Competition between disorder and exchange splitting in superconducting ZrZn_2

7 Pages·0.16 MB·English

by  B. J. Powell

3 LETTER TO THE EDITOR
0
0
2 Competition between disorder and exchange

n splitting in superconducting ZrZn
a 2
J

1
B J Powell†, James F Annett and B L Gy¨orffy
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HHWillsPhysicsLaboratory,UniversityofBristol,TyndallAvenue, Bristol

] BS81TL,UK
n
E-mail: [email protected]
o
c
- Abstract. Weproposeasimplepicturefortheoccurrenceofsuperconductivity
r
p and the pressure dependence of the superconducting critical temperature, TSC,
u in ZrZn2. According to our hypothesis the pairing potential is independent of
s pressure, but the exchange splitting, Exc, leads to a pressure dependence in the
. (spindependent) densityof states (DOS)at the Fermilevel, Dσ(εF). Assuming
t
a p-wave pairing TSC is dependent on Dσ(εF) which ensures that, inthe absence
m of non-magnetic impurities,TSC decreases as pressureisapplied until itreaches
a minimum in the paramagnetic state. Disorder reduces this minimum to zero,
- thisgivestheillusionthatthesuperconductivitydisappearsatthesamepressure
d
asferromagnetismdoes.
n
o
c
[

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Submitted to: J. Phys.: Condens. Matter
v
4
PACSnumbers: 74.20.Rp,74.25.Dw,74.25.Ha,74.62.Fj,74.70.Ad,75.30.Et
6
The coexistence of ferromagnetism and superconductivity is a problem of long
3
1 standing and general interest. Thus its recent discovery in UGe2 [1], ZrZn2 [2]
0 and URhGe [3] is attracting considerable attention. In particular its occurrence
3 in ZrZn is intriguing because, at ambient pressure, ZrZn is a weak ferromagnet
2 2
0
(T ≈28.5K)andbytheapplicationofpressureitcanbetunedthroughaquantum
/ FM
t criticalpoint(QCP)(P ≈21kbar)tobecomeaparamagneticmetal[2]. Thisrevives
a C
m an old suggestion of Fay and Appel [4]. These authors calculated TSC mediated by
paramagnonsin a McMillan like formalismand found that there is superconductivity
-
d in the (triplet) A channel [5] on both sides of the QCP. However, while the broad
1
n description of Fay and Appel agrees with the observations the fine details do not. In
o
FayandAppel’stheorythe transitiontemperaturegoestozeroattheQCP,T then
c SC
rises to a (local) maxima as the model is tuned away from criticality (experimentally
:
v
this corresponds to pressure being varied away from P ). T then falls again away
C SC
i
X from the QCP. They also predicted that TSC would be approximately the same
r magnitude in both the ferromagnetic and paramagnetic sides of the phase diagram
a (although slightly higher on the paramagnetic side).
When superconductivity was observed in ZrZn it was only seen on the
2
ferromagneticsideofthephasediagramandfurtherthemaximuminT wasobserved
SC
at ambient pressure [2]. The experiments show a monotonic decrease in T with
SC
† Presentaddress: DepartmentofPhysics,UniversityofQueensland,Brisbane,Qld4072,Australia.
Letter to the Editor 2
pressure until about 18 kbar. No data has been published in the pressure range 18
kbar < P < 22 kbar, therefore one cannot ascertain where in this range T falls to
SC
zero. Several groups have attempted to explain this either by revisiting the theory of
Fay and Appel and examining specific coupling mechanisms [6] or by considering the
problemwithinaGinzburg–Landauformalism[7]. Bothofthesegroupspredictedthat
T goes to zero at P . Here we will present a simple alternative to these scenarios
SC C
for the coexistence of superconductivity and ferromagnetism in ZrZn . We will show
2
that no variation in the coupling constant with pressure (i.e. proximity to the QCP)
is required to explain the experiments due to the natural enhancement of the A
1
phase transition temperature in the ferromagnetic phase and the extreme sensitivity
of the triplet pairing states to scattering from non-magnetic impurities. On the other
hand our arguments rely on ZrZn being a rare example of a Stoner ferromagnet.
2
Interestingly in our consideration the QCP plays no special role and the pressure at
which superconductivity disappears is predicted to be strongly sample dependent.
We wishtoconsiderthe problemviathesimplestmodelwhichhasthepossibility
of illustrating the relevant physical phenomena: triplet superconductivity and
ferromagnetism. To this end we study a one band Hubbard model with an effective,
attractive, pairwise, nearest neighbour interaction, Uijσσ′. We allow ferromagnetism
toenterviatheStonermodelwhichappearstobeingoodagreementwiththeobserved
behaviour of the ferromagnetic phase of ZrZn [2, 8, 9, 10]. Thus we study the
2
consequences of the following Hamiltonian:
1
Hˆ =− tijcˆ†iσcˆjσ + 2 Uijσσ′cˆ†iσcˆiσcˆ†jσ′cˆjσ′ − σExccˆ†iσcˆiσ(1)
ijσ ijσσ′ iσ
X X X
where cˆ(†) are the usual annihilation (creation) operators for electrons with spin
iσ
σ = ±1 occupying a tight binding orbital centred on the lattice site labelled by i.
To render the model tractable we assume that the sites i form a simple cubic lattice.
For a general triplet pairing state in a field gap equations cannot be derived in
the same way as they can for the zero field case [11, 12]. However, if we specialise
to the case of equal spin pairing (ESP) and neglect the action of the dipolar field
on the orbital motion of the electrons a remarkable simplification occurs as shown in
reference [12] the gap equations are
U (k−k′)∆ (k′)
∆ (k)=− σσ σσ (1−2f ) (2)
σσ 2E (k′) Ek′σ
k′ σ
X
where the quasiparticle spectrum is given by
Ekσ = (εk−µ−σExc)2+|∆σσ(k)|2. (3)
q
Equations(2)and(3)haveseveralsurprisingfeatures. Firstly,thereisacomplete
decoupling of the two spin states; even in the presence of Cooper pairing. Secondly,
theexchangesplittingentersonlyintheroleofachemicalpotential,butwithopposite
signs for the two spin states. It must be stressed that these results are only valid for
ESPstates. However,thelargeexchangesplittinginaferromagnetprobablyprecludes
opposite spin pairing (OSP) states (that is singlet states, S = 0 triplet states or
z
even the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state, which is only stable for
moderateexchangesplittings). Inshort(2)and(3)arejusttheHartree–Fock–Gorkov
Letter to the Editor 3
Figure 1. The DOS from the LDA calculations for ZrZn2 (——) by Santi and
coworkers [10,13]andourtight bindingfittotheLDADOS(–––). Inset: The
experimental data [16] is consistent with a fit to TS0C(P = 0)∼ 1.2 K from the
Abrikosov–Gorkovformula(5).
approximations restricted to the ESP sector of the theory. For Uijσσ′ = Uijδσσ′ no
such restriction is required as OSP states are not a possibility.
As T →TSC, |∆σσ′(k)|→0 and we find the linearised gap equations:
U (k−k′) ε(k′)−µ−σE
∆ (k)= σσ tanh xc ∆ (k′). (4)
σσ k′ 2 ε(k′)−µ−σExc (cid:18) 2kBT (cid:19) σσ
X
Before solving t(cid:0)hese equations num(cid:1) erically we must first choose the hoping integrals,
tij, and the coupling constants Uijσσ′. We fit the hoping integrals with an on site
(t = µ) and nearest neighbour terms only so as to give the same relative density of
ii
statesintheregionoftheFermilevelasisfoundinabinitiobandstructurecalculations
[10,13]. TheDOSfromourfitiscomparedwiththatfoundintheabinitio calculation
in figure 1. Evidently, our one band model cannot reproduce the complex behaviour
oftheDOSoverthefullenergyrangeoftheZrrelatedd-band,nevertheless,withinan
energy range of 20 meV about the Fermi energy it does do so. Since Uijσσ′ depends
on only one parameter: U = U for sites i and j being nearest neighbours (U = 0
ij ij
otherwise) it can be determined by reference to the measured T for clean ZrZn .
SC 2
This is hampered by the extreme sensitivity of triplet pairing to scattering from non-
magnetic impurities [14, 15] and by the lack of data. Using experimental estimates of
theresidualresistivity,ρ,wefindthatwhatlittledatathereis[16]isconsistentwitha
cleansuperconductingcriticaltemperature,atambientpressure,T0 (P =0)∼1.2K
SC
as shown in the inset to figure 1.
We solved (4) numerically with U = 0.88t on a k-space integration mesh of 109
points. SuchafineintegrationmeshisrequiredtoaccuratelyreproducetheDOS.Our
Letter to the Editor 4
1
Ferromagnetic
Metal
K)
T(SC
0.5
A phase
1
A phase
2
0
0 0.01 0.02 0.03
E (eV)
xc
Figure 2. The phase diagram of our model. The critical temperature is shown
forbothA1 andA2 phasesoverarangeofexchangesplittings. Thehatchedarea
indicatestheAphase,whichisthegroundstatewhenExc=0.
method (implicitly) requires an accurate calculation of the D (ε ) as we are varying
σ F
the exchange splitting and thus we are changing D (ε ), so any errors in evaluating
σ F
D (ε ) will lead to significant errors in our calculation of the variation of T with
σ F SC
E .
xc
The results of our numerical calculations are shown in figure 2. The A
1
phase displays superconductivity in only the ↑↑ channel. In the A phase d(k) ∼
1
(k +ik ,i(k +ik ),0), where d(k) is the usual BW vector order parameter for
x y x y
triplet superconductivity [5]. The A phase correspondsto superconductivity in both
2
ESP channels but with different amplitudes in the two channels so that d(k) ∼
(k +ik ,iκ(k +ik ),0) where 0 < κ < 1. The A phase, which is only stable in
x y x y
zero exchange spitting, has the same pairing amplitude for both of the ESP states,
correspondingtotheorderparameterd(k)∼(k +ik ,0,0). Becauseofthecomplete
x y
separation of the up and down sheets the linearised gap equations (4) can be used
to calculate the lower transition temperature, TA2. This transition represents the
SC
formation of a superconducting state in the minority spin state. Our phase diagram
of course assumes that no other phase transitions occur. The existence of the A and
1
A phasesonlyis consistentwithageneralsymmetryanalysis[17]andthe Ginzburg–
2
Landau expansion of the free energy [12, 18].
Tomakecontactwithexperimentwemustallowforthestrongdependenceofthe
superconducting transitiontemperature of a p-wavesuperconductoron non-magnetic
impurity scattering [14, 15]. This is done via the Abrikosov–Gorkovformula:
T 1 ~ 1
SC0
ln =ψ + −ψ (5)
T 2 4πτ k T 2
(cid:18) SC (cid:19) (cid:18) tr B SC(cid:19) (cid:18) (cid:19)
where τ is the quasiparticle lifetime as measuredin transportexperiments and ψ(x)
tr
isthedigammafunction. NotethatwedonotneedtoworryabouttheBaltensperger–
Sarmaequation[19,20],whichaccountsforthereductioninthecriticaltemperatureof
asuperconductorduetoexchangesplitting,asthisisonlyvalidforOSPstates. Thus,
the Abrikosov–Gorkov formula can be used to calculate T as a function of τ , or
SC tr
equivalentlyρviathe Drudeformula. Tomakethe mostofthe availableexperimental
Letter to the Editor 5
1
K)
T(SC
0.5
0
0 0.01 0.02 0.03
E (eV)
xc
Figure 3. The critical temperature of a ESP p-wave superconductor and the
temperature of the A1-A2 transition as a function of exchange splitting in the
presence of disorder. The curves correspond (from the top down) to ρtr = 0,
0.1 µΩcm, 0.2 µΩcm, 0.3 µΩcm, 0.4 µΩcm, 0.5 µΩcm, 0.6 µΩcm, 0.7 µΩcm,
0.8µΩcmand0.9µΩcm.
data we used the Abrikosov–Gorkov formula (5) to investigate the effect of disorder
on the above phase diagram (see figure 3).
The final step needed to make a direct comparison with measurements of T
SC
as a function of pressure is to note that, experimentally, the Curie temperature is a
linear function of pressure [8, 9]. Namely, T (P) = T (0)(1−P/P ). The zero
FM FM C
temperature magnetisationis alsolinearinpressure[8]andthus proportionaltoT
FM
(as is predicted by the Stoner model), giving M(P,T = 0) = M(0,0)(1−P/P ).
C
For a Stoner ferromagnet, the magnetisation is linearly dependent on the exchange
splitting and hence
E (0,0) 1− P P ≤P
Exc(P,T =0)= xc PC C (6)
( 0 (cid:16) (cid:17) P >PC.
We now invoke the fact that T ≫T which implies that E (P,T =T )∼
FM SC xc SC
E (P,T = 0). Thus we can map the results of T (E ) (shown in figure 3) onto
xc SC xc
T (P)whichweshowinfigure4. Itcanbeseenthatalthoughquantitativeagreement
SC
withexperimentisnotachieved,thegeneralfeaturesofexperimentarereproducedby
several of the curves.
Thusweconcludethatwehavedemonstratedtheviabilityofthefollowingsimple
picture. Irrespective of the mechanism of pairing and exchange splitting ZrZn is
2
a p-wave superconductor with a low T ∼ 1.2 K (at ambient pressure). This
SC
superconductivity is not observed in the paramagnetic phase of currently available
samples due to disorder. However, the exchange field enhances T of an A p-wave
SC 1
state and this is the cause of the observed superconductivity in the ferromagnetic
phase. Experiment suggests [2, 8, 9, 10] ZrZn is a rareStoner ferromagnetfor which
2
the exchange splitting is proportional to the magnetic order parameter. Thus when
P >P andthereforeT =M =E =0thereisnomeasurablesuperconductivity.
C FM xc
However, improvement in sample quality will lead to a lowering of the residual
Letter to the Editor 6
1.0
P)
0)/T(SC0.5
=
P
T(SC
0.0
0.0 5.0 10.0 15.0 20.0 25.0
P(kbar)
Figure 4. The critical temperature of a ESP p-wave superconductor and the
temperature of the A1-A2 transitionas afunction of pressureinthe presence of
disorder. Thetheoretical curves arescaledsothat TSC =1atambientpressure.
Theexperimentaldata,takenfrom[2],wasscaledinthesamewayafterstraight
line had been fitted to the data. The curves correspond (moving from top right
tobottomleft)toρtr =0,0.1µΩcm,0.2µΩcm,0.3µΩcm,0.4µΩcm,0.5µΩcm,
0.6µΩcm,0.7µΩcm,0.8µΩcmand0.9µΩcm.
resistivity and thus presents the possibility of the observation of superconductivity
in the paramagnetic state (as is demonstrated by the curves with ρ < 0.3 µΩcm in
tr
figure 4). This explanation is consistent with and lends microscopic support to the
more phenomenological arguments of Walker and Samokhin [7] and Mineev [17].
It is a pleasure to thank Stephen Hayden, Stephen Yates and Gilles Santi
for sharing their results with us and for helpful discussions. We would also like
to thank the Laboratory for Advanced Computation in the Mathematical Sciences
(http://lacms.maths.bris.ac.uk) for extensive use of their beowulf facilities. One of
the authors (BJP) was supported by an EPSRC studentship and, in the final stages
of this work, by the Australian Research Council.
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