Physica C 476 (2012) 54–58

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Physica C

journal homepage: www.elsevier .com/locate /physc

Electronic structure and properties of BaAIGe and SrAlGe

S.J. Youn a,b,⇑, A.J. Freeman b

a Department of Physics Education and Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Koreab Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208-3112, USA

a r t i c l e i n f o

Article history:Received 6 February 2012Accepted 15 February 2012Available online 23 February 2012

Keywords:BaAlGeSrAlGeElectronic structureFermi surface

0921-4534/$ - see front matter � 2012 Elsevier B.V. Adoi:10.1016/j.physc.2012.02.017

⇑ Corresponding author at: Department of PhysInstitute of Natural Science, Gyeongsang NationaRepublic of Korea.

E-mail address: ysj@gnu.ac.kr (S.J. Youn).

a b s t r a c t

The electronic structures of BaAlGe and SrAlGe which are superconductors with hexagonal honeycomblayers have been studied by using a first principles method. Energy bands, Fermi surfaces, and densityof states are presented. The two materials have topologically different Fermi surfaces. BaAlGe has twoFermi surfaces: one has a three dimensional spinning-top-like shape and the other has a cylindrical shapewith two dimensional character. SrAlGe has only one connected Fermi surface. Two gap superconductiv-ity for BaAlGe is suggested from the inherently different character of the two Fermi surfaces. The higher Tc

of SrAlGe than BaAlGe is related to the difference in both the topology of the Fermi surface and the banddispersions along the z direction.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The discovery of superconductivity in MgB2 with Tc = 39 K [1]has attracted continuing attention due to interests in both funda-mental research [2] and practical applications [3]. It crystallizesin the AlB2 type structure with a space group of P6/mmm

D16h;No: 191

� �[4]. Numerous efforts to find a novel superconduc-

tor with higher Tc in the AlB2 family have been unsuccessful so far.There are eight valence electrons in a unit cell which belong to twogroups according to the symmetry of wave functions: r bands orig-inate from B s, px,y orbitals and form a strong honeycomb networkof boron atoms while p bands originate from B pz orbitals and showthree dimensional conductivity. Electron–phonon couplingbetween E2g in-plane phonon mode and in-plane r electrons in aboron layer is the main superconductivity mechanism [5]. Due tothe two inherently different bands, MgB2 has two different super-conducting energy gaps [6,7].

Related to this, studies of superconductivity in the ternary sili-cides, MASi (M = Ca, Sr, Ba, A = Al, Ga), were triggered by CaSi2

which becomes a superconductor (Tc = 14 K) at high pressureabove 16 GPa, while it is a semimetal at ambient pressure [8]. Itundergoes a phase transition with pressure from a trigonal struc-ture with corrugated Si layers to the AlB2 structure. MASi has ninevalence electrons with one more electron than MgB2 [9]. CaAlSi hasthe highest Tc = 7.8 K among the ternary silicides [10]. However, it

ll rights reserved.

ics Education and Researchl University, Jinju 660-701,

was found that the structure consists of 6 slabs, 6H–CaAlSi, whilethe Tc of one slab, 1H–CaAlSi, is 6.5 K [11]. The ternary silicidesare metallic with one partially filled p⁄ band originated from anti-bonding pz orbitals while the r bands are fully occupied [12]; theyexhibit electron-like conductivity which is different from the hole-like conductivity of MgB2 [13]. The d electrons in M metal atoms,for example, of Ca 3dxy and dz2 origin, play an important role whichis different from MgB2 with only s and p electrons [14,15]; M metalatoms also play an active role in determining the lattice dynamics[12]. It is reported that the electron–phonon interaction mediatedby a soft phonon mode vibrating along the z-direction gives rise toBCS type superconductivity [16].

Recently ternary superconductors in the AlB2 family wereextended to include germanides with nine valence electrons [17].Among them, SrAlGe (Tc = 6.7 K) and BaAlGe (Tc = 6.3 K) are thetwo highest Tc superconductors at ambient pressure. Both SrAlGeand BaAlGe crystallize in a hexagonal structure of BaPtSb type withspace group of P�6m2 D1

3h;No: 187� �

, as depicted in Fig. 1. The crys-tal structure is similar to that of MgB2 where the boron layers arereplaced by AlGe layers. There is a horizontal mirror symmetry rh

without inversion symmetry. Due to the rh symmetry, wave func-tions can have either even (r band) or odd parity (p band) withrespect to rh.

The exact understanding of the electronic structure is crucial tounderstand the mechanism of superconductivity. Especially thecharacter of the Fermi surface is important in determining thesuperconducting properties. We report in this paper the electronicstructure of SrAlGe and BaAlGe by using a first principles method.Energy bands, Fermi surfaces, and densities of states will be pre-sented in Section 3. The different character between BaAlGe andSrAlGe will be addressed.

K

H

L

M

A

Γ

(a) (b)Fig. 1. (a) Crystal structure and (b) hexagonal Brillouin zone of MAlGe. The M site in(a) represents a site for Ba or Sr. In (b), symbols are marked only around theirreducible Brillouin zone.

S.J. Youn, A.J. Freeman / Physica C 476 (2012) 54–58 55

2. Method

First-principles calculations were performed using the full-potential linearized augmented plane wave (FLAPW) method[18,19] in the local density approximation (LDA) for theexchange–correlation functional by Hedin and Lundqvist [20].Experimental lattice constants, a = 4.3043 Å and c = 4.7407 Å forSrAlGe and a = 4.3512 Å and c = 5.1401 Å for BaAlGe, are employed[17]. Kmax = 3.7 and lmax = 8 were used for both materials.

Muffin-tin radii are taken to be 2.6, 2.4 and 2.2 a.u for Ba(Sr), Al,and Ge, respectively. Semicore electrons such as Sr 4p and Ge 3dare treated as valence electrons, which are explicitly orthogonal-ized to the core states [21]. Although there is no inversion symme-try, time reversal symmetry plus the crystal symmetry reduce thevolume integration in k space to 1/24 of the whole Brillouin zone[22]. Summations in the Brillouin zone are done with a 11 �11 � 11 mesh in the Monkhorst–Pack scheme during the self-consistent cycles [23], while the density of states is obtained bythe tetrahedron method with a 15 � 15 � 15 mesh [24]. In orderto calculate Fermi velocity and plasma frequency at the Fermi level,eigenvalues from the self-consistent calculations are fitted by aspline method over the whole Brillouin zone [25].

3. Results

The electronic structure of BaAlGe is presented first since it issimilar to the known electronic structure of isoelectronic ternarysilicides [12]. Then results for SrAlGe are presented with emphasison the difference between the two materials.

3.1. BaAlGe

Fig. 2a depicts the energy bands along the symmetry lines of thehexagonal Brillouin zone shown in Fig. 1b. The first band around�9 eV comes from mostly the Ge 4s orbital. The 4.8 eV wide bandfrom �5.8 eV to �1 eV is the r hybridized band between Al s, px,y

and Ge px,y orbitals where the minimum is at A and the maximumis at H. The r band width of MgB2 is about 8 eV, which is muchwider than that in BaAlGe. The narrower band width of BaAlGecan be ascribed to weak covalent bonding between Al and Ge sincestrong ionic bonding between Al and Ge is expected due to theirlarge difference in electronegativity [26].

Fig. 2b exhibits band components around the Fermi level. Onlythe fifth band crosses the Fermi level (EF) while the p band ofmostly Ge pz origin is fully occupied, which is another manifesta-tion of the large electronegativity of Ge. The fifth band along HKis of Al pz origin and displays little dispersion with a width of0.10 eV which gives rise to a peak in the density of states in

Fig. 4. The narrow band width in the z direction is attributed tothe weak inter-layer coupling of the Al pz orbitals in BaAlGe. Thefifth band exhibits different character depending on the locationin the Brillouin zone; Sr dxy character is found along H–A–L whileSr dz2 and s character are found along K–C–M. Around the A point,p and p⁄ bands show parabolic behavior; the p bonding band isopen upward while the p⁄ antibonding band is open downward,as shown in Fig. 2b. Although the p⁄ band also shows antibondingparabolic behavior around C, the p band changes to a weakbonding character at C. One can clearly see the p–p⁄ splitting alongL–H–K in Fig. 2b because the AlGe honeycomb layer is made ofdifferent atoms. When the honeycomb layer is made of the sameatoms like MgB2, the p and p⁄ bands are degenerate along HK.

There are two separate Fermi surfaces as shown in Fig. 3: anelectron-like inner pocket centered at C with the shape of a spin-ning top and the hole-like cylindrical outer sheet. The shapes of theFermi surface can be seen easily in the cross-section of Fig. 3d. Asshown in Fig. 2b, the cylindrical sheet originates from Ba dxy and Alpz orbitals, while the electron pocket is from both the Ba dz2 orbitaland s orbitals (Ba s, Al s, Ge s). Since the symmetry of Ba dz2 is dif-ferent from p orbitals [27], it is nonbonding with other p orbitals.Thus the two Fermi surfaces can be called antibonding and non-bonding for the pocket and cylinder Fermi surfaces, respectively,from the character of component orbitals. Cylindrical Fermi sur-faces are connected to other cylindrical Fermi surfaces in neighbor-ing Brillouin zones by 12 bridges around the K points. A similartopology of Fermi surfaces is also found in ternary silicides[15,28]. The two disconnected Fermi surfaces are also manifestedin the energy band of Fig. 2 where the fifth band crosses the Fermilevel twice along CM. Two Fermi surfaces with different charactermay give rise to a two gap superconductor. Actually, two supercon-ducting gaps were suggested from optical measurements for anisoelectronic silicide, CaAlSi [29]. Note the circular hole around Kwhich can be seen only in the cross-sectional plot of Fig. 3c.

Fermi surface properties are calculated from the fitted energybands by the spline method [25]. In-plane and out-of-plane Fermi

velocities are v2x;y

D E1=2¼ 3:10� 107 cm=s and v2

z

� �1=2 ¼ 2:06�

107 cm=s, respectively. The anisotropy ratio of the conductivity isrxx=rzz ¼ v2

x

� �= v2

z

� �¼ 2:26 where an isotropic scattering rate is

assumed; the theoretical anisotropic ratio of MgB2 is 1.06, whichis much lower than that of BaAlGe [6]. The plasma frequenciesare Xx,y = 3.83 eV and Xz = 2.55 eV. The optical and transportproperties are expected to be anisotropic between the x and zdirections. Since the two Fermi surfaces are separated, contribu-tions from each surface can be calculated, which are summarizedin Table 1. Note that the Fermi velocity in the z direction is greaterthan in the x direction on the pocket Fermi surface, which can beattributed to inter-layer coupling by hybridization between Bas; dz2 and Al s, Ge s orbitals. Thus, interlayer coupling is made fromnon-bonding orbitals such as Ba s; dz2 and Al s, Ge s orbitals, whichcan give rise to a strong electron–phonon coupling along the zdirection.

The character of wave functions can be found from the totaldensity of states (DOSs) and partial DOS in Fig. 4. The total densityof states at the Fermi level is 1.64 states/eV which is higher thanthat of MgB2 (0.72 states/eV) [6]. The higher DOS of BaAlGe com-pared to MgB2 is due to Ba d orbitals as shown in Fig. 4d. An energygap of 0.3 eV is explicitly observed between p and p⁄ bands around�0.7 eV, which can also be seen in the band plot of Fig. 2. The sharppeaks at �1.6 eV and �0.28 eV originate mostly from Ge pz and Alpz orbitals, respectively. Fig. 4d shows that Sr d orbitals contributeconsiderably to the DOS at the Fermi level, like other ternary sili-cides; Sr dxy contributes more than Sr dz2 at the Fermi level, asshown in Fig. 4d.

(a) (b)

Ene

rgy

(eV

)

-10

-8

-6

-4

-2

0

2

4

H KA L H K Γ M

Ge pz

Al pz

Badxy

Ba dz2 Ba s

-4

-2

0

2

4

H KA L H K Γ M

Fig. 2. (a) Energy bands and (b) major orbital component of BaAlGe around EF (set at 0).

(a) (b)

Γ

MM

M M

K

K

M M

L LA

L LA

Γ

(c) (d)Fig. 3. Fermi surfaces of BaAlGe. (a) Top view and (b) side view. In (b), the bodycenter of the hexagonal prism is the C point. Blue and golden colors representelectron-like and hole-like surfaces, respectively. Cross-section in (c) (001) planeand (d) (100) plane. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

Table 1Fermi surface properties of BaAlGe and SrAlGe: density of states, N(EF), averagevelocities, v2

x

� �1=2; v2

z

� �1=2, anisotropy ratio, v2x

� �= v2

z

� �, and plasma frequencies, Xx, Xz

of which units are states/eV/spin, 10 7cm/s, and eV, respectively.

N(EF) v2x

� �1=2 v2z

� �1=2 v2x

� �= v2

z

� �Xx Xz

BaAlGePocket 0.17 2.27 3.75 0.37 0.90 1.48Cylinder 1.46 3.21 1.83 3.07 3.75 2.14Total 1.64 3.10 2.06 2.26 3.83 2.55

SrAlGeTotal 1.38 3.70 2.90 1.62 4.36 3.42

56 S.J. Youn, A.J. Freeman / Physica C 476 (2012) 54–58

3.2. SrAlGe

The r band in Fig. 5 between about �6 eV and �1 eV is thehybridized band between Al s, px,y, and Ge px,y orbitals. The widthof the r band is 5.1 eV where the minimum and maximum arelocated at M and K, respectively, which is a little bit wider than thatof BaAlGe. The little difference of band width between the twomaterials can be attributed to their similar in-plane lattice

constants. The p⁄ band has a band width of 5.2 eV or 0.8 eV widerthan that of BaAlGe, which is attributed to a smaller c/a ratio, 1.10,of SrAlGe compared to 1.18 of BaAlGe (1.14 for MgB2). The fifthband along HK shows the wider dispersion of 0.92 eV than thatof BaAlGe (0.10 eV) which is due to more hybridization of the Alpz orbitals along the z direction. The strong hybridization of theAl pz orbital enhances the electron–phonon coupling along the zdirection which plays an important role in the superconductivityof SrAlGe [16]. Note that the fifth band is below the Fermi levelalong CM which gives a connected Fermi surface in Fig. 6. Bandcomponents around the Fermi level are displayed in Fig. 5b. Thep band of mostly Ge pz origin is also fully occupied in SrAlGe asin BaAlGe.

The topology of the Fermi surface of SrAlGe in Fig. 6 is in sharpcontrast to that of BaAlGe. There is only one Fermi surface, sincethe inner and outer Fermi surfaces are connected. The circular holearound the K point in BaAlGe does not appear in SrAlGe. The cylin-drical 2D character decreases in SrAlGe compared to BaAlGe andshows more 3D character: compare cylindrical Fermi surfaces inFigs. 3d and 6d. In-plane and out-of-plane Fermi velocities are

v2x;y

D E1=2¼ 3:70� 107 cm=s and v2

z

� �1=2 ¼ 2:90� 107 cm=s, respec-

tively. The plasma frequencies are Xx,y = 4.36 eV and Xz = 3.42 eV.Both the Fermi velocities and the plasma frequencies of SrAlGe

0

2

4

6

(a) Total

0

0.5

1

1.5

DO

S (S

tate

s/eV

)

(b) Ba sAl s

Ge s

0

0.4

0.8 (c) Ba pzAl pz

Ge pz

Ba pxAl px

Ge px

0

0.05

0.1

0.15

-10 -8 -6 -4 -2 0 2 4

Energy (eV)

(d) Ba dxyBa dyzBa dz2

Al dGe d

Fig. 4. Total and partial density of states of BaAlGe.

(b)(a)

Γ

MM

M M

K

K

MM

LL A

LL A

Γ

(d)(c)Fig. 6. Fermi surfaces of SrAlGe. (a) Top view and (b) side view. In (b), the bodycenter of hexagonal prism is the C point. Blue and golden colors represent electron-like and hole-like surfaces, respectively. Cross-section in (c) (001) plane and(d) (100) plane. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

S.J. Youn, A.J. Freeman / Physica C 476 (2012) 54–58 57

are larger than those of BaAlGe in both directions as shown inTable 1 which are characteristic of more delocalized bands. Theanisotropic conductivity ratio, 1.62, is smaller than for BaAlGewhich is also a feature of the more delocalized band of SrAlGe;however, it is higher than for MgB2. Since the two Fermi surfacesare connected, contributions from each Fermi surface are not cal-culated. The connected Fermi surface may give rise to a higher Tc

(a)

Ene

rgy

(eV

)

-10

-8

-6

-4

-2

0

2

4

H KA L H K Γ M

Fig. 5. (a) Energy bands and (b) major orbital c

of SrAlGe than BaAlGe since both Fermi surfaces can contributeto superconductivity with one superconducting energy gap. Thus,two factors contribute for the higher Tc of SrAlGe than BaAlGe:the enhanced electron–phonon coupling along the z directiondue to the Al pz orbital and the connected Fermi surface.

The total and partial DOS are displayed in Fig. 7; the total den-sity of states at EF is 1.38 states/eV which is smaller than that ofBaAlGe (1.64 states/eV). The lower total DOS is mainly due to thewide dispersion of the Al pz orbital. There is no gap in the DOS

(b)

Ge pz

Al pz

Srdxy

Sr dz2 Sr s

-4

-2

0

2

4

H KA L H K Γ M

omponents of SrAlGe around EF (set to 0).

0

2

4

(a) Total

0

0.5

1

1.5

DO

S (S

tate

s/eV

) (b) Sr sAl s

Ge s

0

0.2

0.4

(c)Sr pzAl pz

Ge pz

Sr pxAl px

Ge px

0

0.05

0.1

-10 -8 -6 -4 -2 0 2 4

Energy (eV)

(d)Sr dxySr dyzSr dz2

Al dGe d

Fig. 7. Total and partial density of states of SrAlGe.

58 S.J. Youn, A.J. Freeman / Physica C 476 (2012) 54–58

between p and p⁄ in Fig. 7c while two bands are split in the bandplot of Fig. 5. There are peaks at �1.68 eV and �0.86 eV whichcome from Ge pz and Al pz orbitals, respectively. Fig. 7d shows thatSr d orbitals in SrAlGe play an important role at the Fermi surfacelike BaAlGe. However, the contribution from each orbital is re-versed. Sr dz2 contributes more than Sr dxy at EF as shown inFig. 7d, which is consistent with the larger dispersion of the p⁄

band in SrAlGe than in BaAlGe.

4. Summary and conclusion

The electronic structure of BaAlGe is determined to be similar tothat of ternary silicides with two separate Fermi surfaces; the p⁄

band is partially occupied to show metallic behavior while the pband of Ge pz origin is fully occupied. SrAlGe has a wider bandwidth than BaAlGe. Interlayer coupling is made by non-bondingorbitals such as metal (Ba, Sr) s; dz2 and Al s, Ge s orbitals. The Fermivelocity in the z direction is faster than in the x direction on thepocket Fermi surface of BaAlGe, which is due to inter-layercoupling. The topology of the two materials is very different: there

are two Fermi surfaces in BaAlGe while they are connected in SrAl-Ge. A large cylindrical sheet of the Fermi surface originates fromantibonding p⁄ orbitals, while the electron pocket comes from non-bonding orbitals. The anisotropy ratio decreases from BaAlGe toSrAlGe which is due to the greater 3D character of SrAlGe. Thehigher Tc of SrAlGe than BaAlGe is attributed to both the strongelectron–phonon coupling along the z direction and the connectedFermi surfaces.

Acknowledgments

SJY acknowledges the sabbatical research Grant by GyeongsangNational University. AJF is supported by the Department of Energy(DE-FG02-88ER45382).

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