resis-
3
tivity measurements were made using a standard four-
1 0 0 H ||a b
probe technique, producing the ρ(T) curves as shown in
Fig. 1. After initial preparation, samples were charac- 8 0 %
terized in PPMS system, and then glued by GE-varnish
)
to a plastic platform, fitting single axis rotator of the m
35 T DC magnet in National High Magnetic Field Lab- c 5 0 % 3 5 T 0
W 5 0
oratory in Tallahassee, Florida. Sample resistance was m
checked after mounting and found to be identical to the (cid:1) (
2 0 %
initial value. High-field measurements were made in He-
( a )
cryostat with variable temperature control inset (VTI)
allowing for temperatures down to 1.5 K. 0
The stepping motor driven rotator enabled in situ ro- 1 0 0
tation with 0.05o resolution around a horizontal axis in H ||c
singleaxisrotationsystemofvertical35Tmagneticfield.
During this rotation the direction of magnetic field with
)
rFeisgp.e2c.tWtoetchaencprryesctiaslelsytaaylisgninthaepdlairneectoiofnrootfatthioenm, asege- cm 5 0 3 5 T 0
W
neticfieldparalleltothesampleplanewithintherotation m(cid:1) (
plane, defined as θ =0, using angular dependence of re-
sistivity, measured in magnetic field slightly below H . ( b )
c2(cid:107)
ThisalignmentisillustratedinFig.2. Inanidealcaseof
0
thesecondsampleaxiscoincidingwiththerotationaxis,
field-rotation plane should contain c-axis of the sample. 4 0
Theremayhavebeenanon-zerouncontrolledangleϕbe-
H ||a b
tweenthefield-rotationplaneandtheplaneofthenormal 3 0
to the sample, see Fig. 2. We estimate that ϕ<5◦.
)
(T 2 0
2
c H ||c
H
III. RESULTS
1 0
( c )
x = 0 .0 5 4
In Figs. 3 and 4 we show raw ρ(T) data for a set
0
of magnetic fields aligned approximately along c-axis 0 5 1 0 1 5
(θ =90o, top panels) and precisely along the conduct- T ( K )
ing plane (θ =0o, bottom panels), for BaNi122 samples
with x=0.054 and x=0.072, respectively. We show also
FIG. 3. (Color Online) In-plane resistivity ρ vs. temper-
the lines corresponding to 20, 50 and 80% of resistiv- a
ature for slightly overdoped Ba(Fe Ni ) As , x=0.054 in
ity value immediately above the transition, ρ(T ), used 1−x x 2 2
c magnetic fields (a) parallel to the conducting ab plane (right
as criteria to determine the transition temperature as a
to left, 0 T, 1 T, 2.5 T, 5 T, 7.5 T, 10 T, 15 T, 17.5 T,
function of magnetic field and construct the phase dia-
20 T, 25 T, 30 T, 35 T); (b) parallel to the c-axis (right to
grams, bottom panels (c) of Figs. 3 and 4. The use of
left, 0 T, 1 T, 2 T, 4 T, 8 T, 10 T, 12 T, 15 T, 20 T, 30 T,
thesecriteriaisjustifiedbysmallvariationoftheresistive 35T).Linesindicate20,50,and80%oftheresistivityvalue
transition width on application of magnetic field, and its immediately above the superconducting transition. Bottom
independence on the extrapolation, typical problem for panel(c)showsH (T)phasediagramsforbothdirectionsof
c2
onset and offset criteria. magnetic field.
As can be most clearly seen from the bottom panel of
Fig. 3, the shapes of the H (T) phase diagrams in par-
c2
allel and perpendicular field orientations share the same IV. DISCUSSION
features as found in previous studies of other Fe based
systems. The H (T) flattens at low temperatures,
c2,ab A. Angular dependence of H
c2
while H (T) maintains positive curvature down to the
c2,c
lowest temperatures of our experiment. Both these fea-
tures are typical for layered materials, see for example 1. Linearization of the Hc2(θ) dependence
Refs. 37 and 64.
In Figs. 5 and 6 we show field dependences of in-plane To check if Eq. 1 describes our data, instead of com-
resistivity taken at fixed temperatures with inclination monly used data fitting, as shown in the bottom panels
angleθasaparameterforslightlyoverdopedsamplewith of Figs. 5 and 6, we used an approach based on data
x=0.054andstronglyoverdopedsamplex=0.072,respec- transformation so as to make possible deviations clearly
tively. The data analysis will be presented in the next visible. AccordingtoEq.1,theH−2 vs(sin2θ)shouldbe
c2
section. a straight line, and in Figs. 7 and 8 we plot this way the
4
H ||a b 1 0 0 (a ) x = 0 .0 5 4 9 K
1 0 0
2 2 .5 T
8 0 % 8 0 %
)
)
m
cm 5 0 % c 5 0 5 0 % asdfasdf
W 5 0 W 9 0 0
m m
(cid:1) ( 1 7 .5 T 0 (cid:1) ( 2 0 %
2 0 %
( b ) 0
0 1 0 0 (b )
1 3 K
H ||c
1 0 0
)
) m
cm W c 5 0 9 0 0 0 o
W m
m(cid:1) ( 5 0 6 T 0 (cid:1) (
0
x = 0 .0 7 2 ( a ) 0 5 1 0 1 5 2 0 2 5 3 0 3 5
0 H ( T )
3 0
2 0 H ||a b
2 5
9 K
1 5 2 0
(T)Hc21 0 H ||c H(T)c211 05 1 3 K
5 5 H ||a b
( c ) (c ) H ||c
0
0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0
0 2 4 6 8 1 0
(cid:1) (d e g re e s )
T ( K )
FIG.5. (ColorOnline)Fielddependenceofin-planeresistiv-
FIG. 4. (Color Online) In-plane resistivity ρ vs. temper-
a ity ρ(H) of slightly overdoped Ba(Fe Ni ) As , x=0.054
ature for heavily overdoped Ba(Fe Ni ) As , x=0.072 in 1−x x 2 2
1−x x 2 2 sample at T=13 K (panel (a)) and T=9 K (panel (b))
magnetic fields (a) parallel to the conducting ab plane (right
with magnetic field inclination angle θ as a parameter. (c)
toleft, 0T,1T,2.5T,5T,7.5T,10T,15 T,17.5T,20T,
IsothermsH (θ),obtainedat9Kand13K,using80%,50%
22.5T);(b)paralleltothec-axis(righttoleft,0T,1T,2T, c2
and 20% criteria. Solid line shows fit to Eq. 1.
4 T, 6 T). Lines indicate 20, 50, and 80 % of the resistiv-
ity value immediately above the superconducting transition.
Bottom panel (c) shows H (T) phase diagrams for both di-
c2 of our experiment:
rections of magnetic field.
H
H (θ,ϕ)= c2,ab . (2)
c2 (cid:113)
dataforsampleswithx=0.054andx=0.72respectively. (γH2 −1)cos2ϕ sin2θ+1
The data show clear deviation from linear trend, irre-
It is seen that constant ϕ, as determined by our ex-
spectively of the criterion of H determination from the
c2 perimental geometry, does not change the linear relation
resistivity data, with the deviation being the strongest
close to H (cid:107) c−axis or sin2θ=1. To check if the devia- of Hc−22 vs sin2θ, despite changing the magnitude of the
variation, vanishing for ϕ = 90o, corresponding to field
tion from Eq. 1 in Figs. 7 and 8 can be caused by finite
rotation parallel to the conducting plane. Therefore the
inclinationangleϕ(seeFig.2forthedefinition),herewe
linear dependence of H−2 on sin2θ is not affected by a
provide the angular dependence of H for arbitrary ϕ c2
c2 misalignment ϕ.
Choosingthecross-sectionoftheplane,inwhichH isro-
The H (θ) described by the Eq. 1, is a direct con-
tated,withtheabcrystalplane(seeFig.2)asthexaxis, c2
sequence of the linearized GL equation for anisotropic
we obtain in the crystal frame cˆ= (0,0,1) and the unit
materials at H :
vector along the field hˆ = (cosθ, sinθsinϕ, sinθcosϕ). c2
This gives for the angle θ between the field and c axis:
c
cosθ =cˆ·hˆ =sinθcosϕ. We then obtain for geometry −(ξ2) Π Π Ψ=Ψ, (3)
c ik i k
5
( a ) x = 0 .0 7 2 0 .0 0 0 .2 5 0 .5 0 0 .7 5 1 .0 0
0 .0 0 8
1 0 0 8 0 %
H ||c
H ||a b 9 K
)
9 0 0
m
c 0 .0 0 6
W 5 0 5 0 % a d
m
(
r 2]
2 0 % 1 .5 K (T)20 .0 0 4
c
0 H[
/
( b ) 10 .0 0 2
1 0 0
) x = 0 .0 5 4
m 0 .0 0 0
c
W 5 0 9 0 0 0 0 1 3 K
m 0 .0 8
(
r
4 K
0 .0 6
0
0 1 0 2 0 3 0 2)]
H ( T ) T
(
c2 0 .0 4
H ||a b
H[
/
1
2 0
1 .5 K 0 .0 2
)
T H ||c
(
c2 0 .0 0
H 1 0 0 .0 0 0 .2 5 0 .5 0 0 .7 5 1 .0 0
s in 2 (cid:1)
( c ) 4 K
H ||c
0
0 3 0 6 0 9 0 FIG. 7. (Color Online) Angular dependence Hc2(θ), deter-
(cid:1) ( d e g r e e s ) mined from fixed temperature ρ(H) of Fig. 5 using 20%,
50%and80%criteria(toptobottom),forslightlyoverdoped
Ba(Fe Ni ) As , x=0.054 at 9 K (top panel) and 13 K
1−x x 2 2
FIG.6. (ColorOnline)Fielddependenceofin-planeresistiv- (bottom panel). The lines are guides to the eyes. The data
ity ρ(H) of strongly overdoped Ba(Fe Ni ) As , x=0.072 are plotted as H−2(sin2θ), which according to Eq. 1 should
1−x x 2 2 c2
sample at T =1.5 K (panel (a)) and T=4 K (panel (b)) be a straight line. The lines are guides to the eyes.
with magnetic field inclination angle θ as a parameter. (c)
Isotherms H (θ), obtained 1.5 K and 4 K, using 80%, 50%
c2
and 20% criteria. Solid line shows fit to Eq. 1.
anddeviationsfromEq.(1)canbenaturallyexplainedby
violationofthisprocedure. Inaddition,foriron-pnictides
the importance of the paramagnetic effects for magnetic
whereΠ=∇+2πiA/φ ,Aisthevectorpotentialandφ fieldsparalleltotheFe-Asplanewassuggestedtoexplain
0 0
isthefluxquantum;summationisimpliedoverrepeating unusual shape of the H (T)32,60,65). This may also lead
c2
indices. Both sides of this equation are scalars, so that to the deviation from Eq.(1) with the maximum effect
(ξ2) is a second rank tensor with the standard angular expected at low temperatures and for orientations close
ik
dependence which is reflected in Eq. (1). to H (cid:107)ab−planes.
We note that in the original papers, the angular de- Motivatedbytheseconsiderations,wecompileinFig.9
pendence, Eq.(1), has been derived for single band s- the published data for various layered materials, ana-
wave superconductors. It has also been recently shown lyzed by plotting H−2 vs sin2θ. The data are arranged
c2
that this behavior is expected for arbitrary Fermi sur- with decreasing anisotropy from top to bottom. The
face, the superconducting gap modulation and for multi most anisotropic materials, staged graphite intercalation
- band materials.71 However, this conclusion is achieved compounds (top panel, data from Ref. 66) and layered
assuming the explicit factorization of the pairing poten- Sr RuO (data from Ref. 15) closely follow Eq. 1. In-
2 4
tial and order parameter, V(k,k(cid:48)) = V Ω(k)Ω(k(cid:48)) and terestingly, that clear deviations from this behavior in
0
∆(T,k) = Ψ(r,T)Ω(k ), see detailed discussion in the Sr RuO , arising due to unusual limiting mechanism
F 2 4
theory section below. There is no microscopic justifi- in magnetic fields close to H (cid:107) ab,15 is very difficult
cation for such factorization in complex superconductors to recognize in a limited angular range near θ=0, as
6
structure,nodelessintheformercase69 andwithvertical
0 .0 0 0 .2 5 0 .5 0 0 .7 5 1 .0 0
linenodesinthelatter.70 Consideringthatamongallthe
0 .0 1 5
materialsforwhichwewereabletofindpublishedH (θ),
c2
onlypuremulti-bandMgB andNbSe revealcleardevi-
2 2
1 .5 K ations from Eq.(1), it is tempting to relate the observed
deviations with the multi-band superconductivity in the
0 .0 1 0 clean limit. This might be quite natural that in these
2
)] systems the factorization of the pairing potential and of
(T the order parameter does not hold given the complexity
2
c of the in- and inter - band interactions. This explana-
[H0 .0 0 5 tion, however, is not universal, since multi-band effects
/
1 are very pronounced in high purity crystals of KFe As ,
2 2
but no clear deviations from Eq.(1) are found there. On
H ||a b H ||c theotherhand,itishardtoconsideroverdopedBaNi122
0 .0 0 0 as a clean system, since scattering due to substitutional
x = 0 .0 7 2 disorder, especially on the Fe site, is significant in these
0 .0 8 compositions. The observation that the deviations from
the linear plot in MgB diminish with disorder, suggest
2
thatitisk-dependenceofthegapmagnitude,ratherthan
2)] 0 .0 6 4 K multi-bandnatureoftheFermisurfaceitselfisimportant
T fortheunusualangulardependence. Thisconclusionisin
(
2
c linewiththerecentextensionoftheHWtheoryformulti-
[H 0 .0 4 band superconductors with arbitrary Fermi surfaces.71
/
1 Indiscussingtheseresultsweshouldkeepinmind,that
0 .0 2 in all cases, except for Sr2RuO4, the Hc2 was measured
resistively,sothatinevitablyitsdeterminationisapprox-
H ||c imate since the resistive transition as a rule has finite
0 .0 0 width and hence, the H values depend on a criterion
0 .0 0 0 .2 5 0 .5 0 0 .7 5 1 .0 0 c2
chosen. Finite resistivity in the flux - flow regime (most
s in 2 (cid:1) pronouncedinthecleansystems)broadensthetransition
making resistive determination difficult. From this point
of view, assertions of Kim et al.26 that their data allow
FIG. 8. (Color Online) Angular dependence Hc2(θ), deter- to distinguish between two models, GL and two - band
minedfromfixedtemperatureρ(H)ofFig.6using20%,50% Usadel approach by Gurevich,28 are hard to accept.
and 80% criteria (top to bottom), for strongly overdoped
In compounds with relatively high T , the determina-
Ba(Fe Ni ) As , x =0.072 at 1.5 K (top panel) and 4 K c
1−x x 2 2 tion of H from resistive measurements is also compli-
(bottom panel). The lines are guides to the eyes. The data c2
are plotted as H−2(sin2θ), which according to Eq. 1 should catedbythephenomenonofvortexlatticemelting: above
be a straight linec.2 the melting point, the resistivity is close to that of the
normalphaseandH persebecomesinvisibleinresistiv-
c2
ity measurements. This complication in a given material
might affect the measurements stronger near T than at
c
the dependence in the whole range is dominated by the
low temperatures.
anisotropy of the Fermi surface. On the other hand,
two materials in which superconductivity shows strong
multibandfeatures,MgB 26 andNbSe 67 showdistinctly
2 2
different angular dependences. The H−2(sinθ) in pure 2. Theoretical considerations
c2
MgB 68 shows downward bent as field approaches c−
2
axis, θ=90◦, similar but much less pronounced than in Recently, the theoretical justification for the near-
our observations in BaNi122. On the other hand, doped universality of the angular dependence (Eq. 1) has been
Mg(B Al ) closely follows the linear H−2(sinθ) de- offered in Ref. 71 for orbital H (θ) albeit under certain
1−x x 2 c2 c2
pendence, Eq. 1, which may be suggestive that doping restricting assumptions. We reproduce briefly the argu-
diminishesmulti-bandeffectsduetoenhancedinter-band ment of this work. Some deviations from the effective
scattering. For pure NbSe the H−2(sinθ) plot shows mass model have been predicted theoretically for quasi-
2 c2
most clear deviations from linearity among all materials, one-dimensional conductors , see Ref. 72. We also note
with an upward curvature towards θ = 90◦, of oppo- that the theory of H (θ) at the arbitrary angle with re-
c2
site trend to pure MgB and BaNi122. The two angular spect to the crystal axes in multi-band superconductors
2
data sets for profoundly multi-band iron pnictide super- has additional complications. In particular, if H is not
conductors, slightly underdoped, BaK122,60 and heav- parallel to the c−axis and the mass anisotropy ratios in
ilyoverdopedK122,65 generallyfollowlineardependence two bands are different, and the first Landau level solu-
despite profound difference in the superconducting gap tions which are then substituted into the HW integrals
7
cannotsimultaneouslysatisfythetwo-bandEilenberger leadtogradienttermsinGLequationsdifferentfromthe
equations and the gap equations, and the higher Landau standard forms (3) or (4) and, therefore, different from
levels become important.52,76 Perhaps this provides an Eq.(1) angular dependencies, see e.g. Ref.73.
alternativeexplanationofwhyweobservethedeviations
We therefore may conclude that deviations of the ob-
from the conventional scaling only in clean two - band
served angular dependence of H from the form (1) (or
c2
systems.77 Keeping this in mind, we proceed with the deviations of H−2 plotted vs. sin2θ from the straight
generalized description.71 c2
line)signalthatthecouplingpotentialcannotbewritten
We recall that, as HW1 had shown within the weak- in the separable form. On the other hand, the exam-
couplingBCS-Gor’kovtheory,thelinearizedGLequation ple of separable potentials (for any Fermi surface and
−ξ2Π2∆ = ∆ for the isotropic case holds at any T at any order parameter symmetry) shows that there is no
Hc2(T). Howeverξ(T)doesnotdivergeforT →Tc(H)< direct relation between the angular dependence of Hc2,
Tc(0), -instead it is finite and should be evaluated to Fermi surfaces, and order parameter symmetries. How-
satisfythebasicself-consistency(orgap)equationforthe ever, deviations of H (θ) from the form (1) may carry
c2
orderparameter∆. ItiseasilyshownthenthatHc2(T)= such an information. To investigate this question fur-
φ0/2πξ2. theronewouldneedabetterdataonthesedeviations,in
For materials with anisotropic Fermi surfaces and or- particular, criterion-independent determination of H ,
c2
der parameters one can consider Eq.(3) as an ansatz for which is hard to achieve in resistive measurements. On
satisfying the self-consistency equation of the theory. To the theoretical side, of course, one should go beyond the
facilitatethisderivationoneassumesthecouplingpoten- weak coupling and separable coupling potentials.
tialoftheformV(k,k(cid:48))=V Ω(k)Ω(k(cid:48))andtheorderpa-
0
rameter ∆ = Ψ(r,T)Ω(k ). The function Ω(k ) which
F F
determinesthek dependenceof∆isnormalizedsothat
F
(cid:104)Ω2(cid:105) = 1. This convenient approximation has been first
designed by Markovic and Kadanov74 and it works well B. Temperature dependence of Hc2
for one band materials with anisotropic coupling.
For two-band materials one can use a generalization There are two mechanisms that determine the up-
of this procedure. Other than s-wave order parameters per critical field of superconductors. The first one, de-
emerge if the coupling V(k,k(cid:48)) responsible for supercon- termined by the supercurrent flow to screen the mag-
ductivityisnotaconstant(ora2×2matrixofkindepen- netic field, is referred to as orbital limiting and de-
dentconstantsfortwobands). Theformallysimplestway scribed by HW theory.1,2 The upper critical field at
to consider different from s-wave order parameters with- T → 0 limit, H (0), in HW theory is determined by
c2
out going to details of microscopic interactions is to use the slope of the H (T) curve close to T , and as T goes
c2 c
a “separable” potential, V (k,k(cid:48)) = V(0)Ω (k)Ω (k(cid:48)) to zero the curve shows downward deviation from linear
αβ αβ α β
whereV(0)isakindependentmatrix,andlookfortheor- dependence and eventual saturation towards the value
derparaαmβetersintheform∆ (r,T,k)=Ψ (r,T)Ω (k) Hc2(0)≈0.7TcddHTc2 in isotropic case.
α α α
with the normalization (cid:10)Ω2(cid:11) =1 for each band (α,β = Ratherrareexceptions,whentheuppercriticalfieldis
α not determined by the orbital motion, are found in the
1,2 are band indices).
materials in which orbital motion of electrons is ham-
IthasbeenshowninRef.71thatinthisapproximation
pered by either short mean free path, heavy mass of
the ansatz
conduction electrons in heavy fermion materials or weak
−(ξ2) Π Π ∆ =∆ , α=1,2. (4) links between the conducting layers in Josephson struc-
ik i k α α
turesorinnaturallylayeredmaterials,providedthatthe
leads to solutions satisfying the self-consistency equa- magnetic field is aligned precisely parallel to the con-
tions, which provide all components of the tensor ducting layer. In this situation the upper critical field
(ξ2)ik(T). In this way the ansatz (4) is proven cor- Hc2 is determined by Zeeman splitting of electron levels,
rect. Along with (ξ2) (T), the upper critical field and known as Clogston-Chandrasekhar3 paramagnetic limit.
ik
its anisotropy can be evaluated for a given Fermi surface This field is determined by a decrease of paramagnetic
and the order parameter symmetries Ω . energy becoming equal to condensation energy of super-
α
Since (ξ2) is a second rank tensor, we conclude, ex- conductor. In weak coupling BCS superconductors the
ik
actlyasinthecaseofanisotropicGLEq.(3),thatthean- paramagnetic limiting field is determined in T →0 limit
gular dependence of Eq.(1) should hold at any tempera- as Hp=1.84Tc, where Hp is field in Tesla and Tc is in K.
tureandforanyanisotropicFermisurface,anysymmetry As can be seen from Figs. 3 and 4 the upper criti-
of the bands order parameters, and at any temperature. cal fields in H (cid:107) ab configuration are higher than the
In other words, within the approximation of “separable” weak-limit paramagnetic limiting H , equal to 32.2 T
p
coupling potential, the dependence (1) is “universal”. (x =0.054) and 13.8 T (x =0.072). These high values
Clearly, “separable” potentials do not exhaust all pos- may come from strong coupling nature of superconduc-
sibleinteractionsand,therefore,otherformsoftheangu- tivity in iron pnictides, or indeed reflect paramagnetic
lar dependence H (θ) can exist. An example of such a limiting at low temperatures as was suggested in several
c2
potentialhasbeenstudiedinRef.9. Suchpotentialsmay studies.32,60,75
8
V. CONCLUSIONS anisotropic penetration depth and thermal conductivity
measurements.53,54,59
By performing high angular resolution study of the
upper critical field in two overdoped compositions of VI. ACKNOWLEDGEMENTS
ironpnictidesuperconductorBa(Fe Ni ) As , wefind
1−x x 2 2
clear deviations from the anisotropic Ginzburg-Landau WethankA.GurevichandA.G.Lebed’forusefulcom-
form. Implementing linearization plot analysis of our ments. Work at the Ames Laboratory was supported by
and previously published data, we find clear deviations the Department of Energy-Basic Energy Sciences under
from the form only in the case of clean fully - gapped Contract No. DE-AC02-07CH11358. JSB acknowledges
multi-band superconductors, such as NbSe ang MgB , support from NSF DMR 1005293. Work at the National
2 2
but not in dirty MgB and not clean KFe As . We HighMagneticFieldLaboratoryissupportedbytheNSF
2 2 2
speculate, that the dependence may reflect c-axis mod- Cooperative Agreement No. DMR0654118 and by the
ulation of the superconducting gap, as suggested by state of Florida.
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10
2 x 1 0 5
0 .0 3 G IC C 4 R b H g T c = 0 .9 9 K T = 0 .4 4 K
G IC C 4 K H g T c = 0 .7 3 K T = 0 .4 0 K
0 .0 2
1 x 1 0 5
0 .0 1
( a )
0 .0 0 0
1 .5 S r2 R u O 4 T c = 1 .4 K T = 0 .1 0 K 2 0 0
1 .0
1 0 0
0 .5
( b )
0 .0 0
6 M g (B 1 -x A lx )2 2 .0
x = 0 T = 3 0 K
x = 0 T = 2 8 K 1 .5
4 x = 0 .1 2 T = 1 4 K
x = 0 .1 2 T = 2 3 K
1 .0
x = 0 .2 1 T = 1 0 K
2
0 .5 2
( c ) )]
) T
0 0 .0 (
(T 8 N b S e 0 .5 0 c2
2 2
c T c = 7 .2 K T = 4 .2 K H
6 [
H /
1
4 0 .2 5
2
( d )
0 0 .0 0
B a K 1 2 2 T c = 2 8 K T = 2 0 K
3 5
z e ro
m id
0 .0 0 2
3 0 o n s e t
2 5
0 .0 0 1
2 0 ( e )
0 .0 0 0
K F e A s T c = 3 .8 K T = 0 .5 K 0 .6
4 2 2
0 .4
2
0 .2
( f )
0 0 .0
0 3 0 6 0 9 0 0 .0 0 .5 1 .0
(cid:1) ( d e g r e e s ) s i n 2 (cid:1)
FIG. 9. (Color Online) Analysis of the isothermal angular dependence of H on inclination angle to the highly conducting
c2
planeθ,usinglinearizationplotH−2(sin2θ). LeftpanelsshowdigitizedH (θ),rightpanelsplotthesamedataasH−2(sin2θ):
c2 c2 c2
(a) Graphite intercalation compounds66 C RbHg (T =0.99 K, measurements taken at T =0.44 K, open circles) and C KHg
4 c h 4
(T =0.73 K, T =0.40 K, solid squares); (b) Sr RuO (T =1.43 K, T =0.10 K, Ref. 15); (c) Mg(B Al ) , Ref. 26, x=0.12
c h 2 4 c h 1−x x 2
(T =30.8 K, T =14 K, black solid squares, and T =23 K, red solid circles) and x=0.21 (T =25.5 K, T =10 K, blue open
c h h c h
circles);(d)NbSe ,Ref.67,(T =7.2K,T =4.2K);(e)(Ba K )Fe As ,Ref.60,(T =28K,T =20K,usingdifferentcriteria
2 c h 1−x x 2 2 c h
for resistive transition, zero resistance- black triangles, midpoint- red circles, onset - green squares); (f) KFe As , Ref. 65,
2 2
(T =3.8 K, T =0.5 K).
c h
