# Phase Diagram of Ba1-xKxFe2As2

S. Avci  Materials Science Division, Argonne National Laboratory, Argonne, IL
60439-4845, USA  O. Chmaissem  Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439-4845, USA  Physics Department, Northern Illinois
University, DeKalb, IL 60115, USA  D. Y. Chung  S. Rosenkranz  Materials
Science Division, Argonne National Laboratory, Argonne, IL 60439-4845, USA  E.
A. Goremychkin  Materials Science Division, Argonne National Laboratory,
Argonne, IL 60439-4845, USA  ISIS Neutron and Muon Source, Rutherford Appleton
Laboratory, Didcot, OX11 0QX, United Kingdom  J.-P. Castellan  I. S. Todorov
J. A. Schlueter  H. Claus  Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439-4845, USA  A. Daoud-Aladine  D. D. Khalyavin
ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot, OX11
0QX, United Kingdom  M. G. Kanatzidis  Materials Science Division, Argonne
National Laboratory, Argonne, IL 60439-4845, USA  Department of Chemistry,
Northwestern University, Evanston, IL 60208-3113, USA  R. Osborn  Materials
Science Division, Argonne National Laboratory, Argonne, IL 60439-4845, USA
<EMAIL_ADDRESS>
###### Abstract

We report the results of a systematic investigation of the phase diagram of
the iron-based superconductor, Ba1-xKxFe2As2, from $x=0$ to $x=1.0$ using high
resolution neutron and x-ray diffraction and magnetization measurements. The
polycrystalline samples were prepared with an estimated compositional
variation of $\Delta x\lesssim 0.01$, allowing a more precise estimate of the
phase boundaries than reported so far. At room temperature, Ba1-xKxFe2As2
crystallizes in a tetragonal structure with the space group symmetry of
$I4/mmm$, but at low doping, the samples undergo a coincident first-order
structural and magnetic phase transition to an orthorhombic (O) structure with
space group $Fmmm$ and a striped antiferromagnet (AF) with space group
$F_{c}mm^{\prime}m^{\prime}$. The transition temperature falls from a maximum
of 139 K in the undoped compound to 0 K at $x=0.252$, with a critical exponent
as a function of doping of 0.25(2) and 0.12(1) for the structural and magnetic
order parameters, respectively. The onset of superconductivity occurs at a
critical concentration of $x=0.130(3)$ and the superconducting transition
temperature grows linearly with $x$ until it crosses the AF/O phase boundary.
Below this concentration, there is microscopic phase coexistence of the AF/O
and superconducting order parameters, although a slight suppression of the
AF/O order is evidence that the phases are competing. At higher doping,
superconductivity has a maximum $T_{c}$ of 38 K at $x=0.4$ falling to 3 K at
$x=1.0$. We discuss reasons for the suppression of the spin-density-wave order
and the electron-hole asymmetry in the phase diagram.

##  I Introduction

There is now an extensive body of research into the origin of
superconductivity in the iron-based superconductors demonstrating the
importance of the subtle interplay of their electronic properties with
crystalline structure [1,2]. These compounds all contain a common structural
motif, namely a square planar net of iron atoms tetrahedrally coordinated with
pnictogens or chalcogens producing Fe${}_{2}X_{2}$ layers, in which $X$ = As
or Se/Te in the highest $T_{c}$ compounds. They are separated by buffer layers
comprising, for example, rare earth oxides in the so-called 1111 systems, such
as those based on LaFeAsO, or alkaline earths in the so-called 122 systems,
such as those based on BaFe2As2. Like the cuprate superconductors, the buffer
layers can act as charge reservoirs, controlling the carrier concentration and
inducing superconductivity in the iron planes by the introduction of
aliovalent dopants, e.g., LaFeAsO1-xFx [3,4] and Ba1-xKxFe2As2 [5,6], but it
is also possible to dope the pnictogen or chalcogen sites, e.g., BaFe2As2-xPx
[7,8], or the iron planes themselves by substituting other transition metal
ions, e.g., BaFe2-xCoxAs2 and BaFe2-xNixAs2 [9,10].

In this article, we report on a systematic investigation of the phase diagram
of Ba1-xKxFe2As2 from $x=0$ to $x=1.0$ using high resolution neutron and x-ray
diffraction combined with bulk characterization. In spite of the diversity of
doping strategies employed to modify the superconducting properties of the
iron-based superconductors, their phase diagrams show remarkable similarities.
There is typically an undoped “parent” compound that is antiferromagnetic
rather than superconducting[1]. These are fully compensated metals, whose
electronic properties are dominated by multiple iron-derived $d$-bands near
the Fermi level with approximately equal concentrations of hole and electron
carriers [11]. The Fermi surfaces consist of quasi-two-dimensional cylinders,
with two or three hole pockets at the Brillouin zone centers and two electron
pockets at the $M$-points on the zone boundaries, i.e., along the direction of
the nearest-neighbor iron-iron bonds [12]. All the Fermi surface pockets have
similar radii in the undoped compounds, making their electronic structure
particularly susceptible to magnetic instabilities resulting from a nesting of
the disconnected hole and electron Fermi surfaces [13]. Since the nesting
wavevector corresponds to the antiferromagnetic wavevector observed by neutron
diffraction [14-16], it is plausible that the magnetism can be explained by a
purely itinerant model of spin density waves, and ab initio density functional
theory does indeed predict the correct magnetic structure [17], although there
is an ongoing debate about the strength of electron correlations [18,19].

The magnetic structure breaks the tetragonal symmetry with an in-plane
wavevector of either (0,$\pi$) or ($\pi$,0) in the unfolded Brillouin zone
with one iron atom per unit cell. With a finite magnetoelastic coupling, this
would induce an orthorhombic structural transition, which is usually observed
to occur at the same temperature as magnetic order in the parent compounds
[20,21]. However, the addition of both hole and electron charge carriers
through chemical substitution suppresses both transitions. A variety of
scenarios are possible in Ginzburg-Landau treatments of the magnetoelastic
coupling [22,23]. The two phase transitions could be first- or second-order
and occur simultaneously or separately. In most of the iron-based compounds,
the two transition temperatures split with doping [24] and there is a report
of a split transition in Ba1-xKxFe2As2 as well [25]. However, this is
inconsistent with our previously reported neutron diffraction data, which
shows unambiguously that they are coincident and first-order for all $x$
before they are both suppressed at $x\lesssim 0.3$ [6]. Unusually, the two
order parameters, magnetic and structural, are proportional to each other,
apparently indicating a biquadratic coupling that is usually only observed at
a tetracritical point [26,27], not over an extended range of compositions.
Possible explanations for this observation will be discussed in the
conclusions.

Superconductivity emerges before the complete suppression of the
antiferromagnetic/orthorhombic (AF/O) phase and coexists at low-doping levels.
The nature of the competition between AF/O order and superconductivity is a
central question in understanding iron-based superconductivity [28,29]. There
were earlier reports based on local probes that, in Ba1-xKxFe2As2, the
coexistence region is characterized by a mesoscopic phase separation into AF/O
and superconducting droplets [30,31], but our previously reported diffraction
data are only consistent with a microscopic phase coexistence [6], a
conclusion since supported by muon spin rotation ($\mu$SR) experiments [32],
suggesting that the earlier reports may be due to compositional fluctuations
within the samples.

Figure 1: (Color online) Structure of BaFe2As2, which crystallizes in a
tetragonal ThCr2Si2-type structure with the space group symmetry of $I4/mmm$.
Potassium substitutes onto the barium sites.

One of the main reasons for studying Ba1-xKxFe2As2 is that superconductivity
extends up to much higher hole-doping levels, with 0.5 holes/Fe atom, than in
the electron-doped superconductors produced by transition metal substitutions.
In the case of BaFe2-xCoxAs2, superconductivity vanishes at only 0.12
electrons/Fe atom [10]. Furthermore, the maximum $T_{c}$ with hole-doping is
38 K, significantly higher than the maximum $T_{c}$ of $\sim 25$ K obtained
with electron doping. This electron-hole asymmetry in the phase diagram has
been attributed to an enhanced Fermi surface nesting in the hole-doped
compounds, consistent with angle-resolved photoemission spectroscopy (ARPES)
data and band structure calculations [33]. This explanation is also supported
by the evolution of resonant spin excitations, which become incommensurate due
to the mismatch in hole and electron Fermi surface volumes when $T_{c}$ starts
to fall [34]. On the other hand, there is also a strong correlation between
$T_{c}$ and internal structural parameters such as the Fe-As-Fe bond angles
[35,36]. These are known to have an influence on the band structure and the
degree of moment localization, but their role in optimizing superconductivity
and the implications for the gap symmetry is a matter of debate [37,38].

There have been two previous reports of the doping dependence of this series
in addition to our own brief report, which are all in qualitative agreement
[6,36,39]. At room temperature, all members of the Ba1-xKxFe2As2 series
crystallize in a tetragonal structure with the space group symmetry of
$I4/mmm$ (Fig. 1), while low-doped samples also exhibit a low temperature
phase transition to an orthorhombic structure with space group $Fmmm$ [5].
Superconducting samples at higher doping have a maximum $T_{c}$ of 38 K and
remain tetragonal at all measured temperatures down to 1.7 K. However, there
are significant discrepancies in the published reports concerning the critical
dopant concentrations defining the onset of superconductivity and the
suppression of the AF/O phase, with the latter varying from $x\sim 0.3$ [36]
to $\sim 0.4$ [39]. As already mentioned, there have also been disagreements
about the nature of the competition between the three ordered phases at low
doping. We believe that these discrepancies are due to uncertainties in the
actual composition of the synthesized samples, since it is well-known that
potassium is particularly volatile. Controlling the inhomogeneity to within
acceptable limits in order to improve the accuracy of the various phase
boundaries has been a key goal of this work and we estimate that we have been
able to make samples in which $\Delta x<0.01$. We have performed neutron and
x-ray diffraction studies of the magnetic and structural order using high-
resolution powder diffractometers so that the systematic variation of the
lattice parameters and internal structural parameters can be used to estimate
the degree of uncertainty in the average composition and its variation within
the samples.

$x$ | $x$ | $x$ | $x$ |  $T_{c}$ (K) |  $T_{N}$ (K) |  $T_{N}$ (K) |  $T_{s}$ (K) | Magn. Moment ($\mu_{B}$) | $\delta\times 10^{3}$
---|---|---|---|---|---|---|---|---|---
| (fitted) | (nominal) | (ICP) |  | (magnetization) | (neutron) |  |  |
0 |  | 0 | 0 |  | 139(1) | 139.0(1) | 138.17(6) | 0.756(36) | 3.92(4)
0.1 | 0.097 | 0.1 | 0.094(2) |  | 136(1) | 136.5(3) | 136.02(8) | 0.741(21) | 3.68(3)
0.125 | 0.126 | 0.125 | 0.114(2) |  | 130(2) |  | 128.29(6) | 0.697(29) | 3.49(4)
0.15 | 0.15 | 0.15 | 0.139(1) | 4 | 122(2) | 122.1(3) | 122.09(7) | 0.702(21) | 3.35(4)
0.175 | 0.172 | 0.175 | 0.159(2) | 10 | 113(2) | 113.9(1) | 112.1(6) | 0.683(24) | 3.14(5)
0.2 | 0.202 | 0.2 | 0.184(2) | 17 | 100(2) | 102.0(1) | 102.00(2) | 0.652(46) | 2.76(7)
0.21 | 0.209 | 0.24 |  | 18 |  | 96.0(1) | 96.0(3) | 0.610(32) | 2.59(3)
0.22 | 0.225 | 0.22 |  | 23.5 |  | 93.97(1) | 93.93(1) | 0.550(22) | 2.20(9)
0.24 | 0.237 | 0.26 |  | 26 |  | 79.9(1) | 80.0(2) | 0.572(29) | 2.00(3)
0.25 | 0.249 | 0.24 |  | 28.5 |  | 74.9(1) | 74.8(8) | 0.456(22) | 1.43(8)
0.28 |  | 0.28 |  | 34 |  |  |  |  |
0.3 |  | 0.3 | 0.312(4) | 36 |  |  |  |  |
0.4 |  | 0.4 |  | 38 |  |  |  |  |
0.5 |  | 0.5 | 0.476(8) | 34 |  |  |  |  |
0.7 |  | 0.7 | 0.675(3) | 20 |  |  |  |  |
0.9 |  | 0.9 | 0.892(1) | 7 |  |  |  |  |
1 |  | 1 | 1.00(2) | 3 |  |  |  |  |
Table 1: Structural and magnetic phase diagram of Ba1-xKxFe2As2. The nominal
value of x represents the starting composition given by the Ba/Fe ratio. The
fitted value is determined by smoothing the variation in the a-axis lattice
parameter from $0.1\leq x\leq 0.25$ using a power law function. The first
column is the value of $x$ used in the text and figures. The superconducting
transition temperature is determined from magnetization measurements. $T_{N}$
and $T_{s}$ are determined by magnetization and neutron diffraction data as
described in the text. The magnetic moments and orthorhombic order parameters,
$\delta=(a-b)/(a+b)$, are determined from the low-temperature Rietveld
refinements.

In this article, we present the results of Rietveld refinements for the entire
series and use this analysis along with bulk measurements to produce a
comprehensive magnetic and structural phase diagram that provides insight into
the nature of the phase competition that underlies iron-based
superconductivity. Our results show that there is a steeper decrease in
$T_{c}$ and hence a narrower region of phase coexistence of the AF/O order
with superconductivity than previous reports. After a description of our
experimental results, we combine our findings with results reported in the
literature on the electron-doped BaFe2-xCoxAs2 series in order to elucidate
the origin of the electron-hole asymmetry in the phase diagram.

##  II Experimental Details

The synthesis of homogeneous single phase Ba1-xKxFe2As2 samples is known to be
particularly delicate due to unfavorable kinetics, high vapor pressures, and a
significant difference in the chemical reactivity of K and Ba with FeAs that
may result in stabilizing other binary by-products. For this work, the
synthesis and properties of our samples were optimized by the systematic
examination of all reasonable combinations of reaction parameters, e.g.,
purity of the starting materials, reaction containers, temperature, and
duration of heating, etc. Our final samples were produced according to the
following procedure: Handling of all materials was performed in a nitrogen-
filled glove box. Raw materials (BaAs/KAs/Fe2As) were prepared by heating
elemental mixtures at 400∘C, 600∘C, and 850∘C, respectively. The
stoichiometric mixture of these starting precursors for a desired composition
was thoroughly ground to ensure uniform and homogeneous and subsequently
annealed at 1050∘C in a Nb tube sealed in a quartz tube. The closed metal
tubes are needed to eliminate any chemical loss that may otherwise result from
the evaporation of K and As. A large number of high quality samples were
synthesized covering the full phase diagram $0\leq x\leq 1$ with special
emphasis given to the $0.1\leq x\leq 0.25$ range in which the potassium
content was incremented in very small amounts with $\Delta x=0.025$. The
deliberate synthesis of samples with finely tuned K content was necessary in
order to carefully investigate the rapid suppression of magnetism with
increasing K and to elucidate the nature of phase coexistence with
superconductivity within the same sample. Samples with coarse K increments
would otherwise lead to inconclusive results.

Initial characterization of the samples was performed by x-ray diffraction,
magnetization measurements, and Inductively Coupled Plasma (ICP) elemental
analysis. For select samples, neutron powder diffraction experiments were
performed on the HRPD ($x=0$, 0.1, 0.125, 0.15, 0.175, 0.2, 0.21, 0.24, 0.3,
0.5 and 1) and Wish diffractometers ($x=0.22$ and 0.25) at the ISIS Pulsed
Neutron and Muon Source (Rutherford Appleton Laboratory) and on beamline
11-BM, ($x=0.28$) at the Advanced Photon Source (Argonne National Laboratory).
The resolution, $\Delta d/d$, at 2 Åis 0.001 for HRPD and 0.002 for Wish. For
diffraction experiments, the samples were sealed under vacuum for shipment and
re-opened just before the measurements in a helium environment to prevent air
exposure. The nuclear and magnetic structures together with interatomic bond-
lengths and bond-angles were determined by the Rietveld refinement technique
using the comprehensive General Structure Analysis System (GSAS) software
suite [40] and the associated graphical user interface (EXPGUI) [41]. Minute
traces of no more than 0.5-3% by weight of FeAs and Fe2As impurity phases were
observed in some of the samples, too small to affect the analysis.

Our neutron diffraction results show that the synthesis methods has reduced
the compositional uncertainty significantly, which we estimate to be $\Delta
x\lesssim 0.01$. In previously reported phase diagrams, it is not totally
clear whether the potassium contents were nominal or actual measured values.
Because of its volatility, it is always necessary to add excess potassium, so
the eventual stoichiometry is largely governed by the Ba/Fe ratio. Thus, any
perceived discrepancy with other work, such as Ref. [39] is probably due to
differences in the handling and control of the volatile potassium and arsenic
constituents. In this article, we have used a number of methods to
characterize the sample compositions, including direct measurements of the
stoichiometry from ICP analysis. To produce the final compositions used in our
phase diagrams, we started with the nominal $x$ values determined from the
starting Ba/Fe ratio and then smoothed the variation in the a-axis lattice
parameter from $0.1\leq x\leq 0.25$ using a power law function (see Table 1).
In most cases, the agreement with the nominal value was better than 0.005,
with just two samples requiring a significant shift of 0.02 and 0.03,
respectively. In all cases, the adjustments also improved the consistency of
other measurements, such as the variations in transition temperatures and
order parameters. The first column of Table 1 shows the value of $x$ that we
have used in the text and figure labels, derived by rounding the fitted values
to the nearest 0.005, although the fitted values were used in the plots and in
numerical analyses of the doping dependence, e.g., the fit of $T_{c}$ vs $x$.
For $x\geq 0.28$, where the precise composition is not so critical, we have
used the nominal values.

Figure 2: (Color online) Variation of lattice constants, $a$, $b$ and $c$,
with temperature for $x=0$, 0.1, 0.21 and 0.3. The lattice constants, $a$ and
$b$, in the orthorhombic phase are divided by $\sqrt{2}$.

We are also able to monitor fluctuations in composition within a single sample
because HRPD has sufficiently high resolution that it is sensitive to
distributions of the lattice parameter and internal strains caused by
compositional gradients and even particle size broadening [42]. The
diffraction peaks from the (220) reflection shown in Fig. 1 of Ref. [6] show
that there is no change in the linewidths and lineshapes of the diffraction
peaks from the undoped compound up to $x=0.24$, consistent with a high-degree
of compositional homogeneity ($\Delta x\lesssim 0.01$). The regular spacing
between the peak positions is in agreement with the fixed steps in $x$.

##  III Results

###  III.1 Structural Phase Diagram

Figure 3: (Color online) Temperature dependence of the (110) peak in the
vicinity of structural transition temperature $T_{s}$. The bold curve shows
the peak at the estimated $T_{s}$. For $0\leq x\leq 0.24$, the other curves
show the peak at intervals of 2 K around $T_{s}$. For $x=0.3$, the peak is
shown in 20 K intervals between 1.7 K and 120 K.

In the undoped BaFe2As2 compound, there is a structural phase transition at
139 K from the tetragonal ThCr2Si2-type structure with the space group
symmetry $I4/mmm$ to an orthorhombic $\beta$-SrRh2As2-type structure with
space group $Fmmm$ [5]. The structure of both the tetragonal and orthorhombic
phases of Ba1-xKxFe2As2 can be described as a stack of edge-sharing Fe2As2
layers separated by layers of (Ba,K) ions (Fig. 1). The (Ba,K) ions occupy
crystallographic positions that are tetrahedrally coordinated with four
arsenic anions.

With potassium doping, the tetragonal to orthorhombic structural transition
temperature, $T_{s}$, decreases until it is fully suppressed for $x>0.25$. In
Fig. 2, we show the lattice parameters as a function of temperature for $x=0$,
0.1, 0.21 and 0.3. Below $x=0.3$, a significant orthorhombic splitting of the
basal plane $a$ and $b$ lattice parameters is observed. The evolution of the
orthorhombic order parameter defined by $\delta=(a-b)/(a+b)$ is discussed
later. Although the transitions appear to be continuous, we observed small but
sharp volume anomalies at all the structural phase transitions (Fig. 3 in Ref.
[6]) showing that they are weakly first-order in character over the entire
phase diagram.

Figure 4: (Color online) Variation of lattice constants and volume in
Ba1-xKxFe2As2 with $x$ at 1.7 K. Solid lines are guide to the eye. The lattice
constants ($a$ and $b$) and the volume in the orthorhombic phase are divided
by $\sqrt{2}$ and 2, respectively, from those for the $Fmmm$ space group.

The structural transition temperatures in Table 1 were determined by fitting a
power law, $\delta\propto(T_{s}-T)^{\beta}/T_{s}$, close to the transition
temperature, yielding exponents mostly in the range $\beta\sim 0.13$ to 0.2.
Apart from $x=0$, where $\beta=0.129(3)$, in reasonable agreement with Ref.
[27], the exponents will be modified by compositional fluctuations and so are
not reliable estimates of the critical behavior. The exponents at $x=0.21$ and
0.24 were anomalously high ($\beta=0.25$ and 0.30, respectively), which could
reflect a slightly greater degree of compositional variation within those
samples. As a check on these values of $T_{s}$, we used the peak profiles
close to the transition temperature. Fig. 3 shows the temperature dependence
of the tetragonal (110) Bragg peak for $x=0$, 0.1, 0.15, 0.2, 0.24 and 0.3 in
the vicinity of the structural transition. This reflection splits into two
(022) and (202) orthorhombic peaks below $T_{s}$. Close to the transition, the
two peaks cannot be resolved but we can determine $T_{s}$ from the temperature
dependence of the full width at half maximum (FWHM) as they merge. In the
high-temperature phase, the FWHM of $\sim 0.0037(3)$Åis independent of the
composition. These two methods of determining $T_{s}$ agreed within the
errors.

Figure 5: (Color online) Variation of Fe-Fe and Fe-As bond lengths with $x$ at 1.7 K and with temperature for different K substitutions. Blue triangles represent the Fe-As bonds. Black square and red circle symbols represent the Fe-Fe bond lengths merging at $T_{s}$. Solid lines are guides to the eye.  $x$ |  $a$(Å) |  $b$(Å) |  $c$(Å) |  Vol(Å3) |  $z$As | Fe-As(Å) | Fe-Fe(Å) | Fe-Fe(Å) |  As-Fe-As(∘) |  As-Fe-As(∘) |  As-Fe-As(∘)
---|---|---|---|---|---|---|---|---|---|---|---
0 | 5.6157(2) | 5.5718(2) | 12.9424(4) | 404.970(45) | 0.35375(3) | 2.3905(1) | 2.8078(1) | 2.7859(1) | 108.071(3) | 108.718(3) | 111.648(2)
0.1 | 5.5997(1) | 5.5587(1) | 13.0031(4) | 404.755(36) | 0.35405(3) | 2.3919(1) | 2.7998(1) | 2.7793(1) | 108.356(7) | 108.961(7) | 111.110(15)
0.125 | 5.5940(2) | 5.5551(2) | 13.0243(5) | 404.745(52) | 0.35405(3) | 2.3918(2) | 2.7970(1) | 2.7775(1) | 108.438(8) | 109.011(8) | 110.977(16)
0.15 | 5.5890(2) | 5.5517(2) | 13.0404(5) | 404.634(46) | 0.35399(3) | 2.3905(1) | 2.7945(1) | 2.7758(1) | 108.464(3) | 109.013(3) | 110.947(3)
0.175 | 5.5842(2) | 5.5492(2) | 13.0563(6) | 404.598(55) | 0.35386(3) | 2.3901(2) | 2.7921(1) | 2.7746(1) | 108.522(9) | 109.037(9) | 110.864(17)
0.2 | 5.5767(4) | 5.5460(4) | 13.0736(9) | 404.353(82) | 0.35372(5) | 2.3885(1) | 2.7883(2) | 2.7730(2) | 108.578(5) | 109.030(5) | 110.815(4)
0.21 | 5.5750(1) | 5.5462(2) | 13.0749(4) | 404.288(36) | 0.35404(2) | 2.39073(4) | 2.7875(1) | 2.7731(1) | 108.678(2) | 109.102(2) | 110.640(2)
0.22 | 5.5706(5) | 5.5461(4) | 13.0803(12) | 404.126(112) | 0.3543(7) | 2.3925(6) | 2.7855(3) | 2.7733(3) | 108.798(21) | 109.155(21) | 110.47(4)
0.24 | 5.5672(2) | 5.5449(2) | 13.0888(4) | 404.051(39) | 0.35394(3) | 2.3894(2) | 2.7836(1) | 2.7724(1) | 108.750(7) | 109.078(7) | 110.592(14)
0.25 | 5.5636(4) | 5.5476(4) | 13.1027(10) | 404.157(90) | 0.35398(6) | 2.3900(5) | 2.7810(2) | 2.7732(2) | 108.846(17) | 109.073(16) | 110.50(3)
0.3 | 3.9165(2) | 3.9165(2) | 13.1614(5) | 201.877(23) | 0.35383(4) | 2.3878(3) | 2.7694(2) | 2.7694(2) | 109.090(28) |  | 110.237(14)
0.5 | 3.8893(2) |  | 13.3242(6) | 201.554(20) | 0.35376(4) | 2.3859(3) | 2.7501(1) |  | 109.615(11) |  | 109.185(21)
1 | 3.8251(2) |  | 13.7846(5) | 201.691(40) | 0.35314(4) | 2.3833(4) | 2.7047(1) |  | 110.855(12) |  | 106.708(23)
Table 2: Results of Rietveld refinements for Ba1-xKxFe2As2 from neutron powder
diffraction data collected on HRPD at 1.7 K. For $x<0.3$, the space group is
$Fmmm$, in which $a\neq b$, there are two inequivalent Fe-Fe bond distances
and three inequivalent As-Fe-As bond angles. For $x\geq 0.3$, the space group
is $I4/mmm$, in which $a=b$, there is one Fe-Fe bond distance and two
inequivalent As-Fe-As bond angles

Fig. 4 shows the evolution of the lattice parameters and unit cell volume at
1.7 K as a function of $x$. Barium substitution by the smaller potassium
cations reduces the in-plane $a$\- and $b$-lattice parameters while
significantly lengthening the out-of-plane $c$-axis. However, the c-axis
enhancement is not large enough to fully compensate for the shrinking basal
plane axes and the unit cell volume gradually decreases in magnitude upon
increasing the K content until $x\sim 0.5$. The reason for the non-monotonic
behavior of the unit cell volume at high dopant levels is not understood and
will require further investigation.

The behavior of the Fe-Fe and Fe-As interatomic distances (at 1.7 K) are
presented in Fig. 5. The Fe-Fe distances mimic the in-plane lattice parameters
both as a function of K content and of temperature. The similarity in behavior
is explained by the fact that Fe atoms occupy special rigid positions along
the long edges of the lattice. Six As-Fe-As bond-angles can be identified in
each FeAs4 tetrahedron. In the tetragonal $I4/mmm$ structure, these angles can
be grouped into two independent angles: two equivalent angles, $\alpha_{1}$,
and four equivalent smaller ones, $\alpha_{2}$. In the orthorhombic $Fmmm$
structure below $T_{N}$, the angle $\alpha_{1}$ remains unaffected but the
angle $\alpha_{2}$ splits into two pairs of equivalent angles
$\alpha^{\prime}_{2}$ and $\alpha^{\prime\prime}_{2}$ with an angular
separation of $\sim 0.5-0.6^{\circ}$. A sketch showing the different angles is
displayed in Fig. 6. As shown in the same panel, the angle $\alpha_{1}$
increases linearly and continuously throughout the whole phase diagram.
However, starting from BaFe2As2, the angles $\alpha^{\prime}_{2}$ and
$\alpha^{\prime\prime}_{2}$ increase with increasing K until reaching a
critical composition below $x\sim 0.3$ beyond which the structural transitions
are suppressed and the two angles merge into $\alpha_{2}$, which continues to
increase with higher K contents. The refined values of the lattice parameters,
bond lengths and bond angles at 1.7 K are shown in Table 2.

Figure 6: (Color online) Variation of As-Fe-As bond angles with $x$ at 1.7 K
and with temperature for different K substitutions. The top panel shows
$\alpha_{1}$, $\alpha^{\prime}_{2}$ and $\alpha^{\prime\prime}_{2}$ in
orthorhombic setting. Blue circles represent $\alpha_{1}$, red triangles and
black squares represent $\alpha^{\prime}_{2}$ and $\alpha^{\prime\prime}_{2}$
merging into one $\alpha_{2}$ at $T_{s}$. Solid lines are guides to the eye.

###  III.2 Magnetic Phase Diagram

Neutron powder diffraction data reveal the presence of weak magnetic Bragg
reflections that appear below the structural phase transition for all of the
orthorhombic samples. The magnetic peaks shown in Fig. 7, located at 2.45 Åand
3.43 Å, were indexed as 121 and 103 in agreement with the widely reported
antiferromagnetic spin density wave (SDW) ground state [15,16,39,43]. As with
the structural transitions, the antiferromagnetic transition (Néel)
temperatures as a function of doping were determined by power-law fits to
temperature variation of the magnetic moment (see Table 1). These coincide
with the orthorhombic transition for all values of $x$. It is important to
realize that the structural and magnetic orders are identified in the same
measurements, the first from the splitting of the nuclear Bragg peaks and the
second by the intensity of the magnetic Bragg peaks. This means that the
conclusion that the two transitions are coincident does not depend on the
accuracy of the thermometry. As shown in Ref. [6], the two order parameters
determined in this way are directly proportional to each other at all
temperatures over the entire phase diagram, an unusual result that we will
discuss in more detail in the discussion.

Figure 7: (Color online) Neutron diffraction at 1.7 K with the magnetic Bragg
peaks at a $d$-spacing of 2.45 Åand 3.43 Åindicated by the arrows. They are
absent above $T_{N}$, for $x=0$, 0.1, 0.21, 0.22 and 0.25. At $x=0.3$, no
magnetic peaks are observed.

As a further check on our data, the values of $T_{N}$ determined by neutron
diffraction are in excellent agreement with those determined by peaks in the
temperature derivative of the magnetization (Fig. 8 and Table 1). These
magnetization peaks decrease in magnitude because of the progressive
attenuation of the magnetic signal due to increasing K content until they are
no longer detected for $x\geq 0.21$.

A full analysis of the magnetic structure was performed using the allowed
subgroup magnetic symmetries of $Fmmm$. All possible models were tested but
only the magnetic space group $F_{c}mm^{\prime}m^{\prime}$ resulted in a
proper fit to the data. Removal of the time reversal symmetry from two of the
mirror planes resulted in an antiferromagnetic arrangement of the magnetic
moments with a magnetic wave vector Q = (1,0,1); that is, the Fe magnetic
moments are antiferromagnetically coupled in the $x$ and $z$ directions and
ferromagnetically coupled along the $y$ axis. This model is consistent with
similar results previously reported for the parent BaFe2As2 material
[15,16,44].

Figure 8: (Color online) SQUID magnetization measurements for $x=0$, 0.125,
0.15, 0.175, 0.2 and 0.24 in 2 kG applied magnetic field. Insets are the first
derivatives of the magnetization curves, d$M$/d$T$ (10-5), used to determine
the Néel temperatures given by the arrows. For $x>0.2$, the magnetization
anomaly at $T_{N}$ is too weak to be detected.  Figure 9: (Color online)
Dependence of the neutron diffraction intensity for Ba1-xKxFe2As2 at 1.7 K
with $x$. The magnetic Bragg peaks are shown by the arrows. The solid lines
represent the calculated intensity of the Rietveld refinement.

Rietveld refinements of both the atomic and magnetic structures were performed
simultaneously as a function of temperature and doping, allowing the magnetic
moment to be defined in absolute units by normalization of the intensity of
the magnetic Bragg peaks to the structural Bragg peaks. The neutron data
displayed in Fig. 9 were collected at 1.7 K and normalized to the sample mass
and exposure time (measured in beam pulses). The figure qualitatively shows
the intensities of the magnetic (121) and (103) Bragg reflections to remain
roughly unchanged for the $x=0$ and 0.1 samples followed by a monotonic
decrease upon increasing the K content until they nearly vanish at $x=0.24$.
The refinements show that the magnetic moment drops from $\mu=0.75\mu_{B}$ for
the parent BaFe2As2 material to $0.46\mu_{B}$ for $x\sim 0.25$ (see Table 2).
No magnetic peaks are observed beyond this value.

Figure 10: (Color online) Dependence of the square of the magnetic moment,
$\mu^{2}$ (red stars), and the orthorhombic order parameter,
$\delta=(a-b)/(a+b)$ (blue circles), at 1.7 K on the potassium concentration,
$x$. The inset shows a comparison of $T_{N}$ and $\mu^{2}$ vs $x$, showing
that $T_{N}\propto\mu^{2}$. Solid lines in the main panel and the inset are a
fit to $(1-x/x_{c})^{2\beta}$ with $\beta=0.125$.

The doping dependence of the two order parameters making up the AF/O phase is
shown in Fig. 10. We compare $\delta$ and $\mu^{2}$ vs $x$, showing that they
are directly proportional over the entire range of AF/O order. We have not
been able to measure any samples between $0.25\leq x\leq 0.28$, but a power
law fit to $\delta$ and $\mu^{2}$ close to $x_{c}$, i.e., to
$(1-x/x_{c})^{2\beta}$, gives a critical concentration of 0.252 with an
exponent of $\beta=0.125(1)$ for the structural and magnetic order parameters.
The inset to Fig. 10 shows that $T_{N}$ is also proportional to $\mu^{2}$. In
a mean-field model, $T_{N}$ scales as $J\mu^{2}$, where $J$ is the effective
interionic exchange interaction, so this result would seem to indicate that
$J$ is approximately independent of $x$ over this range.

###  III.3 Superconductivity and Phase Coexistence

Figure 11: (Color online) SQUID magnetization measurements (zero field-cooled)
in 0.1 G magnetic field for (a) $x=0.15$ (solid squares), 0.175 (open
squares), 0.21 (solid circles), 0.22 (open circles), 0.25 (solid triangles),
$x=0.3$ (open triangles), and (b) 0.5 (solid triangles), 0.7 (solid circles),
0.9 (solid squares) showing well-defined superconducting transitions.
Magnetization values are normalized to mass of the samples. (c)
Superconducting transition temperatures (onset $T_{c}$) of the underdoped
compounds. Solid line represents the linear regression showing that the
critical concentration for superconductivity is 0.130(3).

Bulk superconductivity is observed for all samples with $x\geq 0.15$. Zero-
field magnetization data shows that superconducting transition temperatures
peaks at $\sim 38$ K for $x=0.4$ before it slowly decreases to 3 K for the end
member KFe2As2 (Fig. 11). By comparing the samples’ magnetic moment with that
of a Sn-powder sample of similar volume, we estimate that these samples are
bulk superconductors with a volume fraction of at least 80%. The uncertainty
is due to variations in the demagnetization factor between samples. All the
superconducting transitions are well-defined and sharp, even close to the
critical concentration where $T_{c}$ is varying rapidly with $x$. Even low
levels of compositional inhomogeneity associated with the uneven distribution
of Ba/K ions would be revealed in magnetic susceptibility measurements by a
broad or stepped-like transition from the normal state to the superconducting
state, so this is further evidence of the sample quality. The increase in
$T_{c}$ at low doping is approximately linear with dopant concentration up to
$x\sim 0.25$, so we have estimated the critical concentration for the onset of
superconductivity using linear regression to be $x_{c}=0.130(3)$.

In Ref. [6], we discussed the behavior of the order parameters below $T_{c}$.
We observed a small reduction in both the magnetic and structural order
parameters of approximately 5% at $x=0.21$ and 0.24, without seeing evidence
of additional phases. In a scenario in which the sample divides into separate
mesoscopic regions of AF/O phase and superconducting phase, this would imply
that 95% of the sample remains in the AF/O phase and only 5% becomes
superconducting. This is inconsistent with the magnetization measurements
showing bulk superonductivity. While it is not possible to rule out the
presence of other phases, the results indicate that there is microscopic phase
coexistence of magnetism and superconductivity, with the reduction in the AF/O
order parameters being due to a competition with the superconducting order
parameter. This competition has been discussed extensively by Fernandes et al
[45], who show that there should be an additional phase boundary, with a
positive slope vs $x$, between the coexistence region and the region of purely
superconducting phase. We have not yet identified any anomalies corresponding
to the phase line below $T_{c}$, so we assume that it rises steeply with $x$.

##  IV Discussion

The overall phase diagram of Ba1-xKxFe2As2 is shown in Fig. 12. We first
discuss the nature of the spin-density wave order and orthorhombic order.
Unlike the electron-doped compounds, where the two transitions split within
increased doping, the two transitions are coincident and first-order in
Ba1-xKxFe2As2. We reported the first-order character of the transition by the
observation of volume anomalies at $T_{s}$ [6]. Similar volume anomalies were
also observed by Tegel et al [46] in unsubstituted SrFe2As2 and EuFe2As2 but,
because of the small magnitude of these anomalies, the authors suggested that
the structural phase transition may be second-order. However, other authors
reported first-order transitions in polycrystalline SrFe2As2 [20] and single
crystals of CaFe2As2 [21] and BaFe2As2 [44]. In the latter reference, a first-
order-like hysteresis was obtained for the intensities of the (101) Bragg peak
when measured on cooling and warming. However, no such hysteresis was observed
by Wilson et al [27], when examining their BaFe2As2 single crystal. The
systematic observation of volume anomalies across the phase diagram is
unambiguous evidence that, at least in this system, all the transitions are
first-order, although weakly first-order with extremely small hysteresis.

Figure 12: (Color online) Phase diagram of Ba1-xKxFe2As2 with the
superconducting critical temperatures, $T_{c}$ (circles), the Néel
temperatures, $T_{N}$ (stars), and the structural transition temperatures,
$T_{s}$ (squares).

Phenomenological theory of magnetoelastic coupling predicts the possibility of
simultaneous first-order transitions that are driven by a linear-quadratic
term in the Ginzburg-Landau expansion [22,23] (see also Ref. [47]). This is
the lowest-order term allowed by symmetry. If the magnetic transition were to
occur at higher temperature than the structural transition (in the absence of
any competition), the magnetic order would drive the structural order in a
simultaneous first-order transition. The converse would produce two split
transitions as seen in most of the iron-based compounds [24]. There is a
report of a split transition in Ba1-xKxFe2As2 based on nuclear magnetic
resonance results [25], but this is a local probe, which cannot necessarily
identify compositional fluctuations. As we have already discussed, the neutron
measurements, which represent true averages over the bulk, are quite
unambiguous that the two transitions are simultaneous, although there could be
some rounding of the transitions at higher doping from small compositional
fluctuations.

The two order parameters are directly proportional to each other as a function
of temperature, which seems to indicate an unusual biquadratic coupling in the
Ginzburg-Landau expansion, rather than a linear-quadratic coupling. This is
usually only observed at a tetracritical point [26,27] where two phase
boundaries intersect, whereas our observations extend over a range of
compositions. There are a number of possible reasons for this. The most
intriguing and exotic idea is that the AF/O order parameters are both
secondary to another order parameter and directly driven by it. This would be
the case in, for example, valley density wave theory, in which a mother
density-wave drives both the magnetic and charge-density-wave orders [48]. A
second explanation is provided by the recent theoretical work of A.
Nevidomskyy [49], which uses a microscopic Kugel-Khomskii model to produce a
biquadratic spin-orbital term in the free energy. A subtle, but ultimately
more conventional explanation, is that the coupling is linear-quadratic after
all, but the proximity to a first-order transition produces a temperature
dependence that is approximately equivalent to biquadratic coupling to first
order [29]. This is in the context of a theory in which Ising-nematic order,
produced by an itinerant model of Fermi surface nesting, drives the structural
transition. Support for this explanation is provided by Fig. 10, where the
doping dependence of the magnetic and structural order parameters at low
temperature indicates a linear-quadratic coupling. Whatever the eventual
explanation, it is clear that this result is key to understanding the nature
of the normal state and the role of nematic order in the eventual
superconductivity.

The strong coupling between AF/O order parameters persists into the regime of
phase coexistence with superconductivity. We have already argued that
Ba1-xKxFe2As2 is characterized by microscopic phase coexistence because
mesoscopic phase separation would result in a significant decrease in the
volume fraction of the AF/O phase below $T_{c}$. The consensus in favor of
microscopic phase coexistence has existed for some time in the electron-doped
superconductors, such as BaFe2-xCoxAs2 [24], where the phase boundary below
$T_{c}$ to a non-magnetic, purely superconducting region has also been
identified. Theoretically, this behavior is consistent with unconventional
$s_{\pm}$ pairing of the Cooper pairs suggesting that itinerant long range
magnetism and superconductivity may coexist and compete for the same electrons
[45]. However, the idea of microscopic phase coexistence was more
controversial in Ba1-xKxFe2As2 because of local probe measurements that seemed
to indicate a phase separation into mesoscopic regions of magnetism and
superconductivity [30,31]. Since the most recent $\mu$SR data are also
consistent with microscopic phase coexistence [32], it appears that the
earlier reports may have been due to compositional fluctuations close to the
phase boundaries and that microscopic phase coexistence has now been
confirmed.

Figure 13: (Color online) (Top panel) Magnetic and structural phase diagram of
electron-doped Ba(Fe1-xCox)2As2 and hole-doped Ba1-xKxFe2As2 with the
superconducting critical temperatures, $T_{c}$ (squares), Néel temperatures,
$T_{N}$ (stars) and structural transition temperatures, $T_{s}$ (circles). The
$x$-axis is normalized to the charge carrier per iron atom. Data for the
electron-doped side where the transition temperatures are represented with
open symbols are taken from Ref [50]. The error bars for $T_{N}$ and $T_{s}$
values in the hole-doped side are within the symbols. The dashed line
enveloping the superconducting dome represents the Lindhard function taken
from Ref [33]. (Bottom panel) Charge carrier dependence of the As-Fe-As bond
angles for both electron- and hole-doping. Solid triangles represent the
results of our neutron diffraction study at 1.7 K for the hole-doped
Ba1-xKxFe2As2. At this temperature one of the As-Fe-As angles splits due to
orthorhombic distortion below $x=0.3$. Therefore, we took the average of these
two splitting angles. The As-Fe-As bond angle data for the electron doped side
is taken from Ref [51]. Solid lines are guide to the eye.

Finally, we discuss the electron-hole asymmetry in the phase diagram, shown in
Fig. 13, where we have added data from the literature [50,51] to allow a
comparison with the more commonly studied electron-doped superconductors. In
this phase diagram, the $x$-axis is normalized to the number of charge
carriers per Fe atom. Neupane et al have recently suggested that this
asymmetry is due to differences in the effective masses of the hole and
electron pockets [33]. This is justified by ARPES data that show that hole
doping can be well described within a rigid band approximation [52]. An ab
initio calculation of the Lindhard function of the non-interacting
susceptibility at the Fermi surface nesting wavevector shows exactly this
asymmetry, with a peak at $x\sim 0.4$ where the maximum $T_{c}$ occurs. Our
recent inelastic neutron scattering measurements of the resonant spin
excitations that are also sensitive to Fermi surface nesting have shown a
similar correlation between the strength of superconductivity and the mismatch
in the hole and electron Fermi surface volumes [34], that is responsible for
the fall of the Lindhard function at high $x$. An overall envelope may be
drawn (dashed line in Fig. 13) to encompass both the hole and electron
superconducting domes of the phase diagram. If anything, the Lindhard function
underestimates the asymmetry, predicting a larger superconducting dome on the
electron-doped side. We attribute this behavior to the fact that the iron
arsenide layers remain intact in the potassium substituted series, whereas Co
substitution for Fe disturbs the contiguity of the FeAs4 tetrahedra and
interferes with superconductivity in these layers.

Interestingly, the maximum overall $T_{c}$ also correlates with the perfect
tetrahedral angle of $\sim 109.5^{\circ}$ as demonstrated in the bottom panel
of Fig. 13. In the plot, average $<$As-Fe-As$>$ bond angles for our
K-substituted series have been extracted from the Rietveld refinements. The
As-Fe-As bond angles for BaFe2-xCoxAs2 are extracted from the literature [51].
The continuity of the bond angles across the electron-doped and hole-doped
sides of the phase diagram is remarkable and the crossing of the two
independent angles at $x\sim 0.4$ to yield a perfect tetrahedron and maximum
$T_{c}$ is clear. This has been remarked before in other systems [35,53]. It
is possible that these two apparently distinct explanations for the maximum
$T_{c}$ are two sides of the same coin. In a theoretical analysis of the 1111
compounds [38], it has been suggested that the pnictogen height is important
in controlling the energies of different orbital contributions to the
$d$-bands and so affect the strength of the interband scattering that produces
superconductivity.

We now turn our attention to the SDW region of the phase diagram. While it is
clear that spin-density-wave order has to be suppressed in order to allow
superconductivity to develop, it is not immediately clear what is responsible
for the suppression. Both the strength of magnetic interactions and
superconductivity, at least in an itinerant model, depend on the same Lindhard
function [54], the former on the peak in the susceptibility at the magnetic
wavevector, and the latter on an integral over the Fermi surfaces. It would
seem therefore that the magnetic transition temperature should also peak at
$x\sim 0.4$. One intriguing reason why it would peak at $x=0$ is because
magnetic order is more sensitive to disorder-induced suppression of the peak
susceptibility whereas superconductivity is more robust. There is some support
for this idea from the observation that isoelectronic doping produces a
similar suppression of magnetic order than seen with hole-doping [51]. On the
other hand, Kimber et al succeeded in rendering the parent BaFe2As2 material
to exhibit zero resistance at 30.5 K by the application of significant
external pressures up to 5.5 GPa [55], i.e., without introducing disorder, but
they remarked that superconductivity needs to be confirmed by other bulk
measurement techniques. Interestingly, the authors also correlate the induced
$T_{c}$ with approaching a perfect tetrahedron angle of 109.5o similar to our
observations for $x\sim 0.4$.

In summary, we have synthesized high quality samples covering the full phase
diagram of Ba1-xKxFe2As2. Using high resolution neutron powder diffraction and
SQUID magnetization measurements, we have investigated the effects of
potassium substitution on superconductivity, structural transformation and
magnetic ordering. Our measurements allowed the construction of a detailed
magnetic and structural phase diagram, which displays a narrower phase
coexistence region than the previous reports. Moreover, neutron diffraction
and the SQUID magnetization data confirmed that magnetic and structural
transitions are coincident with first order transitions. Additionally, we
determined the effects of temperature and substitution on the various internal
atomic and structural parameters. Our results confirm the importance of
obtaining precise structural parameters across the whole phase diagram as a
way of providing insight into the nature of the phase competition that
underlies iron-based superconductivity.

This work was supported by the Materials Sciences and Engineering Division of
the Office of Basic Energy Sciences, Office of Science, U.S. Department of
Energy, under contract No. DE-AC02-06CH11357.

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