Anomalous Thermal Expansion in �-Titanium

P. Souvatzis,1 O. Eriksson,1 and M. I. Katsnelson2

1Department of Physics, Uppsala University, Box 530, SE-75121, Uppsala, Sweden2Institute for Molecules and Materials, Radboud University Nijmegen, NL-6525 ED, Nijmegen, The Netherlands

(Received 1 February 2007; published 6 July 2007)

We provide a complete quantitative explanation for the anisotropic thermal expansion of hcp Ti at lowtemperature. The observed negative thermal expansion along the c axis is reproduced theoretically bymeans of a parameter free theory which involves both the electron and phonon contributions to the freeenergy. The thermal expansion of titanium is calculated and found to be negative along the c axis fortemperatures below �170 K, in good agreement with observations. We have identified a saddle pointvan Hove singularity near the Fermi level as the main reason for the anisotropic thermal expansion in�-titanium.

DOI: 10.1103/PhysRevLett.99.015901 PACS numbers: 65.40.De, 63.20.Dj, 71.20.Be

The most general aspects of the chemical bonding in thetransition d metals can be understood from the Friedelmodel [1], explaining the trends in equilibrium volume,bulk modulus, and cohesive energy. The transition metalsare found to crystallize at low temperatures in the cubic fccand bcc structures, and the hexagonal hcp structure [2],which can be qualitatively explained from a band filling ofitinerant d states [3]. In addition, the Debye model repro-duces the thermal volume expansion with a rather goodaccuracy [4]. Hence, with a seemingly good understandingof the fundamental mechanisms governing the propertiesof the transition metals, the recently observed negativethermal expansion coefficient along the c axis of one ofthese elements, the hcp (�) phase of Ti [5], stands out as anenigma, especially since no other transition metal so farhas been shown to display such a behavior.

The problem of finding connections between the elec-tronic structure of metals and alloys and peculiarities oftheir lattice properties has a long history, starting with the‘‘third Hume-Rothery rule’’ concerning boundaries ofphase stability in noble-metal alloys [6] and its explanationby Jones in terms of touching of the Brillouin zone faces bythe Fermi sphere [7] (for a review of further developmentsof these ideas, see Ref. [8]). The general concept of elec-tronic topological transitions (ETT), introduced by Lifshitz[9], that is, a coincidence of the Fermi level with avan Hove singularity of the electronic density of states(DOS), is of crucial importance for understanding theseinterrelations. Phase transitions and premartensitic anoma-lies of elastic moduli in alkali and alkaline-earth metalsunder pressure provide a clear example of the effects of thevan Hove singularities on the lattice properties [10,11]. Itturns out that the singularity in the electron DOS at theFermi energy N�EF� should be visible also in elastic mod-uli and Debye temperature and, thus, in the thermodynamicproperties of metals at low enough temperatures (theanomalies in phonon spectra with large enough wave vec-tors and thus in high-temperature thermodynamic proper-ties are in general weaker; see Ref. [8] and references

therein). Since the thermal expansion is connected withthe pressure derivatives of the elastic moduli, anomalies inthe thermal expansion might be especially strong. In par-ticular, it can be proven thermodynamically that ETT innoncubic metals should lead to a singular anisotropicthermal expansion at low enough temperatures [5,12].The latter means that in principle it is always possible toprepare a textured material with zero thermal expansion.This conclusion, being interesting in itself, opens newways to find nonmagnetic Invar systems. However, basedon these general considerations alone it is impossible topredict the temperature region where the effect should beobservable, or how far from the point of ETT the effect isstill visible. Here we answer these questions based ondirect microscopic calculations, in a framework of thedensity functional theory, and we address the recentlydiscovered negative thermal expansion of �-Ti [5].

The occurrence of negative thermal expansion at lowtemperatures for noncubic elemental solids has beenknown for quite some time, but only for elements outsidethe transition metal series. For instance, the hexagonalclose packed metals zinc and cadmium [13] have negativethermal expansion coefficients along the a axis (�?) fortemperatures below�75 K, while among the IIIB group ofthe Periodic Table it is tin and indium that have negativethermal expansion coefficients along the basal plane (�?)and orthogonal to the basal plane (�jj), respectively [13].Among the transition metals, Ti, however, stands out.

The analysis presented here is based on first principlesdensity functional theory of the electron and phonon con-tributions to the total energy. We write the Helmholtz freeenergy as

F� ��; T� �V2

Xij

Cij�i�j � Fphon� ��; T� � Fel� ��; T�; (1)

where Cij are the elastic constants, �� the elastic strain, Vthe volume, Fphon the phonon free energy and Fel theenergy of thermal excitations in the electron subsystem.

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In this expression the reference (zero) level is for a crystalat equilibrium conditions at zero temperature. The elasticconstants were calculated from first principles [14].

To evaluate the free energy contribution Fphon, whichcan be expressed as [16,17]

Fphon� ��;T� �Z 1

0d!g�!; ���

�@!2� kBT ln�1� e�@!=kBT�

�;

(2)

the phonon DOS g�!; ��� has to be calculated. This wasdone within the quasiharmonic approximation [18,19],where all anharmonic effects except the thermal expansionare neglected when calculating the temperature depen-dence of the phonons. In practice, the phonon DOS wascalculated by making small displacements of the atoms in asupercell [19]. The directions of the displacements were�110� and �001� with amplitudes that were equal to�0:4%of the lattice constant. The supercell used was a 3 3 2cell. Further details are found in Ref. [14]. The energy ofthermal excitations of electron states Fel was calculated bythe standard expression by Sommerfeld and Frank [20]:

Fel� ��; T� � ���kB�2

6D��F; ���T2; (3)

whereD��F; ��� is the calculated electronic density of statesat the Fermi level.

In Fig. 1 we compare our calculated phonon spectrum(at T � 0) with experimental values (at room temperature).It is worthwhile to mention that a tight-binding calculationof the phonon dispersion for hcp Ti has been publishedrecently [21], where parameters of the model were fitted toexperimental data as well as to first principles calculations.The theoretical phonon dispersion curve in Fig. 1 agreesvery well with the theoretical curves in Ref. [21]. When

comparing the theoretical and experimental [22] curves,we note an overall agreement, although certain differencescan be identified. For instance, along the �-A direction thetheory underestimates the frequencies in the lowest experi-mental branch, whereas the higher branches are repro-duced with better accuracy, especially using the generalgradient approximation (GGA). Also, the calculated lowestbranch along the �-K direction comes out somewhat toolow compared to observations. The phonon DOS was thencalculated with the method of Ref. [23].

By differentiating the free energy (1) with respect to�v d�ln�V�� and �c d�ln�c=a��, it is possible to obtainan expression for the change in volume and structuralproperty as a function of temperature. These changes areexpressed in terms of equilibrium strains, �0

v and �0c at

which @F=@�v � @F=@�c � 0, and can be written in termsof the elastic constants and strain derivatives of the freeenergy:

�0v�T� �

1

V�B11B22 � B212�

��B22

@F�

@�v� B12

@F�

@�c

�(4)

�0c�T� �

1

V�B11B22 � B212�

�B12

@F�

@�v� B11

@F�

@�c

�(5)

where

B11 �2

9

�C11 � C12 �

1

2C33 � 2C13

�(6)

B22 �2

9�C11 � C12 � 2C33 � 4C13� (7)

B12 �1

9�C33 � C13 � C11 � C12� (8)

F� � Fphon � Fel: (9)

Furthermore, by differentiating (4) and (5) with respect tothe temperature the following relations are obtained for thethermal expansion coefficients [5,12]:

�a �1

3V�B11B22 � B212�

���B22 � B12�

@2F�

@T@�v

� �B12 � B11�@2F�

@T@�c

�(10)

�c �1

3V�B11B22 � B212�

���B22 � 2B12�

@2F�

@T@�v

� �B12 � 2B11�@2F�

@T@�c

�(11)

� �1

V�B11B22 � B212�

��B22

@2F�

@T@�v� B12

@2F�

@T@�c

�;

(12)

0 0.25 0.5 0.25 0 0.50

2

4

6

8

10

ω [

TH

z]

Γ K M Γ A

[110] [100] [001]

LDAGGA

FIG. 1. The phonon dispersion of hcp Ti at room temperatureand ambient pressure. The solid curve is the calculated frequen-cies from GGA and the dashed curve represents calculatedfrequencies from LDA. Both calculations are done at the experi-mental volume V0 � 15:91 �A3. The filled circles are the experi-mental data of Stassis et al. [22].

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where �a � �1=a��da=dT�, �c � �1=c��dc=dT�, and � ��1=V��dV=dT�.

By fitting free energies calculated at different strains andat a given temperature to polynomials of first degree in �vand second degree in �c, the equilibrium strains can beobtained from Eqs. (4) and (5), and the thermal expansioncoefficients can be calculated from Eqs. (10) and (12).

In Fig. 2 we show the calculated thermal expansioncoefficients of�-titanium. The most important informationto be extracted from this figure is that the observed nega-tive thermal expansion coefficient along the c axis isreproduced by our theory, where especially the calculationbased on GGA reproduces observations with the highestaccuracy. It should be noted that GGA often is found todescribe chemical bonding with better accuracy than thelocal density approximation (LDA). The temperature in-terval for which �c is negative is roughly 0–170 K, both inthe observations and from theory. The order of magnitudeof �c and �a is also the same when comparing experimentand theory. Figure 2 also shows that theory reproduces,with good accuracy, the volume expansion coefficients ofRef. [24], especially at somewhat elevated temperatures.We also note that based on thermodynamic relations �should approach zero at T � 0 K, which our theoreticalcurves do.

The fact that both the measured and calculated thermalexpansion coefficients along the c axis of Ti are negative atlow temperatures strongly suggests the uniqueness of ele-mental Ti among transition metals, although the absolutevalue of the measured low temperature expansion coeffi-cient is still somewhat uncertain. The measured data ofRef. [5] (filled circles) have in Fig. 2 been scaled (opencircles) to reproduce room temperature values of �, �a,

and �c, and it is found that these scaled values comparebetter with our theory (Fig. 2). Although a slight calibra-tion error in Ref. [5] cannot be excluded, there is goodreason to view the negative value of �c at low temperaturesas a true materials property of �-Ti.

As we will show below, the microscopic origin of thenegative thermal expansion for �c of Ti is due to thecloseness to a saddle point van Hove singularity of the

0 100 200 300

T [ K ]-10

0

10

20

30

α·10

6 [ K

-1]

β

αa

αc

LDAGGA

FIG. 2. Linear thermal expansion for hcp Ti at ambient pres-sure. The solid lines are the theoretical calculation using GGAand the dashed line is from an LDA calculation. The filled circlesare the experimental data of Nizhankovskii et al. [5]. The filledsquares are the experimental volume expansion coefficient ofMal’ko et al. [24]. The filled triangle and the open triangle arethe experimental data of �a and �c, respectively, of Pawar andDeshpande [25]. The open circles are the experimental data ofNizhankovskii et al. scaled to give a volume expansion coeffi-cient in agreement with Ref. [24].

FIG. 3 (color online). Calculated Fermi surface of hcp Ti atT � 0 equilibrium volume for three different lattice constants c,at c � 0:988c0 (a), at c � c0 (b) and at c � 1:012c0 (c). Here c0

corresponds to the (T � 0) equilibrium lattice constant.

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electronic structure. To illustrate this singularity we pro-ceed with an analysis of the Fermi surface. In order to dothis we show in Fig. 3 the calculated Fermi surface at theequilibrium volume for three different values of the out-of-plane lattice constant c. The figure shows that as the clattice constant decreases the inner ellipsoidal surface atthe � point and the Fermi surface centered at the A pointbecome connected along the �A line. The electronic struc-ture as revealed by the Fermi surface shown in Fig. 3 thusdemonstrates the presence of a saddle point van Hovesingularity, which is associated with a singular contributionto the density of states at the Fermi level. It should bestressed that this singularity is close to the Fermi levelfor a range of values of the c axis around c � c0. Thevan Hove singularity gives rise to a singular contributionto the density of states as follows: �N�EF� ��

��������������������EF � Ec1

p��EF � Ec1�, where ��x� fulfills ��x > 0� �

1, ��x < 0� � 0, and Ec1 is the critical point energy [5,12].The energy difference between EF and the energy of thecritical saddle point, at the theoretical equilibrium volumeand at c � c0, has been calculated to be EF � Ec1 �44 meV. Another critical point, associated with the appear-ance of a new ellipsoid around the K symmetry point(not shown in Fig. 3) has been found in the calculations,giving rise to the singular contribution �N�EF� ���������������������Ec2 � EFp

��Ec2 � EF� to N�EF�. However, since Ec2 �EF � 142 meV and j@�Ec2 � EF�=@�j � j@�EF �Ec1�=@�j, it is clear that the saddle point topological tran-sition, at Ec1, gives rise to the strongest singular contribu-tion to N�EF�.

By calculating the derivatives of N�EF� with respectto the two different types of strains, we have foundthat @N�EF�=@�v � 0:75 eV�1 and @N�EF�=@�c ��0:77 eV�1. Since the singularities in N�EF� influencethe elastic moduli, thus affecting the Debye temperature[8], it is natural to attribute the main reason for the aniso-tropic thermal expansion in �-titanium to the saddle pointvan Hove singularity near the Fermi level.

We are grateful to the Strategic Foundation forResearch (SSF), the Swedish Research Council (VR), theSwedish National Supercomputer Center (NSC), UPMAX,and to the Goran Gustafsson Foundation for support.M. I. K. acknowledges a support from Stichting voorFundamenteel Onderzoek der Materie (FOM), TheNetherlands. Valuable discussions with Professor U.Jansson are acknowledged.

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[2] With the exception of the complex structure of Mn at lowtemperatures.

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code [15]. Convergence in sampling of the Brillouin zonewas obtained at 20480 k points (for elastic constants) and486 k points (for phonon calculations). The calculationsemployed both the LDA and the GGA. The phonondensity of states were calculated using a 150 150100 k-point mesh with a 0.05 THz smearing. The calcu-lations have been performed for 15 different volumestrains �v in the range 0 �v 0:013, and at eachvolume strain for three different tetragonal strains �c ��0c � 0:001, �0

c, �0c � 0:001, where �0

c is the tetragonalstrain corresponding to the minimum static lattice energyat a given volume strain �v. In the VASP calculations acutoff of 302 eV was used. To each eigenvalue a Gaussiansmearing of �0:2 eV was applied to speed up the con-vergence of the calculation.

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[16] D. C. Wallace, Thermodynamics of Crystals (Wiley, NewYork, 1972).

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(1931).[21] D. R. Trinkle et al., Phys. Rev. B 73, 094123 (2006).[22] C. Stassis, D. Arch, B. N. Harmon, and N. Wakabayashi,

Phys. Rev. B 19, 181 (1979).[23] The Phon software together with a description of the

program can be found at http://chianti.geol.ucl.ac.uk/dario/.

[24] P. I. Mal’ko, D. S. Arensburger, V. S. Pugin, V. F.Nemchenko, and S. N. L’Vov, Powder Metall. Met.Ceram. 9, 642 (1970).

[25] R. R. Pawar and V. T. Deshpande, Acta Crystallogr. A 24,316 (1968).

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