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Soc Stratification

by: Mina Heaney

Soc Stratification SOC 140

Mina Heaney
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This 14 page Class Notes was uploaded by Mina Heaney on Tuesday September 8, 2015. The Class Notes belongs to SOC 140 at University of California - Davis taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/187316/soc-140-university-of-california-davis in Sociology at University of California - Davis.


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Date Created: 09/08/15
PHYSICAL REVIEW B 77 035135 2008 Rareearth boron bonding and 4f state trends in RB4 tetraborides Z P Yin and W E Pickett Department Of Physics University Of Califarnia Davis Davis CA 95616 Received 19 July 2007 published 31 January 2008 The B B bonding boron rare earth coupling and the changes in 4f states across the lanthanide series in RB4 Rrare earth compounds are studied using the correlated band theory LDA U method A set of boron bonding bands that are well separated from the antibonding bands can be identi ed Separately the dimer B 2pz orbital is nonbonding viz graphite and MgBZ but mixes strongly with the metal 4d or 5d states that form the conduction states The bonding bands are not entirely lled even for the trivalent compounds thus the cation d bands have some lling which accounts for the lack of stability of this structure when the cations are divalent with more bonding states un lled The trends in the mean 4f level for both majority and minority and occupied and unoccupied states are presented and interpreted DOI 101103PhysRevB77035135 I BACKGROUND AND MOTIVATION The tendency of the metalloid boron to form clusters has led to widespread study of the properties of condensed bo ron Of the many classes of compounds that B forms Brich metal borides include classes with very important and in tensely studied properties One example is MgBZ which is the premier phononcoupled superconductorl at 40 Al though this structural class includes several transition metal borides and other simple metal borides such as LaBZ MgBZ is unique in this singlemember class of quasitwo dimensional sp metal with very high superconducting tran sition temperature due to strong covalent B B bonds that are driven metallic2 by the crystal structure and chemistry Another class that has received great attention is the hexaborides MB6 formed from vertexlinked B6 octahedra that enclose the metal ion in the cubic interstitial site This class includes the divalent metals M CaSrBa that are small gap semiconductors12 The stability of this structure was understood decades ago when cluster studies established 4 that the bonding states of linked B6 clusters are lled by 20 electrons which requires two per B6 unit in addition to the B valence electrons There are many trivalent hexaborides as well including lanthanide members which have very peculiar properties unusual magnetic ordering heavy fermion formation and superconductivity6 7quotl113 16 Two monovalent members NaB6 Ref 17 and KB6 Ref 18 have been reported Yet another class that has been known for decades is the metal mostly rare earths tetraboride RB4 family which is richer both structurally and electronically and for which con siderable data are available see for several RB4 Refs 19 22 YB4 Refs 23 27 LaB4 Ref 28 CeB4 Refs 29 31 NdB4 Ref 32 GdB4 Refs 33 38 TbB4 Refs 39 44 DyB4 Refs 45 50 and ErB4 Refs 44 51 and 52 Yttrium and all the lanthanides except Eu and Pm form isostructural metallic tetraborides RB4 with space group P4mbm No 127 described below and pictured in Fig l Presumably Eu is not stable in the tetraboride structure because of its pref erence for the divalent con guration in such compounds The Sr and Ba tetraborides also are not reported A calcium tet raboride with formula CaB1xCx4 x005 was reported53 recently 1098 012120087730351359 035135 1 PACS numbers 7128d 71207b 7120Eh These rareearth tetraborides exhibit an unusual assort ment of magnetic properties While CeB4 and YbB4 f1 and f respectively do not order and PrB4 orders ferromagneti cally at TC25 K36 all of the others R Nd Sm Gd Tb Dy Ho Er Tm order antiferromagneti cally with Neel temperature TN see Table I spanning the range 7 44 K A noteworthy peculiarity is that T N does not obey de Gennes scaling law which says that the magnetic transition temperature is proportional to g 12JJ 1 across an isostructural series where the rareearth atom is the only magnetic component 54 Here J is Hund s rule total angular momentum index and g I is the corresponding Land g factor In the rareearth nickel borocarbide series for ex ample de Gennes scaling is obeyed faithfully55 This lack of 6 H r ea l a 0 o v I 15 2 quot i l l m 390 39 I A I in FIG 1 Color online Structure of RB4 viewed from along the c direction The large metal ion spheres red lie in z0 plane Apical B1 atoms small black lie in z02 and z08 planes Lightly shaded yellow dimer B2 and equatorial B3 dark blue atoms lie in z05 plane The sublattice of R ions is such that each one is a member of two differently oriented R4 squares and of three R3 triangles 2008 The American Physical Society Z P YIN AND W E PICKETT PHYSICAL REVIEW B 77 035135 2008 TABLE I Data on magnetic ordering in the RB4 compounds Refs 19 21 36 and 50 The columns provide the experimental ordering temperatures Tm the ordering temperature T predicted by de Gennes law relative to the forced agreement for the Gd 4 compound the orientation of the moments and the measured ordered moment compared to theoretical Hund s rule moment MB ng K T K Direction Mexp Mth PrB4 24 21 Me 320 358 SIIB4 26 12 084 GdB4 42 42 16 781 794 TbB4 44 24 28 16 964 972 DyB4 203 127 19 Ho 1044 1063 H0134 71 57 1st 12 Ho 104 106 ErB4 154 7 Me 929 960 T111134 117 971st 3 is 735 756 scaling indicates that magnetic coupling varies across the series rather than following a simple RudermanKittel KasuyaYosida RKKYlike56 behavior with a xed Fermi surface Both the ferromagnetic member PrB4 and antiferromag netic ones RB4 show strong magnetic anisotropy For ferro magnetic PrB4 the c axis is the easy axis The situation is more 1 quot for the T 39 1 d which display varying orientations of their momenm below TN an some have multiple phase transitions GdB4 an ErB4 have only one second order phase transition while both TbB4 and DyB4 have consecutive second order phase transi tions at distinct temperatures A yet different behavior is shown by HoB4 and TmB4 which have a second order phase transition followed by a rst order phase transition at lower temperature The magnetic ordering temperatures primary spin orientations and experimental and theoretical effective CurieWeiss magnetic moments have been collected in Table I The variety of behaviors displayed by these tetraborides suggests a sensitivity to details of the underlying electronic structure Unlike the intense scrutiny that the tetraborides have attracted there has been no thorough study of the tet raboride electronic structure which contains a new structural element the boron dimer and an apical boron that is in equivalent to the equatorial boron in the octahedron We pro vide here a detailed analysis beginning with the reference compound YB4 which allows an analysis of the itinerant bands without the complications of 4f orbitals Then we pro ceed to provide an initial look into the trends to be expected in the 4f shells of the rareearth ions II CRYSTAL STRUCTURE The full RB4 structure was rst reported by Zalkin and Templeton31 for the Ce Th and U members These tetra borides crystallize at room temperature in the tetragonal space group P4 mbm DZ with four formula units occupy ing the positions listed in Table II The lattice constants for the reported rareearth tetraborides are presented in Table III The B1 and B3 atoms form B6 octahedra apical and equatorial vertices respectively that are connected by B2 dimers in the zl 2 plane The B6 octahedra which are arrayed in centered fashion in the xy plane within the cell are attened somewhat with distances from the center being 120 A along the c axis and 129 A in the xy plane taking GdB4 as an example Each B2 atom is bonded to two Bl atoms in separate octahedra and to one other B2 atom A suggestive form for the chemical formula then is R2B2B62 The rareearth atoms lie in the large interstitial holes in the z0 plane and form a twodimensional array that can be regarded as fused squares and rhombuses35 The R site symmetry is mm The symmetry of an R site is important for the magnetic properties of the compounds as it dictates the crystal eld splitting of the ion with total angular momentum J LS and thereby the resulting magnetic state at low temperature The R ion is coordinated by seven B atoms in planes both above and below three of them being dimer B2 atoms two 288 A distant and one at a distance of 308 A and four of them equatorial B3 atoms two each at distances of 276 and 284 A Within the unit cell the four R sites form a square of side d0518a370 A oriented at about 15 with respect to the square sublattice of B6 octahe dra The low site symmetries of the apical Bl dimer B2 and equatorial B3 atoms are 4 mm and m respectively The reported lattice constanm for the lanthanides are plot ted in Fig 2 It is evident that most fall on smooth lines re ecting the lanthanide contraction in this system The be havior is representative of trivalent behavior from La to Lu The big exception is Ce which has smaller volume suggest ing that rather than being simple trivalent the 4f electron is participating in bonding Pm with all unstable isotopes has not been reported EuB4 also has not been reported Eu typi cally prefers the divalent state due to the gain in energy of the half lled 4f shell so it is not surprising that it is differ TABLE II Site designations symmetries and atomic positions of the atoms in the RB4 crystal R 4g mm 1 x 0 BI 48 4 0 0 Z B2 4h mm 1 x B3 8 j m x y 0351352 RAREEARTHiBORON BONDING AND 4F STATE PHYSICAL REVIEW B 77 035135 2008 TABLE III Tabulation of the lattice constants and internal structural parameters used in our calculations The regularity of the internal coordinates through this system is clear and makes the irregularity in 231 for Dy of some concern See the text for discussion R a A C A Xx 231 X32 X33 yBS Ref Y 7111 4017 0318 0203 0087 0176 0039 23 La 7324 4181 0317 0209 0088 0174 0039 22 and 28 Ce 7208 4091 0318 0203 0087 0176 0039 22 and 30 Pr 7235 4116 0318 0203 0087 0176 0039 21 Nd 7220 4102 0318 0203 0087 0176 0039 22 and 32 Pm 7193 4082 0318 0203 0087 0176 0039 Sm 7179 4067 0318 0203 0087 0176 0039 21 Eu 7162 4057 0318 0203 0087 0176 0039 Gd 7146 4048 0317 0203 0087 0176 0038 35 Tb 7120 4042 0317 0202 0087 0176 0039 41 and 43 Dy 7097 4016 0319 0196 0086 0175 0039 21 and 51 Ho 7085 4004 0318 0203 0087 0176 0039 21 Er 7071 4000 0318 0203 0086 0177 0038 43 and 51 Tm 7057 3987 0318 0203 0087 0176 0039 22 Yb 7064 3989 0318 0203 0087 0176 0039 22 Lu 7036 3974 0318 0203 0087 0176 0039 22 ent However some divalent tetraborides do form in this of 243 for more precision The basis set was structure eg CaB4 see Sec IV so it cannot be concluded at EuB4 is unstable simply on the basis of divalency Fi nally the small deviation of Yb from the smooth curves sug gesm that it may be mixed or intermediate valent although close to trivalent III CALCULATIONAL METHODS The full potential local orbital FPLO code57 version 518 was use in our calculations Both LDA PW92 of Perdew and Wang and LDAU using the atomic limit functional are used We used a k mesh of 123 in the full Brillouin zone For the density of states DOS plot and Fermi surface plot we used a k mesh 0 NR Lattice Constants A 701 I I I I La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu FIG 2 Color online Plot of experimental lattice constants of RB4 vs position in the Periodic Table atomic number showing a lanthanide contraction of about 5 for a and 3 for c The smooth lines show a quadratic t to the data ls2s2p3s3p3d4s4p 4d4f535p6s6p5d for all metal el ements except Yls2s2p3s3p3d 4s4p535p4d and Cals2s2p 3s3p 4s4p3d For boron atoms we used the basis ls2s2p3d In the LDA U calculations we used values typical for 4f atoms U 8 eV and J 1 eV corresponding to Slater inte grals F1800 F2ll83 F48l4 and F6586 through out all calculations The high symmetry points in the tetrag onal zone are I 000 X00 M0 Z 4003 R 0 and A The experimental structures Table II were used for our calculations The reported value of zBl0l96 for DyB4 dif fers from the others which all have z31020270203 Using a dense sampling k mesh of 24 X 24 X 24 points in the zone we compared energies for GdB4 and DyB4 The results were for the two values of zBl GdB4 E0203 E0l96 263 meV DyB4 E0203 E0l96 17 meV Hence the Dy compound does display some energetic differ ence from the Gd compound and presumably from the oth ers The difference in the B1 position is no more than 003 A and this amount is not enough to affect appreciably the trends we discuss in this paper Whether this softness in the B1 position is related to the structural transition observed in DyB4 Ref 59 is a question for further studies The crystal structure in Fig l was rendered graphically with the XCRYSDEN softwarequot0 0351353 Z P YIN AND W E PICKETT YB4 band structure PHYSICAL REVIEW B 77 035135 2008 Energy snk eV FIG 3 Band structure of YB4 top panel and CaB4 lower panel within 6 eV of the Fermi 1 X M 1 Z CaB4 band structure Energy snk eV F X M F Z IV GENERAL ELECTRONIC STRUCTURE The valenceconduction band structure of YB4 where there are no 4f bands is shown in Fig 3 For LaB4 which differs in volume and conduction d level position the bands are very similar with only slightly differing Fermi level crossings along the MF direction The occupied valence bandwidth is ll eV not all bands are shown in this gure One striking feature of the bands is the broad gap of more than 3 eV along the top and bottom edges RAR of the Brillouin zone Bands along these lines stick together in pairs due to the nonsymmorphic space group and nearly all bands disperse very weakly with k or ky along these edges This gap closes along the kzzw c plane of the zone only for small in plane components of the wave vectors It is such gaps enclosing E that often account for the stability of a crystal structure and the stability of boride structures including this one has been a topic of interest for decades3v4v61v62 The band structure of a divalent cation member CaB4 is also included in Fig 3 for comparison The largest difference bottom portions of the tetragonal Brillouin zone Notice the lack of dispersion along the upper and lower zone edges RAR kz77c and either kX or ky is wa Note also that due to the nonsym morphic space group bands stick together in pairs along XM the zone equator and along RA top and bottom zone edges is the band lling as expected although some band positions differ in important ways near the Fermi level Still the 3d bands of Ca are not quite empty as a band with substantial 3d character lies at EF at R and is below EF all along the RA line CaB4 can be fairly characterized though as having nearly lled bonding B 2 bands and nearly empty Ca 3d bands A Bonding and antibonding bands As mentioned in the Introduction the stability of the hexaborides is understood in terms of ten bonding molecular orbitals of the B6 octahedron This octahedron occurs also in these tetraborides along with one additional B2 dimer that is bonded only in the layer spz Lipscomb3v4 started from this point and argued that each of the B2 atoms in a dimer forms single bonds with two B3 atoms but a double bond with its dimer neighbor so each B2 atom needs four electrons The total of 208 electrons for each set of 62 boron atoms 0351354 RAREEARTHiBORON BONDING AND 4F STATE leaves a de cit of four electrons or a de cit of 8 electrons in the cell This amount of charge can be supplied by four di valent cations with CaB4 as an example Most tetraborides contain trivalent cations however so this is an issue worth analyzing An empirical extended Huckel band structure study53 for CaB4 indeed gave a gap albeit a very narrow one The Huckel method can be very instructive but is not as accurate as selfconsistent density functional methods Our FPLO cal culation on CaB4 shown in Fig 3 gives a metallic band structure However the valence occupied and conduc tion unoccupied bands Huckel and also FPLO are readily identi ed and it is clear that there are disjoint sets of bands with different characters There are the boron bonding bands at EF and below that can be clearly distinguished from conduction bands at and above EF These conduction bands are primarily metal d bands with an interspersed nonbond ing B2 pl band see below If they were 05 eV higher it would result in an insulating band structure in CaB4 The boron antibonding bands lie higher above 5 eV at least and mix strongly with the metal d bands The separation into bonding and antibonding B 2p bands agrees almost with the ideas of Lipscomb34 and con rms his counting arguments However the existence of numerous R B4 compounds and only one divalent member shows that the extra electron is not a destabilizing in uence while it increases the conduction electron density hence the conduc tivity and magnetic coupling In covalently bonded materials it is common to be able to identify the distinction between the bonding bands and the antibonding bands In covalent semiconductors for example they lie respectively below and above the band gap an ab solutely clean separation In the RB4 system the d bands lie within the corresponding bondingantibonding gap and com plicate the picture Analysis of the orbitalprojected bands clari es this aspect The Bl and B3 atoms being engaged in threedimensional bonding within an octahedron and to an other unit octahedron or dimerj have a clear bonding antibonding splitting of a few eV beginning just below EF Likewise the dimer B2 jumpy states display a similar split ting The B2 pl orbital is quite different As is the case in Mng whose planar structure is similar to the local arrangement of a B2 atom p Z bands extend continuously through the gap in the B bonding andor antibonding bands and mix fairly strongly with the rareearth d states in that region There is considerable B2 pl character in the bands near both below and above EF at the zone edge M point as well as the Y 4d character that is evident in Fig 3 So while there is some Bl and B3 characters in the rareearth metal d bands that lie within the boron bondingantibonding gap the amount of B2 pl character is the primary type of B participation in these bands that provide conduction and magnetic coupling B Pseudong in the density of states From the projected DOS of the three types of B atoms of YB4 see Fig 4 one can detect only relatively small differ ences in the distribution of Bl B2 and B3 characters arising PHYSICAL REVIEW B 77 035135 2008 E E 12 a 3 m t g t MI yr 0 a I I I I gt4 Iquot I I I I I 6 712 78 4 4 8 12 EVEFeV FIG 4 Color online Projected density of states per atom of each of the B atoms for YB4 The curves are shifted to enable easier identi cation of the differences The B 2p bondingantibonding gap can be identi ed as roughly from 1 eV to 475 eV from their differing environmenm First note that in the DOS of Bl and B3 there is a peak around 15 eV while there is no such peak for B2 This peak arises from the overlap of 2s and 2170 states of each of the boron atoms forming the B6 octahedra B1 and B3 the 2s character is about three times as large as the 2p7 character and the remaining 23 character is mixed into the lower 2p bands This state is a well local ized B6 cluster orbital and there are two such orbitals octa hedral clusters per cell The bridging B2 atoms do not par ticipate in any such bound state Another difference in characters of the B sites is that in the region below but within 2 eV of the Fermi level the DOS of the dimer B2 atom is signi cantly larger than that of Bl and B3 atoms as can be seen in Fig 5 Together with plots showing the band character not shown this difference re ects the fact that all of the 2p orbitals of Bl and B3 octahedron atoms are incorporated into bonding lled and antibonding empty bands The distinct characteristic of the 0 0 Ln m 0 4 0 N YB 4 PDOS stateseV per atom cgt o H w FIG 5 Color online Enlargement of the partial densities of states of Y 4d and B 2p states per atom near the Fermi level The states at the Fermi level and even for almost 2 eV below have strong 4d character The apical B2 character is considerably larger than that of B1 or B3 in the two peaks below EF but is only marginally larger exactly at E F 0351355 Z P YIN AND W E PICKETT B2 pz state was discussed in the previous subsection All B 2p states do hybridize to some degree with the metal d bands however and all B atoms have some contribution at the Fermi level The full Y 4d DOS not shown establishes that these bands are centered about 4 eV above E F with a bandwidth full width at half maximum of 6 7 eV a full bandwidth would be somewhat larger The largest Y character near E F along symmetry lines is 4dx2 y2 primarily in the bands dispersing up from 05 eV at Z toward F The atbands around 1 eV along FX MF are strongly 4dz2 character indicative of a nonbonding almost localized state in the xy plane Note that these bands disperse strongly upward along 0 0kz and lie 3 4 eV above EF in the kz 7TC plane Thus the 4dz2 orbitals form two nearly separate onedimensional bands along kz and give rise to at parts of some Fermi surfaces see following subsection These bands can be modeled by a tightbinding band tdd0 cos kzc with hopping amplitude tddawl eV Most of the 4dxz 4dyz character and 4dxy character lie above E F and are centered 3 4 eV above E F The B2 2pz state mixes primarily with Y 4dxz 4dyz near the M point near EF and above The B2 2pz orbitals are shifted up somewhat with respect to the 2px and 2py states by the ligand eld effects there is a bonding interac tion within the xy plane only C Fermi surface The Fermi surfaces of YB4 shown in Fig 6 will be rep resentative of those of the trivalent RB 4 compounds although small differences may occur due to elementspeci c chemis try of trivalent rare earths and due to the lanthanide contrac tion The large gap along the RAR edges precludes any Fermi surface on or near most of the k5 face The Fermi surfaces can be pictured as follows Square hole pyramids with only slightly rounded vertices lie midway along the FZ line and similar nested electron pyramids lie along the M A line near the M point A pointed ellipsoid football oriented along kz sits at the Z point Surrounding F is a lenstype electron surface joined to pointed ellipsoids along the 110 directions Finally there are two tortoise shell shaped hole surfaces within the zone located along the FZ lines These surfaces and the small variation through the lan thanide series are surely relevant to the varying magnetic behavior observed in RB4 compounds There are nesting pos sibilities between the bases of the square pyramids for ex ample which will appear as RKKY coupling as the associ ated nesting vectors The ellipsoidal attachments on the zonecentered lens surface may provide some weak nesting V LANTHANIDE SERIES Any effective oneelectron treatment of the electronic structure of 4f electron systems faces serious challenges The root of the dif culty is that the ground state of an open 4f shell has intrinsic manybody character being characterized by the spin S and angular momentum L of all of the 4f electrons and the resulting total angular momentum J fol lowing Hund s rules Although it is possible to delve into the PHYSICAL REVIEW B 77 035135 2008 FIG 6 Color online Fermi surfaces of YB4 Light yellow surfaces enclose holes and dark red surfaces enclose electrons The full tetragonal Brillouin zone is pictured the F point being in the center of each gure the R point is the midpoint along the horizontal edges and the A point lies at the corner see Sec III for speci cation of high symmetry points The wide gap throughout the top and bottom edges of the zone accounts for the lack of Fermi surfaces except for the one football centered at the Z point at the center of the upper and lower faces lower left panel extent to which the LDAU method can reproduce the z components of such con gurations63 that is not the intention here LDA U reliably gets the high spin aspect which con tains much of the physics that determines relative 4f level positions and hence trends across the series There is recent evidence from calculations on rareearth nitrides64 that if spinorbit coupling is neglected and the symmetry is lowered appropriately the high orbital moment Hund s second rule can usually be reproduced The exceptions are the usual dif cult and interesting cases of Eu and Yb Hund s rule ground state of the ion often breaks the local symmetry of the site and if one is exploring that aspect the site symmetry should be allowed to be lower than the crys talline symmetry As stated we are not interested here in those details In the calculations reported here the crystal symmetry is retained The site symmetry of the lanthanide ion is already low mm re ected in its 14fold coordination with B atoms In addition spinorbit coupling has not been included 035135 6 RAREEARTHiBORON BONDING AND 4F STATE DyB4 Energy cnk eV A Band structure Most of the RB4 lanthanide tetraborides follow the usual trivalent nature of these ions and the itinerant parm of their band structures are very similar to those of YB4 and LaB4 The exceptions are REu and Yb which tend to be divalent to achieve a half lled or lled shell respectively By way of illustration of the complexity of the full RB4 bands the full band structure of DyB4 is presented in Fig 7 for ferromagnetic ordering The 4f bands themselves can be identi ed by their at weakly hybridizing nature An en larged picture of the bands within 1 eV of EF is given in Fig 8 The splitting of the majority and minority itinerant bands provides a direct measure of the Kondo coupling of the 4f moment to the band states Note that the sign of this splitting can vary from band to band Figure 8 suggests that the Fermi surfaces will be different in the magnetic tetraborides compared to YB4 in speci c ways For Dy the Fcentered surface splits almost impercep PHYSICAL REVIEW B 77 035135 2008 FIG 7 The full valence band structure of DyB4 and up to 5 eV in the conduction bands This plot is for ferromagnetic alignment of the spin moments with the solid bands being major ity and the lighter dashed lines showing the mi nority bands The atbands in the 45 to 11 eV are 4f eigenvalues as described by the LDA U method tibly The surfaces that cross the FZ line also are relatively unaffected by exchange splitting At the M point however a new surface appears due to the magnetism an electron sur face of minority spin For this band the polarization is op posite to the direction of the Dy spins This gure is speci cally for ferromagnetic alignment while DyB4 actually orders antiferromagnetically see Sec I B Position of 4f levels The mean position of 4f levels is displayed in Fig 9 separated into occupied and unoccupied and majority and minority and trends are more meaningful than absolute en ergies Simple ferromagnetic alignment is used here in order to follow the chemical trends in the simplest manner For the occupied majority states the 4f level drops rapidly from Pr 3 eV to Sm 7 eV then becomes almost at for GdTm around 8 eV For the unoccupied minority states the mean 4f level drops almost linearly from Pr 5 eV to Er eV FIG 8 Band structure of DyB4 on a ne scale around the Fermi ener see Fig 7 The ex change splitting between solid and dashed Energy cnk bands gives a direct measure of the coupling be tween the polarized Dy ion and the itinerant bands see text 0351357 Z P YIN AND W E PICKETT Energy eV 72 4 K 76 8 391 m PHYSICAL REVIEW B 77 035135 2008 VI SUMMARY In this paper we have provided an analysis of the elec tronic structure of trivalent tetraborides using YB4 as the reference compound and compared this with a divalent member aB4 In agreement with earlier observations on the likely bonding orbitals in the B atoms it is found that bond ing states are nearly lled and antibonding states are empty The states at the Fermi level in the trivalent compounds are a combination of the dimer B2 p Z nonbonding orbitals whose bands pass through the bondingantibonding gap and the cation d orbitals Since the extra electron in the trivalent 1 Pr Nd le Sm 13qu 2b rpr Er Tm Y 2 4 5 8 10 12 FIG 9 Color online Calculated mean 4f eigenvalue position symbols connected by lines with respect to EF and the spread in eigenvalues of RB4 compounds The smooth behavior from Pr to Tm except for Eu re ects the common trivalent state of these ions Eu and Yb are calculated to be divalent and deviate strongly from the trivalent trend Ce has a higher valence than 3 accounting for its deviation from the trivalent trend 2 eV and for Tm the 4f level is very close to EF The unoccupied majority levels which become occupied minor ity levels beyond the middle of the series drop more steeply with slope almost 1 eV per unit increase in nuclear charge There are the usual exceptions to these overall trends Ce is very different indicating that it is very atypical the cal culational result is tetravalent and nonmagnetic Both Eu and Yb are divalent in the calculation an extra 4f state is occupied so their mean 4f level position is 6 eV 8 eV for Yb higher than the trivalent line The spread in 4f eigenvalues is also displayed in Fig 9 This spread is sensitive to the speci c con guration that is obtained and also has no direct relation to spectroscopic data although it does re ect some of the internal potential shifts occurring in the LDA U method The distinctive fea tures are the unusually large spread for the occupied majority levels in Dy two electrons past half lled shell and for the unoccupied minority and also unoccupied majority levels in Pr two electrons above the empty shell r does not go into an antibonding state there is no signi cant destabilization of the crystal structure The trends in the energy positions of the 4f states in the rareearth tetraborides have been found to be consistent with expectations based on other rareearth systems as is the fact that Eu and Yb tend to be divalent rather than trivalent In vestigations of the magnetic behavior of rareearth tetra borides will require individual study Nearest neighbor mag netic interactions may involve a combination of 4f4d 2pZ4d4f interactions and longer range RKKY inter actions that may bring in the Fermi surface geometry An other possible coupling path is the direct 4f2pZ4f path The coupling is likely to be even more complicated than in the rocksalt EuO and Eu chalcogenides where competition be tween direct and indirect magnetic coupling paths has re ceived recent attention65 The tetraboride structure is fasci ting in several respects A relevant one if coupling does proceed directly via 4f2pZ4f is that the dimer B2 atom coordinates with three neighboring rareearth ions which will introduce frustration when the interaction has antiferro magnetic sign ACKNOWLEDGMENTS We have bene ted from discussion of the calculations with D Kasinathan K Koepernik and M Richter and from communication about data on DyB4 with E Choi Support from the Alexander von Humboldt Foundation and the hos pitality of IFW Dresden during the early part of this work is gratefully acknowledged This work was supported by Na tional Science Foundation Grant No DMR0421810 1H Nagamatsu N Hakagawa T Muranaka Y Zenitani and J Akimitsu Nature London 410 63 2001 2J M An and W E Pickett Phys Rev Lett 86 4366 2001 3W N Lipscomb J Chem Phys 33 275 1960 4W N Lipscomb J LessCommon Met 82 1 1981 5Y Imai M Mukaida M Ueda and A Watanabe Intermetallics 9 721 2001 6M Takeda T Fukuda F Domingo and T Miura J Solid State Chem 177 471 2004 7CH Chen T Aizawa N Iyi A Sato and S Otani J Alloys Compd 366 L6 2004 8L W Rupp Jr and P H Schmidt J Phys Chem Solids 30 1059 1969 9S Muranaka and S Kawai J Cryst Growth 26 165 1974 10J L Gavilano B Ambrosini H R Ott D P Young and Z Fisk Physica B 281amp282 428 2000 11 J L Gavilano B Ambrosini H R Ott D P Young and Z Fisk Physica B 284288 1359 2000 12K Schmitt C Stl39i39ckl H Ripplinger and B Albert Solid State Sci 3 321 2001 13S Otani T Aizawa and Y Yajima J Cryst Growth 234 431 2002 14CH Chen Y Xuan and S Otani J Alloys Compd 350 L4 2003 0351358 RAREEARTHiBORON BONDING AND 4F STATE 15 P Teredesai D V S Muthu N Chandrabhas S Meenakshi V Vijayakumar P Modak R S Rao B K Godwal S K Sikka and A K Sood Solid State Commun 129 91 2004 16SK Mo GH Gweon J D Denlinger HD Kim J W Allen C G Olson H Hochst J L Sarrao and Z Fisk Physica B 312313 668 2002 17P G Perkins and A V J Sweeney J LessCommon Met 47 165 1976 123A Ammar M Menetrier A Villesuzanne S Matar B Cheva lier J Etourneau G Villeneuve J RodgiruezCarvajal HJ Koo A I Smirnov and MH Whangbo Inorg Chem 43 4974 2004 19Z Fisk and M B Maple Solid State Commun 39 1189 1981 20J Etourneau J LessCommon Met 110 267 1985 21J Etourneau J P Mercurio A Berrada and P Hagenmuller J LessCommon Met 67 531 1979 22Z Fisk A S Cooper P H Schmidt and R N Castellano Mater Res Bull 7 285 1972 23B Jager S Paluch W Wolf P Herzig O J Zogal N Shit sevalova and Y Paderno J Alloys Compd 383 232 2004 24S Otani Y Xuan Y Yajima and T Mori J Alloys Compd 361 L1 2003 25S Otani M M Korsukova T Mitsuhashi and N Kieda J Cryst Growth 217 378 2000 26T Tanaka and Yoshio Ishizawa J Phys C 18 4933 1985 27J Gu39nster T Tanaka and R Souda Phys Rev B 56 15962 1997 28K Kato I Kawada C Oshima and S Kawai Acta Crystallogr Sect B Struct Crystallogr Cryst Chem B30 2933 1974 29S V Meschel and O J Kleppa J Alloys Compd 226 243 1995 30A Zalkin and D H Templeton Acta Crystallogr 6 269 1953 31A Zalkin and D H Templeton J Chem Phys 18 391 1950 32P Salamakha A P Goncalves O Sologub and M Almeida J Alloys Compd 316 L4 2001 33B J Mean K H Kang J H Kim I N Hyun M Lee and B K Cho Physica B 378380 598 2006 34S Ji C Song J Koo KB Lee Y J Park J Y Kim JH Park H J Shin J S Rhyee B H Oh and B K Cho Phys Rev Lett 91 257205 2003 35M T Garland J P Wiff J Bauer R Guerin and JY Saillard Solid State SCI 5 705 2003 K H J Buschow and J H N Creighton J Chem Phys 57 3910 1972 37S W Lovesey J Fernandez Rodrguez J A Blanco and P J Brown Phys Rev B 70 172414 2004 33 J A Blanco P J Brown A Stunault K Katsumata F Iga and S Michimura Phys Rev B 73 212411 2006 39B J Mean K H Kang J H Kim I N Hyun M Lee and B K Cho Physica B 378380 600 2006 4OJ C Gianduzzo R Georges B Chevalier J Etourneau P PHYSICAL REVIEW B 77 035135 2008 Hagenmuller G Will and W Schafer J LessCommon Met 82 29 1981 41F Elf W Schafer and G Will Solid State Commun 40 579 1981 42Z Heiba W Schafer E Jansen and G Will J Phys Chem Solids 47 651 1986 43G Will W Schafer F Pfeiffer and F Elf J LessCommon Met 82 349 1981 44G Will Z Heiba W Schafer and E Jansen Ann Isr Phys Soc 140 130 1986 45D Okuyama T Matsumura H Nakao and Y Murakami J Phys Soc Jpn 74 2434 2005 46R Watanuki H Mitamura T Sakakibara G Sato and K Su zuki Physica B 378380 594 2006 47T Matsumura D Okuyama H Nakao and Y Murakami Photon Factory Activity Report 2004 No 24 Part B 2005 unpub lished 423E M Dudnik Y A Kochierzhinsky Y B Paderno N Y Shit sevalova E A Shishkin and I E Kir yakova J LessCommon Met 67281 1979 49W Schafer G Will and K H J Buschow J Magn Magn Mater 3 61 1976 50R Watanuki G Sato K Suzuki M Ishihara T Yanagisawa Y Nemoto and T Goto J Phys Soc Jpn 74 2169 2005 51G Will and W Schafer J LessCommon Met 67 31 1979 52L I Derkachenko V N Gurin and M M Korsukova J Solid State Chem 133 296 1997 53R Schmitt B Blaschkowski K Eichele and H J Meyer Inorg Chem 453067 2006 54D R Noakes G K Shenoy D Niarchos A M Umarji and A T Aldred Phys Rev B 27 4317 1983 55P C Can eld S L Bud ko B K Cho W P Beyennann and A Yatskar J Alloys Compd 250 596 1997 56M A Rudennan and C Kittel Phys Rev 96 99 1954 T Kasuya Prog Theor Phys 16 45 1956 K Yosida Phys Rev 106 893 1957 57K Koepernik and H Eschrig Phys Rev B 59 1743 1999 53J P PerdeW and Y Wang Phys Rev B 45 13244 1992 59D Okuyama T Matsumura H Nakao and Y Murakami J Phys Soc Jpn 75 198 2006 60A Kokalj Comput Mater Sci 28 155 2003 The code is avail able from httpWWWxcrysdenor 61T Lundstrom Pure Appl Chem 57 1383 1985 62A Vegas L A MartinezCruz A RamosGallardo and A Romero Z Kristallogr 210 574 1995 63M D Johannes and W E Pickett Phys Rev B 72 195116 2005 64P Larson W R L Lambrecht A Chantis and M van Schilf gaarde Phys Rev B 75 045114 2007 65J Kune and W E Pickett J Phys Soc Jpn 74 1408 2005 0351359 Up From the Holler Living in Two Worlds at Home in Neither By TAMAR LEWIN The New York Times May 19 2005 PIKEVILLE Ky Della Mae Justice stands before the jury in the Pike County Courthouse arguing that her client s land in Greasy Creek Hollow was illegally grabbed when the neighbors expanded their cemetery behind her home With her soft Appalachian accent Ms Justice leaves no doubt that she is a local girl steeped in the culture of the old family cemeteries that dot the mountains here in East Kentucky quotI grew up in a holler I surely didquot she tells jurors as she lays out the boundary con ict Ms Justice is indeed a product of the Appalachian coalmining country where lush mountains ank rustcolored creeks the hollows rising so steeply that there is barely room for a house on either side of the creeks Her family was poor living for several years in a house without indoor plumbing Her father was absent her older halfbrother sometimes had to hunt squirrels for the family to eat Her mother malried again when Della was 9 But the stepfather a truck driver was frequently on the road and her mother who was mentally ill often needed the young Della to care for her Ms Justice was always hungry for a taste of the world beyond the mountains Right after high school she left Pike County making her way through college and law school spending time in France Scotland and Ireland and beginning a highpowered legal career In just a few years she moved up the ladder from rural poverty to the highachieving circles of the middle class Now at 34 she is back home But her journey has transformed her so thoroughly that she no longer fits in easily Her change in status has left Ms Justice a little off balance seeing the world from two vantage points at the same time the one she grew up in and the one she occupies now Far more than people who remain in the social class they are born to surrounded by others of the same background Ms Justice is sensitive to the cultural significance of the cars people drive the food they serve at parties where they go on vacation all the little clues that indicate social status By every conventional measure Ms Justice is now solidly middle class but she is still trying to learn how to feel middle class Almost every time she expresses an idea or explains herself she checks whether she is being understood asking quotDoes that make sensequot quotI think class is everything I really doquot she said recently quotWhen you re poor and from a low socioeconomic group you don t have a lot of choices in life To me being from an upper class is all about confidence It39s knowing you have choices knowing you set the standards knowing you have connections quot Broken Ties In Pikeville the site of the HatfieldMcCoy feud Ms Justice is a Hatfield memories are long and family roots mean a lot Despite her success Ms Justice worries about what people might remember about her especially about the time when she was 15 and her life with her mother and stepfather imploded in violence sending her into foster care for a wretched nine months quotI was always in the lowest socioeconomic groupquot she said quotbut foster care ratcheted it down another notch I hate that period of my life when for nine months I was a child with no familyquot While she was in foster care Ms Justice lived in one end of a doublewide trailer with the foster family on the other end She slept alongside another foster child who wet the bed and every morning she chose her clothes from a box of handmedowns She was finally rescued when her father heard about her situation and called his nephew Joe Justice Joe Justice was 35 years older than Della a successful lawyer who lived in the other Pikeville one of the welltodo neighborhoods on the mountain ridges He and his wife Virginia had just built a fourbedroom contemporary home complete with a swimming pool on Cedar Gap Ridge Joe Justice had never even met his cousin until he saw her in the trailer but afterward he told his wife that it was quotabhorrentquot for a close relative to be in foster care While poverty is common around Pikeville foster care is something much worse a sundering of the family ties that count for so much So Joe and Virginia Justice took Della Mae in She changed schools changed address changed worlds in effect and moved into an octagonal bedroom downstairs from the Justices 2 yearold son quotThe shock of going to live in wealth with Joe and Virginia it was like Little Orphan Annie going to live with the Rockefellersquot Ms Justice said quotIt was not easy Iwas shy and socially inept For the first time I could have had the right clothes but I didn39t have any idea what the right clothes were I didn t know much about the world and I was always afraid of making a wrong move When we had a school trip for chorus we went to a restaurant I ordered a club sandwich but when it came with those toothpicks on either end I didn39t know how to eat it so I just sat there staring at it and starving and said I didn t feel wellquot Joe and Virginia Justice worried about Della Mae39s social unease and her failure to mingle with other young people in their church But they quickly sensed her intelligence and encouraged her to attend Berea College a small liberal arts institution in Kentucky that accepts students only from low income families Tuition is free and everybody works For Ms Justice as for many other Berea students the experience of being one among many poor people all academically capable and encouraged to pursue big dreams was lifealtering It was at Berea that Ms Justice met the man who became her husband Troy Price the son of a tobacco farmer with a sixthgrade education They married after graduation and when Ms Justice won a fellowship the couple went to Europe for a year of independent travel and study When Ms Justice won a scholarship to the Universitv of Kentuckv law school in Lexington Mr Price went with her to graduate school in family studies After graduating fth in her law school class Ms Justice clerked for a federal judge thenjoined Lexington39s largest law firm where she put in long hours in hopes of making partner She and her husband bought a townhouse took trips ate in restaurants almost every night and spent many Sunday afternoons at real estate open houses in Lexington39s elegant older neighborhoods By all appearances they were on the fast track But Ms Justice still felt like an outsider Her coeditors on the law review her fellow clerks at the court and her colleagues at the law rm all seemed to have a universe of information that had passed her by She saw it in matters big and small the casual references to Che Guevara or Mount Vesuvius that meant nothing to her the food at dinner parties that she would not eat because it looked raw in the middle quotI couldn39t play Trivial Pursuit because I had no general knowledge of the worldquot she said quotAnd while I knew East Kentucky they all knew a whole lot about Massachusetts and the Northeast They all knew who was important whose father was a federal judge They never doubted that they had the right thing to say They never worried about anythingquot Most of all they all had connections that fed into a huge web of people with power quotSomehow they all just knew each otherquot she said Knitting a New Family Ms Justice s life took an abrupt turn in 1999 when her halfbrother back in Pike County called out of the blue to say that his children Will and Anna Ratliff who had been living with their mother were in foster care Ms Justice and her brother had not been close and she had met the children only once or twice but the call was impossible to ignore As her cousin Joe had years earlier she found it intolerable to think of her esh and blood in foster care So over the next year Della Mae Justice and her husband got custody of both children and went back to Pikeville only 150 miles away but far removed from their life in Lexington The move made all kinds of sense Will and Anna now 13 and 12 could stay in touch with their mother and father Mr Price got a better job as executive director of Pikeville s new support center for abused children Ms Justice went to work for her cousin at his law rm where a exible schedule allowed her to look after the two children And yet for Ms Justice the return to Pikeville has been almost as dislocating as moving out of foster care and into that octagonal bedroom all those years ago On a rare visit recently to the hollows where she used to live she was moved to tears when a neighbor came out hugged her and told her how he used to pray and worry for her and how happy he was that she had done so well But mostly she winces when reminded of her past quotLast week I picked up the phone in my officequot she recalled quotand the woman said who she was and then said You don t remember me do you And I said Were you in foster care with me That was crazy Why would I do that It39s not something I advertise that I was in carequot While most of her workweek is devoted to commercial law Ms Justice spends Mondays in family court representing families with the kind of problems hers had She bristles whenever she runs into any hint of class bias or the presumption that poor people in homes heated by kerosene or without enough bedrooms cannot be good parents quotThe norm is people that are born with money have money and people who weren39t don39tquot she said recently quotI know that I know that just to climb the three inches I have which I ve not gone very far took all of my effort I have worked hard since I was a kid and I ve done nothing but work to try and pull myself ou quot The class a person is born into she said is the starting point on the continuum quotIf your goal is to become on a national scale a very important person you can t start way back on the continuum because you have too much to make up in one lifetime You have to make up the distance you can in your lifetime so that your kids can then make up the distance in their lifetimequot Coming to Terms With Life Ms Justice is still not fully at ease in the other welltodo Pikeville and in many ways she and her husband had to start from scratch in finding a niche there Church is where most people in town find friends and build their social life But Ms Justice and Mr Price had trouble finding a church that was a comfortable fit they went through five congregations starting at the Baptist church she had attended as a child and ending up at the Disciples of Christ an inclusive liberal church with many af uent members The pastor and his wife transplants to Kentucky have become their closest friends Others have come more slowly quotPartly the problem is that we re young for middleclass people to have kids as old as Will and Annaquot Ms Justice said quotAnd the fact that we re raising a niece and nephew that s kind of a ag that we weren39t always middle class just like saying you went to Berea College tells everyone you were poorquot And though in terms of her work Ms Justice is now one of Pikeville39s leading citizens she is still troubled by the old doubts and insecurities quotMy stomach s always in knots getting ready to go to a party wondering if I m wearing the right thing if I ll know what to doquot she said quotI39m always thinking How does everybody else know that How do they know how to act Why do they all seem so at easequot A lot of her energy now goes into Will and Anna She wants to bring them up to have the middle class ease that still eludes her quotWill and Anna know what it39s like to be poor and now we want them to be able to be just regular kidsquot she said quotWhen I was young I always knew who were the kids at school with the involved parents that brought in the cookies and those were the kids who got chosen for every special thing not ones like me who got free lunch and had to borrow clothes from their aunt if there was a chorus performancequot Because Ms Justice is selfconscious about her teeth quotthe East Kentucky overbitequot she says ruefully she made sure early on that Anna got braces She worries about the children39s clothes as Sociology 140 Social Strati cation Dr Shu Review Practice Questions for Midterm N E 4 V39 0 9 Compare and contrast the Marxian Weberian and the Structural Functionalism s perspectives on social class 0 What do they say about social class 0 What do they agree 0 What do they disagree Change social class to power in question 1 What are the implications of income distribution on the American class system 0 Describe the pattern of income distribution in the United State 0 What are the social forces that gave rise to such a pattern 0 What are the implications for the American class structure Use the supplyside and demandside explanations of income inequality to analyze gender based income gap 0 Describe the amount of gende based income inequality 0 What are the supplyside and demandside explanations for such a gap 0 How do these explanations account for genderbased inequalities in income Change gender to race and ethnicity in question 4 Compare and contrast the four theoretical perspectives on poverty we discussed 0 Describe these four perspectives o In what ways are these perspectives similar 0 In what ways are these perspectives different Are women consistently in disadvantageous positions in all three dimensions of social distribution income prestige and power 0 Describe gender differences in income prestige and power 0 Point out whether women are ALWAYS in disadvantageous positions 0 Brie y explain the reasons for such a pattern Change women to racial minorities in question 7


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