the fast li - ion conductor 
 with the nominal composition li7la3zr2o12 ( llzo ) is receiving much 
 scientific attention since its discovery in 2007 .
it has a garnet - based structure , and it occurs in at least 
 two structural modifications . at room temperature , 

llzo is tetragonal ( i41/acd ) while the cubic modification ( ia-3d ) is stable above approximately 150 c .
geiger et al . argued that the better conducting cubic phase can 
 be stabilized at room temperature ( rt ) through the incorporation of 
 small amounts of al .
the 
 stabilizing effect of al has now been confirmed by a 
 number of subsequent investigations .
the exact role al plays in cubic al - bearing llzo is 
 important because llzo shows a high ionic conductivity of about 10 s / cm at rt .
this is approximately 2 orders of magnitude 
 higher than that for the lower symmetry al - free tetragonal llzo phase . 

llzo also has good chemical and thermal stability , as well as a wide 
 energy potential window making it an excellent candidate for use as 
 an electrolyte in an all - solid - state lithium - ion battery .
as has been shown recently , ionic conductivity also seems 
 to depend 
 on the amount of al incorporated during synthesis .
further work , however , is needed to quantify this effect . for this 
 purpose , the chemical and physical properties governing li diffusion have to be understood in detail ; in particular this includes 
 the important question as to which crystallographic sites the al ions preferably occupy in the cubic phase of llzo .
considerable experimental research has been undertaken to obtain 
 information about al in llzo , including its local coordination 
 and site partitioning behavior . in this context ,
al magic 
 angle spinning ( mas ) nuclear magnetic resonance ( nmr ) spectroscopy 
 is a key method .
several nmr studies have proposed that the resonance 
 observed at a chemical shift ranging from 64 to 68 ppm corresponds 
 to al located at the 
there is uncertainty , however , about the interpretation and assignment 
 of the other measured nmr lines that have chemical shifts ranging 
 from approximately 78 to 82 ppm .
the interpretations given so far 
 include al residing at nontetrahedral li sites 
 ( 4-fold to possibly 6-fold coordinated ) and tetrahedrally coordinated 
 sites in the neighborhood of la or zr vacancies 
 in al - rich llzo .
in particular , mechanochemically 
 prepared llzo samples with a high amount of al , but reduced 
 in la and zr content , even indicate two 
 magnetically inequivalent tetrahedral sites .
neutron powder diffraction ( npd ) measurements were interpreted as 
 indicating that al is located at an octahedrally coordinated 
 site in the garnet framework .
summarizing 
 the various published experimental results , there is 
 no clear understanding as to which sites are occupied by al in cubic llzo .
this also concerns the detailed interpretation of al nmr mas spectra and diffraction results . to address the 
 important role of al in llzo and to obtain a better understanding 
 of the various experimental results
, we undertook a density functional 
 theory ( dft ) investigation of al - bearing cubic llzo with the aim to 
 calculate ( relative ) al nmr parameters such as chemical 
 shifts and electric quadrupole coupling constants .
in order to understand the computational models used in this investigation , 
 we provide a short description of the crystal structure of cubic llzo 
 garnet ( figure 1 ) .
the yellow dodecahedrally coordinate 
 la ( at the wyckoff position 24c ) and 
 orange octahedrally coordinate zr ( 16a ) .
the blue spheres correspond to tetrahedrally coordinated ( 24d ) li , green spheres to octahedrally coordinated 
 ( 48 g ) li , and red ones to distorted 4-fold 
 coordinated ( 96h ) li .  garnet  is the common name for a 
 large number of 
 natural and synthetic metal oxide and fluoride phases .
conventional oxide garnets have the general 
 formula a3b2c3o12 and 
 crystallize with cubic symmetry ia-3d . in the case of llzo , the o ions , located at 
 general crystallographic positions , 96h ,
form an 
 oxygen - ion framework with interstices occupied by the a cations ( la ) at an 8-fold coordinated position 24c ( point 
 symmetry 222 ) , by b cations ( zr ) at a 6-fold coordinated 
 position 16a ( point symmetry 3 ) , and by c 
 cations ( li ) at a 4-fold coordinated 24d position ( point symmetry 4 ) .
in addition to these cation 
 sites , there are other interstices within the oxygen framework that 
 are empty in the conventional garnet structure , such as the 6-fold coordinated 48 g positions 
 ( point symmetry 2 ) and a general 96h position , sometimes 
 described as 4-fold coordinated with two additional longer bonds greater 
 than 2.8  in length , 5-fold coordinated or 6-fold coordinated ( point symmetry 1 ) . these interstices can be filled 
 by  extra  cations ( such as li ) , giving 
 rise to compositions with nonstandard garnet stoichiometry . an important 
 property is the partial occupancy of the structural sites ( 24d , 48 g , and 96h ) where 
 li is located and also delocalization of li ions throughout .
they concentrate on the basic structural properties and on the li diffusion , but none of them considers al ions 
 explicitly . for our first - principle calculations ,
the ions in 
 llzo were arranged 
 on the basis of crystal structure descriptions in the literature .
three different structural models were used 
 with all having a body centered ( i - type ) bravais 
 lattice with a = b = c = 12.972  and  =  =  = 90. three 
 garnet compositions were chosen , namely li44al4la24zr16o96 with al at 24d , 48 g and 96h , li56al4la20zr16o96 with al solely at 24c , and 
 li60al4la24zr12o96 with al solely at 16a .
the li ions were distributed over the 24d and the 
 96h sites following xu et al .
no significant effect on the results was observed by choosing 
 a different li arrangement . the highest symmetry possible 

the first model was used 
 to understand the behavior of al at various possible 
 sites occupied by li .
the calculations were made with 
 17 different local al positions located among the 24d , 96h , and 48 g sites . 

the various al positions were 
 fixed during relaxation and all al ions are equivalent 
 in the unit cell .
all calculations are based on dft methods 
 using the all - electron 
 full potential linearized augmented plane wave ( lapw ) method as implemented 
 in the wien2k code . the perdew burke ernzerhof ( pbe ) 
 generalized gradient approximation ( gga )
the atomic positions were optimized by minimization 
 of the forces ( below 2 mry / au ) acting on the atoms simultaneously 
 with the self - consistent - field cycle as implemented by marks .
iterative 
 diagonalization  using an efficient preconditioning ( inverse 
 of h - s ) and the block - davidson method .
the radii of the atomic spheres ( rmt ) for the li , la , zr , o , and al ions were chosen to 
 be 1.46 , 2.27 , 1.93 , 1.71 , and 1.40 au , respectively .
the cutoff for 
 the plane wave rmtkmax = 6.0 and the maximum fourier expansion of charge density 
 cutoff gmax = 12 ( au ) were applied .
the separation parameter between the valence and core 
 states was chosen to be 6.0 ry .
the computational considerations 
 were checked by increasing rmtkmax and the number of k - points , 
 but no significant changes with respect to the energy , geometry , and 
 the electric field gradient ( efg ) around al were observed .
the behavior of al at different crystallographic sites 
 was analyzed qualitatively by weighing the various al positions , xi , with 
 their corresponding energy , e(xi ) .
the various local energies for al at and around the crystallographic positions where li can 
 also be located , were described by fitting a polynomial , describing 
 a possible diffusion pathway .
the ocuppation probabilities for an 
 al ion along this path are described using a normalized 
 boltzmann factor given bywhere k is boltzmann s 
 constant , and where xi+1 indicates the states next to xi , xi1 indicates 
 the states before xi with 
 an interval of 0.01  , and t is 298.15 k. the 
 site preference energy , e , is defined as e = e(xi )
 e , where e is the global energy minimum and corresponds 
 to al located at 24d .
it has the lowest 
 total energy for all the calculated arrangements of al in llzo . in nmr spectroscopy
, the chemical shift ,  , of a nucleus , i , describes the nuclear shielding effect of an applied 
 ( external ) static magnetic field and the locally induced magnetic 
 field arising from the surrounding electrons with a certain probability 
 of presence near the corresponding nuclear site i. the magnitude of the resulting effective field , beff , is given by beff = b0(1  i ) , where i is 
 a second - rank nuclear shielding tensor , 1 is the unit 
 matrix and b0 is a uniform external field 
 along the z - axis .
the resonance nmr frequency , i , is then given by vi = ( i/2 ) beff , where i is the magnetogyric ratio of the nucleus under observation . 
 the isotropic chemical shift , iso ( henceforth  ) , 
 describes the relation between the nmr resonance frequency for the 
 nucleus of interest , i , and the 
 corresponding resonance frequency for a reference compound , ref , giving  = 10  ( vi  vref)/vref . because of the orientation in 
 systems with periodic boundary conditions , all anisotropic interactions 
 causing line broadening are projected onto the axis of rotation where 
 they collapse at the magic angle .
to 
 compute  , we used the relaxed geometry and applied the nmr 
 module of the wien2k code .
these 
 chemical shift calculations are based on an all - electron linear response 
 method where one obtains the induced current density considering the 
 perturbation of the ground state wave functions due to the external 
 magnetic field .
the resulting magnetic shielding is then obtained 
 by integration of the all - electron current according to biot 
because there is agreement in 
 the literature that the al nmr resonance at 66  2 ppm represents 
 tetrahedrally coordinated al at 24d in 
 llzo ( 24d = 68 ppm ) , we calculate 
 the difference of the chemical shifts (  ) for al at other positions xi with a xi as follows besides the chemical shift , the interaction 
 of the quadrupole moment , q , of the al nucleus ( spin - quantum number i = 5/2 ) with a 
 sufficiently large electric field gradient 
 ( efg ) , which is produced by a nonspherical charge distribution around 
 the nuclear site , also affects the central nmr transition .
second 
 order quadrupolar effects , influencing the shape of the nmr central 
 line , can not be ( completely ) eliminated by magic angle spinning . from 
 the simulation of spectra recorded under mas conditions , the quadrupole 
 coupling constantand the corresponding asymmetry 
 parametercan be estimated . here
, e is the positive elementary charge , h denotes planck s 
 constant , and vxx , vyy and vzz are the elements of the traceless efg tensor v with |vzz|  |vxx|  |vyy| .
the shape of the nmr central line depends sensitively on 
 the asymmetry of the electronic charge density close to the nucleus . 

the parameter  describes the deviation of the efg from axial 
 symmetry and it can take values between 0 and 1 .
second order quadrupole 
 interactions also affect the position of the nmr line in terms of 
 a quadrupolar shift . a larger external magnetic field
, however , lowers 
 the effect . here , we compute the efg from the all - electron charge 
 distribution without further approximation and cq is obtained using the nuclear quadrupole 
 moment q(al ) = 1.616  10 m determined from the slope of the linear regression 
 proposed by body et al .
the most energetically favorable position of al in llzo is at the tetrahedrally coordinated crystallographic 
 special 24d position .
al can be displaced 
 toward a neighboring vacant general 96h site , which 
 leads to a distortion of its oxygen coordination polyhedron .
calculated 
 al  o bond lengths for this coordination at 0 k are between 
 1.76  and 1.79  and the tetrahedral volume is 2.86  ( see table 1 ) .
our calculations show that al at 96h is 4-fold coordinated , and distorted from tetrahedral coordination , 
 with al  o bond lengths varying from 1.76 to 1.92  , because 
 of the displacement of al towards a vacant 24d site .
additionally , there are two o ions further away that have on average approximately 1  longer 
 bond distances .
if they would be included in the local coordination 
 sphere around al , a distorted octahedral coordination 
 polyhedron with a volume of 16.33  would result . 

al could also possibly be located at the special 
 48 g site with three different pairs of al  o 
 bonds and calculated bond lengths of 1.93  , 2.00  and 
 2.12  , respectively , yielding a coordination volume of 10.67 
  .
any possible al at the 16a site would also be 6-fold coordinated with again three 
 pairs of al 
o bonds with lengths of 1.90  , 2.03  , 
 and 2.11  and a coordination volume of 10.69  .
al at the 24c site would have four 
 shorter al  o bond lengths equal to 1.94  and four longer 
 bonds of 2.84  length .
assuming , on the other hand , 
 8-fold coordination around the 24c site , the resulting 
 polyhedron would have a volume of 22.86  .
the various 
 hypothetical al coordination polyhedra at 0 k , along 
 with their associated site preference energy , are shown in figure 2 .
various coordination polyhedral around al located 
 at the 24d , 48 g , 96h , 16a , and 24c sites .
( nonbonded 
 red spheres indicate additional o ions not considered 
 as next nearest neighbors in the first coordination sphere .
their 
 distances are given in table 1 . ) their associated 
 site preference energies in cubic llzo .
the calculated site preference energies , e , for al in llzo are as follows : 24d >
al at the 
 24d site has the global energy minimum and is 1.38 
 ev more 
 stable than al at 96h and 1.60 ev more 
 stable than al at 48 g .
the latter two 
 sites are energetically similar with an energy difference of just 
 0.22 ev .
the site preference energy for al at 16a and 24c is 0.41 and 1.74 ev higher , respectively , 
 when referenced to the global energy minimum .
the site preference 
 energies for al at various structural positions between 
 48 g and 24d are shown in figure 3 .
site preference energies , e , for 
 al at various structural positions ( gray crosses ) at 
 and near 
 the 48 g , 96h and 24d sites in llzo garnet .
the fits were used 
 to calculate the boltzmann distribution , w(xi ) ( see text ) , discussed below .
the different 
 distributions of al at the various crystallographic sites 
 are normalized to 1 and indicate the room temperature probability 
 of finding al at a distinct position . calculated al nmr parameters for al at 
 the different structural 
 sites in llzo are shown in table 2 .
the corresponding 
 polyhedra ( figure 4 ) around al , slightly displaced from 24d and 96h ( figure 3 ) , were used to analyze the experimental 
 nmr spectra .
the calculations show that al coordinations 
 given by 1 to 5 ( figure 4 ) have  
 values between 3.7 and 0.2 ppm , cq values between 1.38 and 5.01 mhz and  
 values between 0.20 and 0.70 .
al for coordinations given 
 by 9 to 13 have  values between 10.0 and 14.4 ppm , cq values between 2.4 and 3.3 
 mhz and  values between 0.24 and 0.70 . on the other hand ,
al located at 48 g has an extremely large calculated 
  value of 133.5 ppm , 118.5 ppm at 24c , and 108.5 ppm at 16a .
the latter three sites have 
  values that are much larger than any experimentally observed 
 nmr resonances .
oxygen coordination polyhedra around al for 
 al  o 
 distances less than 2.6  at 17 local positions starting with 
 24d ( 1 ) and going to 96h ( 12 ) and 
 beyond and ending at 48 g ( 20 ) in cubic llzo .
in 
 this dft study , we investigated the crystal - chemical role of 
 al in llzo .
specifically , we address the question of 
 the coordination and the site distribution behavior of al .
the results of the calculations can be used , together with published 
 experimental results , to achieve a better understanding of the nature 
 of al incorporation in llzo .
we find that al at the 24d site is energetically the most stable 
 state , followed by al at 96h and 48 g , whereby the site preference energies for the latter two 
 are similar .
based on 
 our dft calculations , al should be exclusively located 
 at 24d at 0 k , which would give rise to a single 
 nmr resonance .
thus , the question arises why additional resonances 
 are observed in the al mas nmr spectra ( figure 5 ) . before discussing our results
, we briefly outline the nmr 
 results 
 from the various studies with respect to the al site 
 occupation in llzo . in all recent studies the resonance at approximately 
 66
other nmr lines also indicate 4-fold coordinated al ions , because their chemical shift values of 78 to 82 ppm are usually 
 indicative of tetrahedral coordination .
geiger et al . assigned the 
 resonance at 81 ppm to al residing at the 96h sites .
in particular , the spectra presented 
 in ref ( 6 ) indicate 
 that several nmr lines contribute to a signal occurring between 75 
 to 85 ppm .
it should be noted that the latter has been reported for 
 llzo samples with a high amount of al and lower than 
 stoichiometric la and zr contents .
such samples were prepared by mechanochemical 
 activation combined with subsequent annealing at moderate temperatures .
the position corresponds to al with the coordination polyhedra in figure 4 and the site preference energies in figure 3 .
calculated values are 
 referenced 
 to the global minimum given by  = 68 ppm at position 1 . literature :  =  
  24d ; calculation :  
 =   68 ppm
the gray areas represent the variation in chemical shift values , 
  , ( 13 to 18 ppm ) in literature .
observed a stronger nmr signal 
 at 64 ppm and 
 a less intense line at 78 ppm for garnets with al contents 
 up to 0.30 per formula unit ( pfu ) .
however , at concentrations above 
 0.60 al pfu the resonance at 78 ppm becomes the most 
 intense . at even higher al contents
a new nmr line at 
 81 ppm emerges , which notably appears in la- and zr - deficient llzo 
 samples .
, in addition to al 
 mas nmr experiments , 
 also al mqmas nmr measurements .
the latter spectra show a resonance at  = 81 ppm , a coupling 
 constant , cq , of 3.3 
 mhz and an asymmetry parameter ,  , of 0.7 .
the second 
 resonance observed at  = 68 ppm shows a stronger quadrupolar 
 interaction with cq = 
 5.0 to 5.2 mhz and a lower  value between 0.0 and 0.1 that 
 suggests axial symmetry at the site .
it should be noted that the al mas nmr spectrum was recorded at 156.26 mhz , thus providing 
 a sufficiently high resolution to simulate its line shape .
such a 
 simulation yields a resonance with  = 81 ppm , cq = 3.3 mhz and  = 0.7 , and for 
 the second signal values of  = 70 ppm , cq = 5.5 mhz and  = 0.5 . the calculated 
 nmr parameters ( table 2 ) 
 are in good agreement with an assignment of al to 96h for the resonance at 81 ppm .
the calculated nmr parameters 
 for the nmr line at 68 ppm that is assigned to al at 
 24d , however , disagree with those determined from 
 the mqmas nmr experiments . for the latter
, it has been proposed that 
 possibly more than one al site is reflected by this broad 
 resonance .
they made al mqmas nmr spectroscopic 
 measurements to investigate this resonance at 68 ppm for an al - doped 
 llzo garnet synthesized at 850 c .
their line shape analysis 
 yielded two nmr lines with the first having  = 68.4 ppm and cq = 4.8 mhz and the second 
  = 73.5 ppm and cq = 6.9 mhz .
both resonances were assigned to tetrahedrally coordinated 
 al at 24d in two different garnet phases . 

the two tetrahedral sites have slightly different distortions and 
 thus different  and cq values .
report that this interpretation is consistent 
 with their high - resolution xrpd results , and they proposed that slow 
 al diffusion within the lattice is responsible for the 
 disordering over the two sites .
for the first step , we calculated the 
 nmr parameters for al at all crystallographic sites , 
 via 24d , 48 g , 96h , 16a , and 24c . according to our 
 results ,
the resonance at 80  2 ppm should be assigned to al residing at the 96h sites and the resonance 
 at 66  2 ppm to al occupying 24d .
there is good agreement between the calculated cq value for al at 96h , but disagreement between cq values for al at 24d .
proposed a large cq value for al at 24d , but it can not 
 be easily explained why the symmetric 24d site should 
 yield a cq value approximately 
 two times larger than al at the more distorted 96h site . to understand 
 this issue better , we calculated the nmr parameters 
 for al at all positions at and around 24d and 96h that are occupied by a certain probability . 

shifting al away from 24d and 96h leads to a distribution of slightly different local oxygen 
 coordination environments .
thereby , broad al mas nmr 
 resonances could result , which would reflect a distribution of slightly 
 different  and cq values .
we note that our calculations for  for al at 24d do not agree with experiment , but  
 was obtained from the experimental spectra by using only one or two al resonances to simulate this broad feature at 66  2 ppm .
the nmr signal located at 78 to 82 ppm is also asymmetric in shape . 

this is possibly due to the slightly different geometries of local 
 al coordinations , as given by their calculated probabilities , 
 toward the 48 g minimum versus those in the direction 
 of 24d .
this situation could produce two overlapping 
 resonances having similar values of  and cq ( see also the agreement with experimental 
 results and the discussion in ref ( 6 ) ) .
although experiment and calculations 
 are in broad agreement in 
 terms of the crystal chemical role of al in llzo , it 
 is not clear based on the dft calculations alone , why al is not located exclusively at 24d as suggested by the site energies .
al at 96h and 48 g must also be considered , which , 
 however , would lead to an additional nmr line which is usually not 
 observed experimentally .
( for the sake of completeness , let us note 
 that an additional nmr line has been observed at 93 ppm for al - doped 
 li6.5la2.5ba0.5zrtao12   and for some of the al - doped llzo samples prepared via mechanosynthesis 
 ; the prominent nmr line at 65 ppm , however , is absent in these 
 cases . ) here , it must be stated that the exact thermodynamic state 
 of llzo , as obtained in the various sintering experiments , is not 
 known .
it is possible that metastable structural states are obtained 
 that can depend on a number of experimental factors ( dopant concentration , 
 sintering temperature and time , heating rate , grain sizes , starting 
 materials , etc . ) .
thus , al could potentially be incorporated 
 metastably at 96h and 48 g as well 
 in cubic llzo , because both sites have similar site energies .
the different site energies for al at 24d and 96h could provide an explanation for differences 
 in published nmr spectra of llzo . the relationship between al concentration and variations in the intensities of the different 
 resonances ( and thus al site occupancies ) , as observed 
 by dvel et al .
, can be interpreted crystal chemically . when 
 an al ion is located at 24d , because 
 of its large effective charge radius , it could create a larger inaccessible 
 region around 24d compared to the situation for al at 96h .
this leads to a reduction of entropy and a loss of 
 energy possibly making the 96h site energetically 
 more accessible for al with increased al contents in contrast to the 24d site .
they showed 
 by using molecular dynamics simulations that the introduction of vacancies 
 in llzo does indeed reduce the free energy .
lastly , we consider the coordination geometry 
 around the 96h site in llzo , which has been described 
 differently .
it is , of course , a matter of 
 definition as to what constitutes a bond in a first coordination sphere . 
 according to li et al . in their description of al at 96h , the average al  o bond distance is d = 2.08  , with a difference of 0.38  between 
 the length of the shortest and the longest bonds .
o = 2.69  and 
 2.80  , thus being 0.46  and 0.57  longer compared 
 to the longest bond in strict 4-fold coordination .
based on our calculations , 
 al at 96h has a locally 4-fold distorted 
 coordination ( figure 4 ) .
it is worth mentioning 
 that long - range structural properties , as determined by the diffraction 
 experiment , such as neutron powder diffraction , can differ from those 
 measured via spectroscopy , for example , nmr , which probes structure 
 at shorter length scales .
based on our dft results , 
 we propose that al could 
 have a number of slightly different local 4-fold coordinations around 
 the crystallographic 24d and 96h sites in cubic llzo garnet .
, in a 
 neutron diffraction study of cubic llzo garnet , proposed that al is located at the 48 g site in octahedral 
 coordination . in terms of calculated site energies , this site could 
 be partially occupied .
however , the calculated al nmr chemical shift 
 value for such coordination has not been observed in experimental 
 nmr spectra .
 we 
 investigate theoretically the site occupancy of al3 + in 
 the fast - ion - conducting cubic - garnet li73xal3+xla3zr2o12 ( ia-3d ) using 
 density functional theory . by comparing calculated 
 and measured 27al nmr chemical shifts 
 an analysis shows 
 that al3 + prefers the tetrahedrally coordinated 24d site and a distorted 4-fold coordinated 96h site . 
 the site energies for al3 + ions , which are slightly 
 displaced from the exact crystallographic sites ( i.e. , 24d and 96h ) , are similar leading to a distribution 
 of slightly different local oxygen coordination environments . 
 thus , 
 broad 27al nmr resonances result reflecting the distribution 
 of different isotropic chemical shifts and quadrupole coupling constants . 
 from an energetic point of view 
 , there is evidence that al3 + could also occupy the 48 g site with its almost 
 regular octahedral coordination sphere . 
 although this has been reported 
 by neutron powder diffraction , the nmr chemical shift calculated for 
 such an al3 + site has not been observed experimentally . 
Introduction
Computational Methods
Results
Discussion
Conclusion