PHYSICAL REVIEW B 85, 094105 (2012)

Uniaxial versus hydrostatic pressure-induced phase transitions in CaFe2As2 and BaFe2As2

Milan Tomic, Roser Valentı, and Harald O. JeschkeInstitut fur Theoretische Physik, Goethe-Universitat Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany

(Received 28 June 2011; revised manuscript received 25 January 2012; published 16 March 2012)

We present uniaxial pressure structural relaxations for CaFe2As2 and BaFe2As2 within density functionaltheory and compare them with calculations under hydrostatic pressure as well as available experimental results.We find that CaFe2As2 shows a unique phase transition from a magnetic orthorhombic phase to a nonmagneticcollapsed tetragonal phase for both pressure conditions and no indication of a tetragonal phase is observed atintermediate uniaxial pressures. In contrast, BaFe2As2 shows for both pressure conditions two phase transitionsfrom a magnetic orthorhombic to a collapsed tetragonal phase through an intermediate nonmagnetic tetragonalphase. We find that the critical transition pressures under uniaxial conditions are much lower than those underhydrostatic conditions manifesting the high sensitivity of the systems to uniaxial stress. We discuss the origin ofthis sensitivity and its relation to experimental observations.

DOI: 10.1103/PhysRevB.85.094105 PACS number(s): 74.70.Xa, 61.50.Ks, 71.15.Mb, 64.70.K−

I. INTRODUCTION

The discovery of superconductivity in iron pnictides1

has initiated an enormous amount of activities related tothese materials. Superconductivity can be triggered eitherby chemical doping or by application of pressure on theundoped parent compounds. One of the families that has beenintensively studied under pressure is the 122 family AEFe2As2

(AE = Ca, Sr, and Ba). CaFe2As2 at ambient pressureundergoes a first-order phase transition from a tetragonal toan orthorhombic phase at 172 K accompanied by a magnetictransition. Initial reports on pressure experiments showed thatat P ∼ 0.23 GPa the orthorhombic and antiferromagneticphases are suppressed and the system superconducts at lowtemperatures.2,3 Moreover, a compressed tetragonal phase—also called “collapsed” tetragonal phase—was identified athigher pressures. Subsequent susceptibility and transportmeasurements under hydrostatic conditions showed at lowtemperatures and P ∼ 0.35 GPa a sharp orthorhombic tocollapsed tetragonal phase transition but no signature ofsuperconductivity.4 In contrast, recent neutron diffractionexperiments on CaFe2As2 under uniaxial pressure along thec axis5 indicate for pressures above 0.06 GPa and at lowtemperatures the presence of an intermediate nonmagnetictetragonal phase between the magnetic orthorhombic andthe nonmagnetic collapsed tetragonal phases. This phasewas identified by the authors as the phase responsible forsuperconductivity at T = 10 K. Other reports based on muonspin-relaxation measurements suggest the existence of super-conductivity in the orthorhombic phase, raising the questionwhether superconductivity and magnetism can coexist.6

BaFe2As2 shows an even more complex behavior underpressure. At ambient pressure it undergoes a phase transitionfrom a metallic tetragonal phase to an orthorhombic antifer-romagnetic phase at T = 140 K. Under pressure the gradualappearance of a superconducting dome has been observed byvarious groups7 though the role of nonhydrostatic conditions isnot yet well understood.8 Recent synchrotron x-ray diffractionexperiments under pressure9 observe at T = 300 K a tetrago-nal to collapsed tetragonal phase transition at P = 22 GPaunder hydrostatic conditions, while this transition appearsalready at P = 17 GPa under nonhydrostatic conditions.On the other hand, the authors of Ref. 10 find at a lower

temperature of T = 33 K that BaFe2As2 undergoes a phasetransition from a magnetic orthorhombic to a nonmagneticcollapsed tetragonal phase at P = 29 GPa and report ananomaly in the As-Fe-As bond angles at 10 GPa that theyascribe to be of electronic origin. In contrast, high-pressureneutron diffraction experiments11 performed at T = 17 K finda tetragonal phase at 3 and 6 GPa.

These experimental results show that the onset of supercon-ductivity as well as the appearance of several structural phasesat low temperatures in CaFe2As2 and BaFe2As2 are extremelysensitive to the pressure conditions5,12–14 and are a subject ofintensive debate. In view of this strong controversy we presentab initio density functional theory results for the electronic,magnetic, and structural behavior of both systems under uni-axial and hydrostatic pressure conditions. Previous theoreticalapproaches which have examined the properties of the 122family under hydrostatic pressure have employed either fixedvolume structural optimizations15 or molecular dynamics16

Recently, anisotropic pressure studies on BaFe2As2 based onground state geometry calculations of more than 300 structuresat different fixed volumes were reported in Ref. 17. Ourapproach consists of constant pressure structural relaxationswhich allows us to probe the low-temperature portion of thephase diagram in a relatively simple and straightforward way.With this approach we can treat nonhydrostatic conditionswhich are at the heart of this work.

We find that uniaxial pressure along the c axis significantlyreduces the transition pressures in both systems. CaFe2As2

shows for both pressure conditions an orthorhombic tocollapsed tetragonal transition, though the transition is lessabrupt when uniaxial pressure is applied. For BaFe2As2 weobserve two phase transitions from orthorhombic to collapsedtetragonal through an intermediate nonmagnetic tetragonalphase. An analysis of the electronic band structure featuresnear the critical pressures reveals the origin of the sensitivityof the systems to pressure conditions.

II. METHODS

Calculations were performed using the Vienna ab initiosimulations package (VASP)18 with the projector augmentedwave (PAW) basis19 in the generalized gradient approximation

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MILAN TOMIC, ROSER VALENTI, AND HARALD O. JESCHKE PHYSICAL REVIEW B 85, 094105 (2012)

(GGA). Structural relaxations under hydrostatic pressure werecarried out with the conjugate gradient (CG) method asimplemented in the VASP package. The energy cutoff was setto 300 eV and a Monkhorst-Pack uniform grid of (6 × 6 × 6)points was used for the integration of the irreducible Brillouinzone. For relaxations with the CG algorithm two cycles wereperformed in order to minimize the error caused by the Pulaystress. Note that the reported bond compressions of up to 7%at 50 GPa do not affect the precision of the PAW basis. In orderto perform relaxations under uniaxial pressure we modified thefast inertial relaxation engine20 (FIRE) algorithm to be able tohandle full structural relaxations with an arbitrary stress tensor.

III. CaFe2As2 RESULTS

In Fig. 1 we show the evolution of lattice parameters,volume, and Fe-Fe, Fe-As distances under hydrostatic and(c-axis) uniaxial pressure for CaFe2As2. We find a first-orderphase transition from a magnetic (stripe order) orthorhom-bic phase to a nonmagnetic collapsed tetragonal phase atPc = 3.05 GPa (Pc = 0.48 GPa) under hydrostatic (uniaxial)pressure and at zero temperature we do not observe anyintermediate tetragonal phase under uniaxial stress.5

In hydrostatic conditions, a and b expand at the orthorhomicto collapsed tetragonal phase transition with b abruptlyincreasing in value, while c shows a significant collapse of6.5% [Fig. 1(a)] and the unit cell volume shows a sharpdrop of about 4.3% [Fig. 1(c)]. The sudden expansion of b

can be explained as a consequence of the Pauli principle16:As long as an antiferromagnetically ordered moment existsin the orthorhombic phase, Fe 3d orbitals may overlap tosome degree along the b direction, but in the paramagneticstate of the collapsed tetragonal phase, the same orbitals onneighboring Fe sites repel each other. The value of c/at = 2.58with at = a/

√2 indicates the onset of a collapsed tetragonal

phase. Our results are in good qualitative agreement withexperimental3 observations, except for the overestimation of

the critical pressure (P expc = 0.3 GPa) also found in previous

theoretical studies.15,16 Following the changes of the latticeparameters at Pc, the in-plane Fe-Fe distances show a sharpincrease at Pc while the out-of-plane Fe-As distance decreases[Fig. 1(e)]. Using the generalized Birch-Murnaghan p − V

equation of state21 we obtained a bulk modulus B = 70 ±3 GPa at ambient pressure, while at Pc the bulk modulus jumpsfrom 56 ± 3 to 105 ± 2 GPa. In order to obtain these estimateswe performed a series of fits for every phase separatelyconsidering every pressure point of our data as a referencepressure. In this way we obtain the bulk modulus as a functionof pressure.

In contrast to the hydrostatic case, when uniaxial pressureis applied [Fig. 1(b)] the a and b lattice parameters expandsignificantly while c is compressed up to Pc = 0.48 GPawhere a drop for c is observed while a and b continue toexpand monotonously. The volume reduces by 3.4% and theratio c/at = 2.56 at Pc [Fig. 1(c)] denotes the entrance toa collapsed tetragonal phase, where magnetism is suppressedcompletely. The phase transition shifts to smaller Pc comparedto hydrostatic pressure, which is in very good agreementwith experiments under nonhydrostatic pressure conditions.5

Nevertheless, the authors of Ref. 5 find for pressures above0.06 GPa a stabilization of the high-temperature tetragonalstructure down to temperatures below the superconductingtransition. This phase is not seen in our calculations whichmay be related to the fact that at very low temperatures thetetragonal phase may be disappearing again (Fig. 1 of Ref. 5).

In order to understand the differences in behavior observedbetween the hydrostatic and uniaxial pressures, we showin Fig. 2 the orbital weighted band structure and kz = 0,ky = 0, and kx + ky = 0 Fermi surface cuts of CaFe2As2

under hydrostatic [Figs. 2(b) and 2(c)] and uniaxial pressure[Figs. 2(d) and 2(e)] at pressures below and above the phasetransition. Band structures and Fermi surfaces were calculatedusing the full-potential local orbital (FPLO) basis.22 The bandstructure and Fermi surface cuts at ambient pressure are also

5.4

5.6

5.8

0 1 2 3 4 5

10

11a,b (Å)

c (Å)

P (GPa)

O cT (a)

hydrostatic

abc

5.4

5.6

5.8

0 0.5 1 1.5

10

11a,b (Å)

c (Å)

P (GPa)

O cT (b)uniaxial || c

abc

300

320

340

0 1 2 3 4 5

1.8

2

V (Å3) c/a

P (GPa)

O cT (c)

hydrostatic

Vc/a

300

320

340

0 0.5 1 1.5

1.8

2

V (Å3) c/a

P (GPa)

O cT (d)uniaxial || c

Vc/a

2.4

2.7

0 1 2 3 4

d (Å)

P (GPa)

O cT

(e)

hydrostatic

dFe−Fe || adFe−Fe || bdFe−As

2.4

2.7

0 0.5 1

d (Å)

P (GPa)

O cT(f)uniaxial || c

dFe−Fe || adFe−Fe || bdFe−As

FIG. 1. (Color online) Structure of CaFe2As2 under hydrostatic and uniaxial pressure. Lattice parameters (a) and (b), volume and axis ratio(c) and (d), and selected bond lengths (e) and (f).

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FIG. 2. (Color online) Band structure and kz = 0, ky = 0, and kx + ky = 0 Fermi surface cuts of CaFe2As2.22 For the orbital character, x

and y point along the nearest neighbor Fe-Fe connections.

shown for comparison [Fig. 2(a)]. We use the orthorhombicspace group Fmmm for all band structure plots in order tofacilitate comparison. The behavior of the electronic structurein the vicinity of the Fermi energy is crucial for understandingthe transition. Right below Pc both pressure conditions showa high density of Fe dxz, dyz, and dxy states at EF [see inFigs. 2(b) and 2(d) the �-Z path and near M] which are pushedaway above Pc [Figs. 2(c) and 2(e)] and the hole pocketsat � disappear, suppressing possible nesting conditions. Thecompression along c enforces the interlayer As pz-As pz

bonding23 which can be related to the proximity of the Aspz band to EF near Pc. Uniaxial stress is for this process moreeffective than hydrostatic pressure since a similar electronicbehavior is reached at much smaller pressures as observed in

Figs. 2(d) and 2(e). In agreement with Ref. 24 the collapsedtetragonal phase sets in as soon as the Fe magnetic momentgoes to zero. Note that the changes in the electronic structureat the phase transition in the uniaxial pressure case [Figs. 2(d)and 2(e)] are more subtle than for hydrostatic pressure, inagreement with the somewhat less abrupt change of the latticeas shown in Figs. 1(b), 1(d), and 1(f).

Also, the shape of the Fermi surface derived from Fig. 2 inthe collapsed tetragonal phase agrees well with the de Haas vanAlphen measurements performed for CaFe2P2 (c/at = 2.59),where a highly dispersive topology in the c axis as wellas the absence of the hole pocket at the � point has beenreported25 [compare Figs. 2(c) and 2(e) with Figs. 2 and 3of Ref. 25]. The isoelectronic substitution of As by P in

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5.4

5.6

5.8

6

0 10 20 30

10

11

12a,b (Å)

c (Å)

P (GPa)

O T (a)hydrostatic

T cT

abc

300

350

400

0 10 20 30

1.6

1.8

2

2.2V (Å3) c/a

P (GPa)

O T (c)hydrostatic

T cT

Vc/a

2.4

2.7

3

0 10 20

d (Å)

P (GPa)

O T (e)hydrostatic

T cT

dFe−Fe || adFe−Fe || bdFe−As

5.4

5.6

5.8

6

0 1 2 3 4

10

11

12a,b (Å)

c (Å)

P (GPa)

O T (b)uniaxial || c

T cT

abc

300

350

400

0 1 2 3 4

1.6

1.8

2

2.2V (Å3) c/a

P (GPa)

O T (d)uniaxial || c

T cT

Vc/a

2.4

2.7

3

0 1 2 3

d (Å)

P (GPa)

O T

(f)uniaxial || c

T cT

dFe−Fe || adFe−Fe || bdFe−As

FIG. 3. (Color online) Structure of BaFe2As2 under hydrostatic and uniaxial pressure. Lattice parameters (a) and (b), volume and axis ratio(c) and (d), and selected bond lengths (e) and (f).

CaFe2As2 corresponds to application of chemical pressure andshows similar features to the collapsed tetragonal phase ofCaFe2As2 obtained after application of (hydrostatic) pressure.The similarity of chemical pressure and applied pressurehas already been discussed in Refs. 25 and 11. In fact,comparison of our obtained c/at = 2.58 ratio and As positionzAs = 0.1358 (hydrostatic) in the collapsed tetragonal phaseof CaFe2As2 with the measured c/at = 2.59 and P positionzP = 0.1357 of CaFe2P2 shows the high resemblance betweenboth crystal structures.

IV. BaFe2As2 RESULTS

We now proceed with the analysis of BaFe2As2. In Fig. 3we present the changes in lattice parameters, volume, andatomic distances under hydrostatic and uniaxial pressuresfor BaFe2As2. Similar to CaFe2As2, the critical pressuresunder uniaxial stress are reduced with respect to hydrostaticconditions. This observation was also reported by recentconstant volume density functional theory calculations onBaFe2As2 under nonhydrostatic pressure.17 BaFe2As2, con-trary to CaFe2As2, shows two phase transitions. At Pc1 =11.75 GPa (Pc1 = 0.72 GPa) we find a phase transitionfrom an antiferromagnetic orthorhombic to a nonmagnetictetragonal phase under hydrostatic (uniaxial) conditions. Asecond smooth phase transition to a collapsed tetragonal phaseis obtained for Pc2 = 28.6 GPa (Pc2 = 3.17 GPa) [Figs. 3(c)and 3(d)].9 High-pressure neutron diffraction experiments11

as well as previous theoretical calculations under hydrostaticpressure conditions also find a phase transition to an intermedi-ate tetragonal phase15,16 but recent synchrotron x-ray diffrac-tion experiments under nonhydrostatic conditions see no sig-nature of an intermediate tetragonal phase at low temperatures.Nevertheless, an anomaly in the As-Fe-As bond angles at P ∼10 GPa (Ref. 10) as well as a loss of magnetic moment14 havebeen reported. This could be related to the phase transition thatwe find at Pc1 = 11.75 GPa where magnetism is suppressed.

At higher pressures the agreement of the onset of the collapsedtetragonal phase with the x-ray diffraction data10 is very good.Clearly the phase transitions in BaFe2As2 are less abrupt thanin CaFe2As2. The ambient pressure bulk modulus is estimatedat 67 ± 4 GPa, in good agreement with experimentallyreported values10 of 82.9 ± 1.4 and 65.7 ± 0.8 GPa at 33

2.32

2.34

2.36

2.38

2.4

0 1 2 3 4 5 6 7 8 9

d Fe−

As

(Å)

P (GPa)

(a)Ref. [26]Ref. [26]theory

2.2

2.25

2.3

2.35

2.4

2.45

0 10 20 30 40 50

d Fe−

As

(Å)

P (GPa)

(b)Ref. [10], T=33 K, P upRef. [10], T=300 K, P upRef. [10], T=300 K, P downtheory

FIG. 4. (Color online) Comparison of T = 0 calculated (squares)and measured Fe-As bond distances in BaFe2As2 as a function ofpressure. (a) dFe-As from XAFS experiment at T = 298 K (triangles,circles).26 (b) dFe-As from x-ray diffraction at T = 33 K (circles) andT = 300 K (triangles).10 P up and down indicate application andrelease of pressure.

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FIG. 5. (Color online) Band structure and kz = 0, ky = 0, and kx + ky = 0 Fermi surface cuts of BaFe2As2.22

and 300 K, respectively. At Pc1 , the bulk modulus abruptlyincreases from 98 ± 4 to 128 ± 3 GPa and at Pc2 it jumpsfrom 150 ± 3 to 173 ± 2 GPa. This is in very good agreementwith the experimental estimate of B = 153 ± 3 GPa forthe collapsed tetragonal phase.10 We also analyzed the Fe-As bond compressibility for P = 9 GPa (hydrostatic) andfound κ = 3.5 × 10−3 GPa−1 which is in excellent agreementwith κ = 3.3 × 10−3 GPa−1 obtained in an extended x-rayabsorption fine structure (EXAFS) experiment.26 In Fig. 4(a)we show the comparison of the measured pressure dependenceof the Fe-As bond distances26 with our results. Due to differenttemperatures (experiment is performed at room temperature,theory at T = 0) our distances are shorter by about 0.02 A(0.8%), but the overall agreement is good. In Fig. 4(b) weshow the comparison to the x-ray diffraction measurement ofthe Fe-As bond distances10 over a large pressure range. The

comparison is particularly good at low and at high pressures; inthe tetragonal phase (11.75 to 28.6 GPa) the deviations are a bitlarger.

In Fig. 5 we present the orbital weighted band structureand kz = 0, ky = 0, and kx + ky = 0 Fermi surface cuts ofBaFe2As2 under hydrostatic [Figs. 5(b) and 5(c)] and uniaxialpressure conditions [Figs. 5(d) and 5(e)] at two pressures belowand above the orthorhombic to tetragonal phase transition atPc1 = 11.75 GPa (Pc1 = 0.72 GPa). Similar to CaFe2As2 weobserve below Pc1 a high density of Fe dxz, dyz, and dxy states atEF which is pushed down (less drastically than in CaFe2As2)for pressures above Pc1 . The hole pockets disappear at the �

point and the Fe magnetic moment goes to zero. Here the Aspz band seems to be little affected at the critical pressure. Incontrast, at Pc2 = 28.6 GPa (Pc2 = 3.17 GPa) (band structurenot shown) the As pz band is pushed toward the Fermi level

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indicating a strong As pz-As pz bonding while entering thecollapsed tetragonal phase. These results show that underperfect hydrostatic or perfect uniaxial pressure conditionsneither the intermediate tetragonal phase nor the collapsedtetragonal phase fulfill Fermi surface nesting conditions. Infact, we find that the structural parameters measured in Ref. 11are similar to our calculated parameters far below Pc1 in theorthorhombic phase (except for the orthorhombic distortion),where well defined hole pockets are found at the � point.

V. CONCLUSIONS

In summary, we have presented finite pressure densityfunctional theory calculations which allow the investigation ofnonhydrostatic pressure conditions. Our finite pressure relaxedstructures show good agreement with the available experimen-tal data (volume, bond lengths, and compressibilities) thoughour magnetic moments in the orthorhombic phase are largerthan the observed experimental values. We expect that thisoverestimation affects mostly the values of Pc. Comparisonof our calculated Fe-As bond distances at different pressureswith measured distances26 shows good agreement. Also,available de Haas van Alphen measurements performed forCaFe2P2

25 agree well with our predicted Fermi surface shapesfor CaFe2As2 in the collapsed tetragonal phase. This overallagreement with experimental observations demonstrates thatthe presented constant pressure calculations provide a reli-able theoretical prediction of structures under nonhydrostaticpressure conditions, allowing for arbitrary stress tensors infuture studies. Such calculations can complement experiments

and help identify the precise degree of hydrostaticity. We findthat uniaxial stress along the c axis considerably reduces thecritical pressures for CaFe2As2 and BaFe2As2. This behaviorcan be understood by the fact that the phase transitions arestrongly dictated by the electronic properties in the vicinityof the Fermi energy, as shown by our electronic structureanalysis. While CaFe2As2 undergoes a magnetic orthorhombicto a nonmagnetic collapsed tetragonal phase for both pressureconditions and no indication of an intermediate tetragonalphase is observed under uniaxial stress, BaFe2As2 shows twophase transitions from a magnetic orthorhombic to a collapsedtetragonal phase through an intermediate nonmagnetic tetrag-onal phase for both pressure conditions. All nonmagneticphases show a disappearance of the hole pockets at the �

point suppressing possible Cooper pair scattering channelsbetween electron and hole pockets. Such scattering channelshave been discussed to be important for the superconductivityin BaFe2As2.27 More experiments need to be done in order tounderstand the origin of the superconducting phase observedin these materials under various pressure conditions.

ACKNOWLEDGMENTS

We would like to thank A. I. Coldea for useful discussions,the Deutsche Forschungsgemeinschaft for financial supportthrough Grant SPP 1458, the Helmholtz Association forsupport through HA216/EMMI and the center for scientificcomputing (CSC, LOEWE-CSC) in Frankfurt for computingfacilities.

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