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Physica C 451 (2007) 24–30

Effect of doping on cohesive and thermophysical properties of MgB2

Nupinderjeet Kaur a,*, Rajneesh Mohan a, N.K. Gaur a, R.K. Singh a,b

a Department of Physics, Barkatullah University, Bhopal 462026, Indiab Institute of Professional, Scientific Studies and Research, Chaudhary Devi Lal University, Sirsa 125055, India

Received 28 July 2006; received in revised form 13 September 2006; accepted 29 September 2006Available online 28 November 2006

Abstract

The effects of doping Al and Mn on the cohesive and thermophysical properties of MgB2 have been investigated using a Rigid IonModel (RIM). The interatomic potential of this model includes contributions from the long-range Coulomb attraction and the short-range overlap repulsion and the van der Waals attraction. This model has been applied to describe the temperature dependence ofthe specific heat of MgB2, Mg1�xAlxB2 (x = 0.1–0.9) and Mg1�xMnxB2 (x = 0.01–0.04) in the temperature range 5 K 6 T 6 1000 K.The calculated results on cohesive energy (/), Bulk modulus (BT), molecular force constant (f), Restrahalen frequency (m0), Debye tem-perature (HD) and Gruneisen parameter (c) are also reported for these materials. Our results on Bulk modulus, Restrahalen frequencyand Debye temperature are closer to the available experimental data. The comparison between our calculated and available experimentalresults on the specific heat at constant volume for MgB2 and Mg1�xAlxB2 (x = 0.1–0.4), particularly, at lower temperatures has shownalmost an excellent agreement. The trend of variation of the specific heat with temperature is more or less similar in pure and dopedMgB2.� 2006 Elsevier B.V. All rights reserved.

PACS: 62.60.Dc; 65.40.�f; 65.90.+i

Keywords: Cohesive energy; Debye temperature; Specific heat; Thermal properties; Thermophysical properties

1. Introduction

The discovery of superconductivity in MgB2 [1] evokedan unprecedented interest in experimental and theoreticalwork to understand the underlying mechanism and toexplore the potential applications. This discovery has putthe diborides as the possible candidates for high tempera-ture superconductivity. Further, MgB2 exceeds the theoret-ical limit for phonon-mediated superconductivity, whichraises the possibility of another coupling mechanism [2].The measurement of isotope effect in MgB2 [3,4] stronglyadvocates for the phonon-mediated mechanism. For anew superconductor, chemical substitution is regarded as

0921-4534/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.physc.2006.09.011

* Corresponding author. Tel.: +91 755 2489028; fax: +91 755 2677223.E-mail address: nuppy@rediffmail.com (N. Kaur).

one of the powerful approaches not only to reveal the nat-ure and physical properties of the parent compound, butalso to enhance the potentiality of the material for variousapplications. Although, various cation substitutions inMgB2 have been reported, but it has been found that thesubstitution of other elements into MgB2 for both Mgand B atoms is very difficult and only a few substitutionsof Mg by Al and Mn were successful [5,6].

The study of thermophysical properties of pure anddoped MgB2 materials has emerged as one of the fascinat-ing fields of research in the recent years for describing cohe-sive, thermal, elastic, and numerous other physicalproperties. The prediction of the experimental data hasbeen done theoretically by developing a model on the basisof the structure of the materials under consideration. Thecentral concern of the present paper is to describe the ther-mophysical properties of pure and doped intermetallic

N. Kaur et al. / Physica C 451 (2007) 24–30 25

diborides (MgB2) on the basis of a theoretical model devel-oped by us [7–9].

For this purpose, we have applied a Rigid Ion Model(RIM), which includes the effects of the van der Waals(vdW) interactions to study the thermodynamical proper-ties of MgB2, Mg1�xAlxB2 (x = 0.1–0.9) and Mg1�xMnxB2

(x = 0.01–0.04). The model has yielded almost satisfactoryprediction of the available experimental data.

2. Theory of RIM approach

We have formulated the Rigid Ion Model (RIM) byincluding the effects of the long-range Coulomb attraction,the short-range Hafemeister–Flygare (HF) type overlaprepulsion and the van der Waals (vdW) interactions. Itsmodel potential (/) consists of the following interatomicinteractions:

/ ¼ /Ckk0 ðrÞ þ /R

kk0 ðrÞ þ /vdWkk0 ðrÞ ð1Þ

where Coulomb contribution /Ckk0(r) is given by

/Ckk0 ðrÞ ¼ �

e2

2

Xkk0

ZkZk0r�1kk0 ð2Þ

with rkk 0 is the separation between the two atoms k and k 0.The overlap repulsive energy /R

kk0 ðrÞ according to Hafe-meister–Flygare [10] type interaction extended upto thesecond neighbour ions, is expressed as

/Rkk0 ðrÞ ¼ nb1bkk0 expfðrk þ rk0 � rkk0 Þ=q1g

þ n0

2b2 bkk expfð2rk � rkkÞ=q2g½

þ bk0k0 expfð2rk0 � rk0k0 Þ=q2g� ð3Þ

Here rkk0 and rkkð¼ rk0k0 Þ are the first and second neighbourseparation, respectively. rkðrk0 Þ are the ionic radii of k(k 0)ions. n(n 0) is the number of nearest (next nearest) ions.(b1, b2) and (q1, q2) are the hardness and range para-meters, respectively. The Pauling coefficients [11] areexpressed as

bkk0 ¼ 1þ ðZk=NkÞ þ ðZk0=N k0 Þ; ð4Þ

where Zk(Zk 0) and Nk(Nk 0) are the valence and number ofelectrons in the outermost orbit.

The contributions from the van der Waals (vdW) attrac-tions /vdW

kk0 ðrÞ due to the dipole–dipole (d–d) and dipole–quadrupole (d–q) interactions are written as

/vdWkk0 ðrÞ ¼ �

Xkk0

ckk0r�6kk0 �

Xkk0

dkk0r�8kk0 ; ð5Þ

where ckk 0 and dkk 0 are the vdW coefficients due to d–d andd–q interactions, respectively. They are defined as followsand their values are determined using the Slater–Kirkwood[12] variational method:

ckk0 ¼3e�h2m

akak0 ðak=N kÞ1=2 þ ðak0=Nk0 Þ1=2h i�1

; ð6Þ

dkk0 ¼27e�h2

8makak0 ðak=N kÞ1=2 þ ðak0=Nk0 Þ1=2

h i2

� ðak=N kÞ1=2 þ 20

3ðakak0=NkNk0 Þ þ ðak0=N k0 Þ

� ��1

;

ð7Þwhere e and m are the charge and mass of the electron,respectively. ak(ak 0) are the polarizabilities of k(k 0) atoms.Nk(Nk 0) are the effective number of electrons responsiblefor polarization of k(k 0) ions.

The model parameters, hardness (b) and range (q)parameters are determined from the equilibrium condition

d/dr

� �r¼r0

¼ 0 ð8Þ

and the Bulk modulus

B ¼ 1

9Kr0

d2/dr2

� �r¼r0

; ð9Þ

where K is the crystal constant (dependent on crystal struc-ture). r0 is the equilibrium interatomic separation.

The other thermophysical properties are calculatedusing the expressions given below

The Bulk modulus is given by

BT ¼ f =ð3Kr0Þ; ð10Þwhere, K is the crystal constant and f is molecular forceconstant given by

f ¼ 1=3b/00Rkk0 ðrÞ þ ð2=rÞ/0Rkk0 cr¼r0; ð11Þ

with /Rkk0 ðrÞ as the Hafemeister–Flygare [10] type interac-

tions defined in Eq. (3).This force constant (f), in turn, gives the Restrahlen fre-

quency (m0) as

m0 ¼ ð1=2pÞðf =lÞ1=2; ð12Þ

where, l is the reduced mass of the system.The Restrahlen frequency (m0) gives us the Debye tem-

perature (HD) as

HD ¼ ðhm0Þ=kB; ð13Þwhere h and kB are Planck and Boltzmann constants,respectively.

With the help of Debye temperature obtained from theabove equation, the specific heat is calculated using thefamiliar expression

Cv ¼ ð12p4=5ÞfNkBðT=HDÞ3gZ HD=T

0

fðx4exÞ=ðex � 1Þ2dxg;

ð14Þwhich is the Debye T3 approximation.

The values of the Gruneisen parameter (c) are calculatedfrom the relation

c ¼ �ðr0=6Þf/000ðrÞ=/00ðrÞgr¼r0: ð15Þ

Table 1The model parameters (b1, b2, q1 and q2), cohesive and thermophysical properties of pure MgB2 and doped Mg1�xMxB2 (M = Al, Mn)

Conc. (x) Model parameters / (eV) f (104 dyne cm�1) BT (1012 dyne�1 cm2) m0 (THz) HD (K) c

B–B M–B

b1 (10�12 erg) q1 (A) b2 (10�12 erg) q2 (A)

MgB2 (Pure) 308.606 0.5054 61.390 0.6234 �109.99 13.816 107.63 13.593 653.061 0.7159110b 15.50c 746c

15.61d 750 ± 30d

Al doped

0.1 319.802 0.4948 64.068 0.6070 �110.881 14.582 113.70 13.931 669.339 0.747715.40a 740 ± 15a

0.2 329.127 0.4854 67.822 0.5924 �111.852 15.272 119.36 14.224 683.394 0.769513.52a 650 ± 15a

0.3 338.571 0.4764 71.879 0.5787 �112.788 15.959 125.20 14.506 696.935 0.790114.15a 680 ± 15a

0.4 348.328 0.4676 75.766 0.5626 �113.561 16.541 131.04 14.733 707.852 0.806614.36a 690 ± 15a

0.5 357.776 0.4598 80.814 0.5536 �114.560 17.311 136.87 15.036 722.408 0.82780.6 367.498 0.4521 85.769 0.5422 �115.404 17.978 142.70 15.287 734.446 0.84510.7 377.343 0.4447 91.055 0.5313 �116.219 18.639 148.52 15.528 746.027 0.86170.8 387.199 0.4377 96.681 0.5210 �117.011 19.293 154.34 15.760 757.167 0.87740.9 397.166 0.4310 103.483 0.5118 �117.802 19.971 160.15 15.995 768.486 0.8934

Mn doped

0.01 312.053 0.5031 31.558 0.6198 �110.044 13.996 108.64 13.678 657.132 0.72800.02 314.410 0.5019 31.840 0.6177 �110.625 14.111 109.63 13.731 659.678 0.72940.03 317.240 0.5003 32.213 0.6143 �110.718 14.279 110.65 13.809 663.429 0.74080.04 319.416 0.4988 32.468 0.6121 �110.976 14.393 111.62 13.861 665.910 0.7438

These results have been compared with available experimental data.a Ref. [18].b Ref. [16].c Ref. [19].d Ref. [3].

26N

.K

au

ret

al.

/P

hy

sicaC

45

1(

20

07

)2

4–

30

N. Kaur et al. / Physica C 451 (2007) 24–30 27

Using Eqs. (1), (10)–(15), we have calculated the thermo-physical properties as listed in Table 1.

The values of the model parameters are listed in Table 1and used to compute the cohesive energy for MgB2,Mg1�xAlxB2 (x = 0.1–0.9) and Mg1�xMnxB2 (x = 0.01–0.04) from Eq. (1). We have also calculated the molecularforce constant (f ), Bulk modulus (BT), Restrahalen fre-

0 200 400 600 800 10000

5

10

15

20

25pure MgB2

Cv(

J/m

olK

)

T(K)

0 10 20 30 40 500

500

1000

1500

2000 expt cal

Cv (

mJ/

mol

K)

T(K)

Fig. 1. Variation of the specific heat of MgB2 in the temperature range5 K 6 T 6 1000 K. Inset shows the comparison of experimental data(filled circles (d)) with calculated results (solid line (—)).

0 200 400 600 800 10000

5

10

15

20

25 x=0.1

T(K)

T(K)

Cv(

J/m

olK

)

1 1010-1

100

101

102

103

104

105

T(K)

calexpt

Cv (

mJ/

mo

lK)

C(J

/mo

lK)

0 200 400 600 800 10000

5

10

15

20

25x=0.3

Cv(

J/m

olK

)

T(K)

C(J

/mo

lK)

100

calexpt

10-1

100

101

102

103

104

105

Cv (

mJ/

mo

lK)

1 10 100

Fig. 2. Variation of the calculated specific heat (Cv) with temperature forexperimental data (filled circles (d)) with calculated results (solid line (—)).

quency (m0), Debye temperature (HD) and specific heat atconstant volume (Cv) and Gruneisen parameter (c) for pureMgB2 and the doped Mg1�xAlxB2 (x = 0.1–0.9) andMg1�xMnxB2 (x = 0.01–0.04) using Eqs. (10)–(15). Theresults thus obtained are presented and discussed below.

3. Results and discussion

3.1. Model parameters

The input data (r0,B) for Mg1�xAlxB2 and Mg1�xMnxB2

at room temperature for different compositions (x) areobtained by using the Vegard’s law [13] and the data onthem for MgB2 [14–16], AlB2 [15] and MnB2 [17] materials.Using these input data and the van der Waals coefficients(ckk 0 and dkk 0) evaluated from the Slater and Kirkwoodmethod [12] for MgB2, Mg1�xAlxB2 (x = 0.1–0.9) andMg1�xMnxB2 (x = 0.01–0.04), we have obtained the modelparameters (b1, b2, q1 and q2) as a function of concentra-tion (x) using the equilibrium condition and Bulk modulusrelation (Eqs. (8) and (9)). The calculated values of theseresults are listed in Table 1. The model parameters arevarying almost linearly with the Al and Mn concentrations.The higher values of the hardness parameters indicatethe high hardness of the metal diborides. The values ofthe hardness parameters (b1, b2) are increasing with the

T(K)

T(K)

10-1

100

101

102

103

104

105

Cv (

mJ/

mo

lK)

0 200 400 600 800 10000

5

10

15

20

25 x=0.2

v

T(K)

0 200 400 600 800 10000

5

10

15

20

25x=0.4

v

T(K)

1 10 100

calexpt

calexpt

10-1

100

101

102

103

104

105

Cv (

mJ/

mo

lK)

1 10 100

Mg1�xAlxB2 for x = 0.1–0.4. Inset in them shows the comparison of

28 N. Kaur et al. / Physica C 451 (2007) 24–30

concentration of Al and Mn, which depicts that the hard-ness of the doped compounds is increasing. On contrary,the values of the range parameters (q1 and q2) are decreas-ing linearly with increasing concentration (x).

3.2. Cohesive and thermal properties

These model parameters (b1, b2, q1 and q2) have beencalculated from Eqs. (8) and (9) using the values of latticeparameters (r0) [15,17], bulk modulus (B) [16,17] and ther-mal expansion coefficient [14] for MgB2, Mg1�xAlxB2 andMg1�xMnxB2 for different compositions (x) and listedthem in Table 1. We have computed the cohesive and ther-mal properties of MgB2, Mg1�xAlxB2 (x = 0.1–0.9) andMg1�xMnxB2 (x = 0.01–0.04) using the expressions givenin the previous section. Their calculated values along with

0

5

10

15

20

25 x=0.5

T(K)

T(K)

T(K)

0

5

10

15

20

25 x=0.7

0

5

10

15

20

25 x=0.9

Cv(

J/m

olK

)

Cv(

J/m

olK

)

Cv(

J/m

olK

)

0 200 400 6

0 200 400 600 800 1000

0 200 400 600 800 1000

Fig. 3. Variation of the calculated specific heat (Cv) w

the available experimental data [3,16,18,19] are listed in thesame Table 1. The results obtained by us are in closerresemblance with the available experimental data.

It is also seen from Table 1 that the magnitude of thecohesive energy increases from 110.881 eV for (x = 0.1)to 117.802 eV for (x = 0.9) for Al doping and from110.044 eV for (x = 0.01) to 110.976 eV for (x = 0.04) forMn doping. Also, it is evident from this table that Al dopedMgB2 has higher magnitude of the cohesive energy thanMn doped MgB2. The systematic trend of variation andthe magnitude of some thermal properties are comparablewith the experimental values (at 300 K), which reveals thesuitability and appropriateness of our model.

The molecular force constant (f) and the isothermal bulkmodulus (BT) both increase with the increase of the dopingconcentration (x) of Al and Mn (see Table 1) and this feature

0

5

10

15

20

25x=0.6

T(K)

T(K)

0

5

10

15

20

25x=0.8

Cv(

J/m

olK

)C

v(J/

mo

lK)

00 800 1000

0 200 400 600 800 1000

0 200 400 600 800 1000

ith temperature for Mg1�xAlxB2 for x = 0.5–0.9.

N. Kaur et al. / Physica C 451 (2007) 24–30 29

shows that the bond is becoming stronger with Al and Mndoping. The high value of the bulk modulus for x = 0.9 con-centration indicates the high hardness of the compound atthis concentration. The Restrahalen frequency (m0) isdirectly proportional to the molecular force constant (f )and varies with concentration accordingly. It can also beobserved from Table 1 that the Gruneisen parameters forthe present dopants increase with the increase of concentra-tion (x). The computed Restrahalen frequency and bulkmodulus at 300 K for Mg1�xAlxB2 (x = 0.1–0.4) are in rea-sonably good agreement with available experimental data[18]. Also, other values, listed in Table 1, although seem tobe more or less realistic, but could not be compared due tothe lack of experimental data on them.

The Debye temperature (HD) of the crystal is involved inmany phenomenon associated with the lattice vibrations. Wehave, therefore, computed Debye temperatures for MgB2,Mg1�xAlxB2 (x = 0.1–0.9) and Mg1�xMnxB2 (x = 0.01–0.04) and listed them in Table 1. The higher values of Debyetemperature indicate the presence of higher phonon frequen-cies in these materials. The computed Debye temperaturesfor MgB2 and Mg1�xAlxB2 (x = 0.1–0.4) are in reasonablygood agreement with the available experimental data [18,19].

3.3. Thermodynamical properties

The specific heat of a material has two major contribu-tions – the first one is the lattice (phonon) contribution and

0 200 400 600 800 10000

5

10

15

20

250.01

Cv(

J/m

olK

)

T(K)

0.03

0

5

10

15

20

25

Cv(

J/m

olK

)

0 200 400 600 800 1000T(K)

Fig. 4. Variation of the calculated specific heat (Cv) wit

the second is the electronic contribution [20]. During thenormal and superconducting transition, the lattice contri-bution remains the same but the electronic contributionundergoes a sudden change and as a result the C–T curveshows an anomalous jump at Tc followed by a rapiddecrease with the temperature [21].

We have computed the variation of the lattice contribu-tion to the specific heat in temperature range5 K 6 T 6 1000 K for MgB2. The values of the specificheat at constant volume (Cv) for MgB2 at different temper-atures are depicted in Fig. 1 and they are compared withthe available experimental data [22] as shown in the insetof this figure. The trend (see Fig. 1) revealed by the calcu-lated curve is almost similar at lower temperatures as isexhibited by the experimental points (closed circles) withinthe temperature range 5 K 6 T 6 50 K. It is clear from thefigure that the specific heat for MgB2 calculated by us iscloser to the experimental results obtained by Yang et al.[22] using an adiabatic continuous heating calorimetry.The specific heat in the normal state of the material is usu-ally approximated by the contribution of the lattice andelectronic specific heat. In our model, we have consideredonly the lattice contribution. Thus, the deviations in thevariation of specific heat between the measured and calcu-lated values might be ascribed to the exclusion the elec-tronic contribution.

We have computed the lattice contribution to the spe-cific heat for the Al doped MgB2 in the temperature range

0.02

0.04

0

5

10

15

20

25

Cv(

J/m

olK

)

0 200 400 600 800 1000

T(K)

0 200 400 600 800 1000

T(K)

0

5

10

15

20

25

Cv(

J/m

olK

)

h temperature for Mg1�xMnxB2 for x = 0.01–0.04.

30 N. Kaur et al. / Physica C 451 (2007) 24–30

5 K 6 T 6 1000 K using the expression given in the previ-ous section. The results obtained are shown graphically asa function of temperature in Figs. 2 and 3. The results forx = 0.1–0.4 (as shown in Fig. 2) are compared with theexperimental data [18] available in the temperature range5 K 6 T 6 100 K and are shown in the inset of theFig. 2. We find that there is closer resemblance in the trendof variation of the specific heat (as shown in Fig. 3) in thetemperature range 5 K 6 T 6 1000 K for Mg1�xAlxB2

(x = 0.5–0.9). At the higher temperatures the specific heatcurves are almost flat and they, generally, overlap eachother for x = 0.1–0.9.

The specific heat increases with the Al doping in MgB2

but at higher temperature range their values are almostconstant for all the concentrations (see Figs. 2 and 3). Itis evident from the results that the effect of Al doping isnot much as there is very small change in the value of thelattice specific heat at higher temperatures. Due to theunavailability of the experimental data (to our knowledge)on the specific heat of Mg1�xAlxB2 for x = 0.5–0.9, we areunable to compare them with experimental data. Here, it isinteresting to note that the variations of specific heat withtemperature for these concentrations follow the trends sim-ilar to those as revealed at other concentrations for whichthe experimental results are available (see Fig. 2).

We have also computed the specific heat of Mn dopedMgB2 for concentration x = 0.01–0.04 and presented themin Fig. 4 in the temperature range 5 K 6 T 6 1000 K. Itcan be seen from the figures that the curve follows thetrends similar to those revealed by MgB2, Mg1�xAlxB2

(x = 0.1–0.9) and other diborides [7–9]. Due to the lackof the experimental results, we could not compare theseresults and till then they will serve as a guide to the exper-imental workers, in future.

4. Conclusions

On the basis of an overall discussion, finally it may beconcluded that the Rigid Ion Model modified to includethe reasonable vdW effects and the second neighbourthrough the HF type potential by us is adequately capableof giving a satisfactory prediction of the cohesive and ther-mophysical properties of pure and doped MgB2. It isnoticed that the contribution of short-range overlap repul-sion is, generally, less than 10% of the total cohesiveenergy. This feature is indicative of the fact that the majorcontribution to the cohesion in these materials is due to the

Coulomb attraction along with the supplementary contri-bution from the vdW attraction in these materials.

Acknowledgements

The authors are thankful to the University Grants Com-mission (UGC), New Delhi and the Council of Scientificand Industrial Research (CSIR), New Delhi for providingthe financial support.

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