ISAS
SISSA
SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI
INTERNATIONAL SCHOOL FOR ADVANCED STUDIES
A rst prin iples study of the thermal expansion
in some metalli surfa es
Thesis submitted for the degree of \Do tor Philosophi"
CANDIDATE SUPERVISOR
Introdu tion 1
1 Motivation of the work 5
1.1 Latti e dynami s . . . 5
1.2 Mole ular Dynami sand the Quasi Harmoni Approximation. . . 7
1.3 Berylliumvs. Magnesium . . . 8
1.4 Be(0001) surfa e . . . 11
1.5 Mg(1010) surfa e . . . 12
2 First prin iples methods 15 2.1 The ele troni problem . . . 16
2.1.1 Density Fun tional Theory . . . 16
2.1.2 Lo alDensity Approximation . . . 17
2.1.3 Plane Waves and Pseudo Potentialapproa h . . . 18
2.1.4 Smearing te hnique . . . 19
2.1.5 Non Linear Core Corre tion . . . 19
2.2 Density Fun tionalPerturbation Theory . . . 20
2.3 Se ond order and Linear response . . . 21
2.4 Third order for metals . . . 22
3 Stru ture and dynami al properties 25 3.1 Be and Mgbulk . . . 25 3.2 Be (0001) surfa e . . . 27 3.3 Be(1010) surfa e . . . 30 3.4 Mg(1010) surfa e . . . 37 4 Thermal expansion 39 4.1 Be and Mgbulk . . . 39 4.2 Be (0001) surfa e . . . 40 4.2.1 Our al ulation . . . 40
4.2.2 Comparison with previous theoreti al results . . . 44
4.2.3 Comparison with the experiments . . . 45
4.2.4 First prin iples mole ulardynami s . . . 46
Aknowledgments 53
In purely harmoni rystals, ioni mean positions do not hange upon in reasing tem-perature, thus rystal thermal expansion is a onsequen e of anharmoni terms in the interatomi potential. Anharmoni e e ts are expe ted to be larger at material surfa es than inthebulk,be auseofthelowersymmetry,andasa onsequen ethermalexpansion of a surfa e is expe ted to be enhan ed.
Re ently this subje t has attra ted mu h attention, and enhan ed thermal e e ts have been experimentally observed on several surfa es, as, for examples, in: Ni(001) [1℄, Pb(110) [2℄, Ag(111) [3℄, and Be(0001) [4℄surfa es. However insome ases the situation is still debated, not only be ause interpretation of experimental measurements is not sraightforward, but also be ause of ontrasting theoreti al results, as for example, for the Ag(111) surfa e [5℄. Finally while good agreement between stati al ulations and experiment is generally found for low temperature stru tures, some systems have been singledoutinwhi hthein lusionofthermale e tsseemstobeimportanteventointerpret data taken at roomtemperatue,as inBe(0001) [4℄, and Rh(001)[6℄surfa es.
Nowadays the equilibrium stru ture of a large lass of materials an be determined theoreti ally withgreata ura y,thanks tostate ofthe art rstprin iples omputational methods essentially based on Density Fun tional Theory (DFT)[7,8℄. Furthermore, the study ofthermalbehaviourfrom rst prin iplesis nowathandalsofor omplex systems, as surfa es,thanks to the availabilityof adequate omputationalpowerand tothe devel-opmentofsophisi atedte hniquessu hasCar-Parrinellomethod[9℄orDensityFun tional Perturbation Theory [10,11℄.
Di erent approa hes have been used to study thermal e e ts within rst-prin iples methods. Essentiallythey are based onMole ularDynami (MD) simulationsand Quasi Harmoni Approximation (QHA), other approa hes, su h as the self- onsistent phonon s heme [12℄, are presently not feasible from rst prin iples. MD simulations a ount exa tly forinteratomi potentialanharmoni ity,but treationi degrees offreedom lassi- ally,and an givereliableresults onlynear oraboveDebyetemperature, wherequantum e e ts arenegligible. Systemsthat an besimulated,nowadays,typi ally onsistofabout a hundred atoms and the time s ale of the simulation an rea h a few pi ose onds. The knowledgeof the frequen ies of the vibrational modes of a system allows the appli ation of the QHA s heme. It provides a omplementary approa h, valid at low temperature, for determiningthe temperaturedependen e ofstru tural properties. Moreover, the pos-sibility ofin ludingvibrationalmodes al ulatedatarbitrarywavelengths, should permit the study of more realisti systems.
prop-however littleis known, up tonow, about its usefullness atthe surfa e. The presen e of larger anharmoni ity than in the bulk ould, infa t, lead to a failure of this approxima-tion. Furthermore, toapplyQHA s heme vibrationalfrequen ieshavetobeevaluatedas afun tionofthevalueofthestru turalparametersdes ribingthe system,andup tonow, within a rst prin iples approa h, QHA has been used to study only systems des ribed byasmallnumberofstru tural parameters(bulkmaterialsorsurfa es inwhi honlyone layer is involved inthe thermalexpansion). The aimof the present work is toassess the validityof QHA inthe study of surfa e thermalexpansion, and toextend this s heme to systems more omplexthan those studied sofar.
Vibrationalfrequen ies an be omputed exa tly ineverypointof the BrillouinZone thanks to Density Fun tional Perturbation Theory (DFPT) [10,11℄. However, in order tostudy eÆ iently systems des ribed by alarge numberof parameters, itisne essary to al ulatethederivativeofthevibrationalfrequen ies. Thisquantityisrelatedtothethird variationof the totalenergy, and an be al ulated, thanks toanextensionof the \2n+1 Wigner'srule",withintheDFPT framework. Wehavedeveloped apra ti aland eÆ ient methodforthe al ulationof thethirdderivativeofthe totalenergyinametalli system, within the DFT and DFPT approa hes. This method has been expli itly implemented forthis workand has made feasiblethe study withinthe QHAof systems inwhi h many stru tural parametersare involved in the thermalexpansion.
As far as materials are on erned, in this work we fo us our attention on the ther-malexpansionof Be(0001),Be(1010), andMg(1010)surfa es. BerylliumandMagnesium are parti ularly indi ated to study the QHA, infa t, be ause of the light atomi masses, quantum e e ts areexpe ted tobemorepronoun edindeterminingthe lowtemperature stru ture, than in other materials. Moreover failure of phenomenologi al models in de-s ribing Be surfa e dynami al properties [15,16℄ make ne essary the use of an ab initio approa h to deal with this metal.
Beryllium (0001) surfa e has re ently attra ted mu h experimental [15,17,18℄ and theoreti al [19{22℄ interest, be ause of its large outward relaxation. Re ently [4℄ a large thermalexpansionofthe Be(0001)top layer,has been experimentallyobserved. The rst interlayer spa ing expands to 6:7%, with respe t to the bulk, at 700 K, and this result was somehow unexpe ted, however early al ulation [4℄, done within an oversimpli ed QHAs heme,resultsinagreementwith theexperiment. Theapproximationsused inthis al ulation have been riti ized by some authors [5℄, and this debate has attra ted our attention.
In this work we show an ab initio study of the thermal expansion of Be(0001) sur-fa e[23℄. Ourresults des ribe very wellsurfa e phonondispersionsand the bulkthermal expansion. A very a urate samplingof the vibrational modes, and a areful he king of all the approximations used is done. Furthermore, in order to assess the validity of the QHAwe ompareour resultswith rst prin iplesMDsimulations. Inspiteofthehigh a - ura yof our al ulations, they do not reprodu e the large measured thermalexpansion, and weargue that the a tualsurfa e ould be less ideal than assumed.
numeri al derivation of the vibrational frequen ies. This approa h annot be used for Be(1010),andMg(1010)surfa es,asthesesurfa esaremoreopenthanthe(0001)one,and many layers relax, and are involved in the thermal expansion. As previously mentioned these systems an bestudiedeÆ ientlyby omputinganalyti allythirdderivativesofthe total energy, and are investigated inthis work.
A re ent experiment for Mg(1010) surfa e [24℄ reports a negative thermal expansion of the rst interlayer spa ing, (where the surfa e region is still expanding as a whole), and this isone of the few ases reported so far. Our al ulation on rm this result, and predi ts the same behaviour alsointhe ase of Be(1010) surfa e.
Motivation of the work
The a urate study of thermal expansion, within a rst prin iples approa h,has be ome nowadays feasible. In this work we are essentially involved in the study of the thermal expansion of metalli surfa es within the QHA, and in this hapter we will show the usefulness of this s heme.
As mentioned in the Introdu tion one of the main motivations of the present study is to assess the validity of the QHA for the study of surfa e thermal behaviour, sin e it has not been widely used so far, and little is known about its a ura y in the surfa e ase. Moreover it has never been used to study systems des ribed by a large number of stru tural parameters.
In Se . 1 we review some on epts in latti e dynami s, and in Se . 2 we ompare two possible s hemes to deal with thermal expansion: Mole ular Dynami s and Quasi Harmoni Approximation. Inthe remaining ofthe hapter wedis uss somere ent exper-imentalresults on erning Be and Mg surfa es, that haveattra ted our attention.
1.1 Latti e dynami s
The study of the motion of a rystal for positions not far from equilibrium an be done within the harmoni approximation[25℄.
The underlying approximation is the Born-Oppenheimer or adiabati one. This is a widely used approximation in solid state physi s, and onsists in de oupling "fast" ele troni degrees offreedomfrom the"slow" ioni ones {byvirtue ofthe greatdi eren e of masses{inthe S hrodinger equationofintera tingele trons and ions. In the adiabati s hemeele trons are onsidered asquantumparti lesmovinginthe potentialof the xed nu lei, and they follows adiabati ally the ions, always remaining in their ground state. The ioni motion is then des ribed by an Hamiltonian, whose interatomi potential is the ground state energy of the system of intera ting ions and ele trons, in whi h ioni positionsare treatedas parameters:
H =T ion +E(fR l s g); (1.1) whereT ion
isthekineti energyofthe ions,E(fR l s
lassi allyor with aquantum me hani al approa h.
Sin e we are studying the system for on guration near the equilibrium we an ex-pandthe energy inaTaylorseries with respe t tothe atomi displa ements aroundtheir equilibriumpositions R 0 l s : E(fR l s g)=E(fR 0 l s g)+ 1 2 X l sl 0 s 0 2 E u s (R l )u s 0 (R l 0 ) 0 u s (R l )u s 0(R l 0 )+O(u 3 ); (1.2) where u s (R l ) = R l s R 0 l s
is the displa ement from equilibrium, and the linear term vanishesbe ause ofequilibrium. The harmoni approximation onsists intrun atingthis expansiontothe se ondorder, thus the lassi alequations of motion are:
M s u s (R l )= E u s (R l ) = X l 0 s 0 C s; s 0 (R l R l 0)u s 0 (R l 0 ); (1.3) whereM s
isthe massof the s th
atom,u s
isthe th
omponentin arthesian oordinates ofu s , and: C s; s 0 (R l R l 0) = 2 E u s (R l )u s 0(R l 0) 0 : (1.4) C s; s 0
is the interatomi for e onstant matrix, that depends onR l
R l
0
be auseof the translationalinvarian e of the latti e. Otherproperties are that:
C s; s 0 ( R l )=C s 0 ; s (R l ); X l ss 0 C s; s 0 (R l )=0 8 ; (1.5) This se ond property omes from the invarian e of the energy of the system under a ontinuous translation of the entire rystal withoutdistortion.
Duetotranslationalinvarian eof thelatti e,thesolutionofthe in nitesetof oupled Eqs.1.3 are normalmodes hara terizedby ave torqinre ipro alspa e andhavingthe form: u s (R l ;t)= 1 p M s v s (q)e iqR l i!t ; (1.6)
The equationsof motions be ome: ! 2 v s (q)= X s 0 D s; s 0 (q)v s 0 (q) (1.7)
whereD(q) is the dynami al matrix de ned as: D s; s 0 (q) = 1 p M s M s 0 X l C s; s 0 (R l )e iqR l (1.8) D(q)is a 3N at 3N at hermitean matrix {N at
is the number of atoms per unit ell{ and has the followingproperties:
D 0 (q)=D 0 ( q); D 0 (q) =D 0 (q): (1.9)
Any harmoni vibration of the rystal an be expressed as a linear superposition of normal-mode vibrations, and, given a q point, eq.1.7 de nes the 3N
at
orresponding normal modes and their vibrational frequen ies. A omplete des riptionof the harmoni properties of a rystal an thus be obtained from the knowledge of the interatomi for e onstants de ned in eqs. 1.4 or 1.8. The quantum me hani al problem, in the harmoni approximation an be solved in a ompletely analogous way [26℄.
1.2 Mole ular Dynami s and the Quasi Harmoni Ap-proximation
In a purely harmoni rystal, ion mean positions do not hange upon in reasing tem-perature, thus rystal thermal expansion is a onsequen e of anharmoni terms in the interatomi potential. Thermal behaviour of a system an be studied in di erent ways, and in this work we have used Mole ular Dynami (MD) simulations, and al ulations based on the Quasi Harmoni Approximation(QHA).
ToperformaMDsimulationmeansto omputethetime evolutionofasystem a ord-ingtotheNewtonianequationsofmotion. InafreeevolutionofasuÆ ientlylargesystem (mi ro anoni al run) the temperature, T, of the system is relatedto the time average of the kineti energy by:
3 2 K B T =h 1 2 mv 2
i. MDsimulationsa ount exa tlyfor interatomi potentialanharmoni ity, buttreats ioni degrees of freedom lassi ally,and an give reli-able results onlynear oraboveDebye temperature, wherequantum e e ts are negligible. Withina rstprin ipleapproa hMDsimulations anbeaheavy omputationaltask,and as a onsequen e omputationally treatable systems may result not large enough to be physi ally realisti . Systems that an be simulated,nowadays, typi ally onsist of about a hundred atoms and the time s ale of the simulation an rea ha few pi ose onds.
TheQHAprovidesa omplementaryapproa h,validbelowthemeltingtemperatureto determine the temperature dependen e of stru tural properties. In this approximation, the equilibrium stru tural parameters, a = (a
1 ;a
2 ;a
3
;:::), of a rystal, at any xed temperature T, are obtained by minimizing the Helmholtz free energy F of a system of purely harmoni os illatorshavingthe frequen iesof the vibrationalmodes of the rystal !
q
(a) in that on guration:
F(T;a)=E tot (a)+F vib (T;a)= =E tot (a)+ X ;q ( h! q (a) 2k B T +k B Tln 1 e h ! q (a) 2k B T ) ; (1.10) F(T;a) a =0; (1.11) where E tot
(a) is the stati energy of the rystal. To apply this model, the knowledge of !
q
(a) as a fun tion of the stru tural parameters is requested, thus in luding some anharmoni e e ts.
Appli ation of the QHA s heme is diÆ ult, from a omputational point of view, be-vib
al ulationmust be performed atvarious values of the stru tural parameters.
As an example,in the simple ase of anisotropi thermalexpansion of an h p rystal (the system is des ribed by two parameters that are the length of the two axis a and ) one an omputephonondispersionsinagrid ofpointsinthe (a; )spa eand interpolate inbetween the se ond term of eq. 1.10. On the other hand if the system isdes ribed by alarge number ofstru tural parameters (we willshowthat this is the ase forh p(1010) surfa es)the Free energy annotbe dire tly al ulatedvarying every parameter involved inthe expansion.
Inthegeneral aseone an pro eedbeingableto omputethe vibrationalfor es F
vib ai
. Asa rst approximation,assumingalineardependen e ofF
vib
onitsarguments,eq. 1.11 be omes: F vib a i + E tot a i (a) =0: (1.12)
Withina rstprin iples approa h E
tot a
i
(a) anbe al ulated ineverypointofthe spa ea thankstothe HellmannFeynmanntheorem [27℄,thuseq. 1.12 an beeasilysolved. On e obtained in this way an estimate of the equilibrium parameters of the system, an exa t solutiontothe problem ould inprin iplebeobtained by al ulatingvibrationalfor es in the new point and iteratingthe pro edure toself onsisten y.
Usinga slightlydi erent notationeq. 1.7 an be rewritten: ! 2 q v q =D(q)v q ; (1.13)
withthis notationthe derivative of the vibrationalfree energy is given by: dF vib da i = h 4 X ;q 1 ! q tgh 1 h! q 2k B T ! hv q j d da i D(q)jv q i P ;s jv ;s j 2 m s : (1.14) Sin e the dynami al matrix D(q) is the se ond derivative of the total energy with re-spe t to atomi displa ements, to al ulate the derivativeof F
vib
with respe t to atomi displa ementsthe al ulationof the third derivativesof the total energy is ne essary. 1.3 Beryllium vs. Magnesium
Be and Mg are simple metals with the same outermost ele troni on guration but dif-ferent properties. They are respe tively the 2
nd and the 3 rd element in the 2 nd olumn ofthe periodi table. In both ases the outermost ele troni on gurationof the isolated atom onsists of a lled s-shell (Be:2s
2
, Mg:3s 2
) and so, in order to form a bond, out-ermost ele trons must hybridize with p states. At room temperature and atmospheri pressure Be and Mg are stable in the hexagonal lose pa ked (h p) rystalline stru ture (Fig. 1.1). A more pronoun ed ovalent hara ter of the bonding in Be re e ts in some physi al properties that make Be di erent from Mg and from the other elements of the
1.3. BERYLLIUM VS.MAGNESIUM 9
c
a
Γ
K
M
A
H
L
Figure 1.1: S hemati drawingof theh p rystalstru ture,and ofthe orrespondingBrillouin zone.
The bonding energy of Be is surprisinglystrong when ompared with lose neighbors in the Periodi Table ( ohesive energy Be: 3.32 eV/atom, Mg: 1.51 eV/atom, Li: 1.63 eV/atom [28℄). Also the Debye temperature is very high (Be: 1440 K, Mg: 400 K, Li: 344 K [29℄), and thus quantum e e ts should be more pronoun ed in determining zero temperature stru tural properties.
There are some eviden e of dire tional bondingin Be. First of allBe =a ratio isone of the most ontra ted (' 4%) for h p metals, unlike the nearly ideal one of Mg ( ideal =a '1.63, Mg: 1.62, Be: 1.57), thus out-of-plane neighbours have shorter bonds than in-plane neighbours. Another eviden e of this anisotropy is that the ontribution tothe density of ele troni states oming from p
x
and p y
states is di erent from the one from p
z [30℄.
Berylliumdensityofele troni statesresemblessomewhatthatoneofasemi ondu tor, sin e ithas aminimumnear the Fermienergy,while the oneof Mgisnearly freeele tron like ('
p
) as shown in Fig. 1.2. In Fig. 1.2 the omparison between al ulated Be and Mgele troni dispersionisalsoshown . NotethatBerylliumbandsdisplayadire tgapin alargepartoftheshownBZ,unlikeMg. Mghasa lledstateat withenergy' 1:3eV, while the orresponding Be state is above the Fermi energy and its band is nearly at. This band is the sour e of both the low density of states near the Fermi energy and the high peakabove the Fermi energy.
Thevibrationalmotionofatomsina rystalisrelatedtothedire tion-ionintera tion and to the s reening properties of the ele troni distribution whi h an give rise to non- entralfor es between ions. Formany non-transitionmetalslatti e dynami s an be or-re tly des ribedintermsofpairwise entralfor es,andthis anbedone,approximatively, for bulk Mg [31℄. On the ontrary without taking into a ount non- entral for es it is impossibletodes ribeeven qualitativelyBe bulkphonondispersion[31{33℄. Forinstan e bygrouptheoreti alanalysis it an beshown, quitegenerally,thatatK=(0;2=3;0)2=a, in an h p rystal, there are three vibrational modes polarized in the basal plane. One of the modes is doubly degenarate, the other two are non degenerate. If the dynami al properties depend only on pairwise entral for es the frequen y of the degenerate mode
Be
Mg
-5
0
Electron energy (eV)
Γ
K M
Γ
A
H L
A K HM L
E
F
DOS
-10
-5
0
Electron energy (eV)
E
F
Figure 1.2: Comparison between Be and Mg ele troni band stru ture al ulated by rst-prin iple methods. Density of States(DOS) is also shown. In the DOS panel the free ele tron like DOS of the orresponding material is also shown. Be and Mg have the same outermost ele troni on guration, but while Mg DOS is nearly free ele tron like, Be DOS \resembles"
0
5
10
15
20
Frequency (THz)
Γ
K
K
5
K
3
K
1
0
2
4
6
8
Γ
K
K
1
K
5
K
3
Be
Mg
Figure 1.3: A urate al ulation of Berylliumand Magnesium bulkphonondispersionsof the three modespolarizedinthebasalplane inthe -Kdire tion. Theordering ofthethree modes at K forBe, isin ompatiblewith apairwise entralfor e model(see text).
one found in Be,unlikeMg (Fig. 1.3). 1.4 Be(0001) surfa e
Be ause ofits large outward relaxation[17℄ Beryllium(0001) surfa e has attra ted mu h experimental [15,17,18℄ and theoreti al [19{22℄ interest. It is a lose-pa ked surfa e, similartothe(111)surfa e ofanf rystal. Itiswellestablishedthatthe rst interlayer separation on most metalsurfa es is ontra ted atroomtemperature, Be(0001) is one of the fewex eptions reported sofar.
Until very re ently there was a substantial disagreement between experimental and rst-prin iples theoreti al [19,21,22℄ results on the amount of topmostinterlayer expan-sion in this system, theoreti al al ulations giving roughly half of the observed value. In a re ent letter, Pohl and oworkers [4℄ re on ile experiment and theory on this point showingthatlowtemperature(110K)low-energyele trondi ra tion(LEED) determina-tions of the rst interlayer separation ( d
12
'+3:3% at T=110K) are in agreement with rst-prin iples al ulationsandthatreporteddis repan iesatroomtemperatureoriginate from a large thermal expansion of the top layer, rea hing 6:7% at 700 K.These ndings are very interesting andpuzzling, sin e surfa e phonons shownosign ofenhan ed anhar-moni ity [4,15℄, however the al ulation of the surfa e thermal expansion [4℄, within a simpli edquasiharmoni approa hre entlyintrodu edinRef.[6,34℄,resultsinverygood
Table 1.1: Mg(1010) surfa e: LEED-IV measurements[24℄ of the two outer layers thermal expansion. d 12 (%) d 23 (%) T=120K -16.4 +7.8 T=300K -20.1 +9.4 T=400K -20.6 +10.6
Asmentionedpreviously,withinthequasi-harmoni approximationthedetermination ofd
12
(T) anbedoneby minimizingthe freeenergy ofthesystem (eq.1.10),with respe t tod
12
. Sin ethe evaluationofthe freeenergyrequires theknowledgeof thephonon spe -trumalloverthe Brillouinzone,that isaheavy omputationaltask,somesimpli ations have been introdu ed in Ref. [6,34℄ . In parti ular only three modes at the zone enter are al ulatedandthefrequen y soobtained areusedtosamplethe vibrationaldensity of statesinthe freeenergy al ulation. The validityof su h anapproa h has been riti ized by some authors[5℄, be ause of the very poorsampling of vibrationalmodes adopted.
In Chap. 4 we address this surfa e by performing full BZ sampling and omparing ourresultwith rst prin iplesMDsimulations. Careful he kingofalltheapproximation used willalsobe shown.
1.5 Mg(1010) surfa e
A s hemati drawing of the (1010) surfa e is reported in Fig. 1.4. This surfa e is more open than the(0001) one, and in prin iple, the trun ated bulk an be terminated in two ways,eitherwithashort rstinterlayerseparation,d
12
,asshowninFig.1.4,orwithalong one, (that an be obtained removing the top-layer fromthe short-terminated surfa e).
Re ently LEED measurementshave determined the termination and interplanar sep-aration of the lean Mg (1010) surfa e as a fun tion of temperature [24℄. The preferred terminationisthe one withshortd
12
and many layers areinvolved inthe relaxationinan os illatoryway: while the rst interlayer ontra ts the se ond expands and so on. Mea-surement (Tab.1.1) indi atesanegativethermalexpansionof the rst interlayerspa ing, and these resultshave not been on rmed by theory,yet.
InChap.4wereport onour study, withinQHAapproa h,ofthe thermalbehaviourof thissurfa e. Thiskindofstudy has beenpossible thankstothedevelopmentof amethod to al ulate third order derivativeof the totalenergy ina metalli system.
side view
top view
[1210]
[1010]
[0001]
d
12
d
23
Figure 1.4: S hemati drawingofthe(1010) surfa eofan h p rystal. The trun atedbulk an be terminated in two ways, either with a short rst interlayer separation d
12
, or with a long one. Here only the termination with the shortest d
12
is plotted, the long-termination an be obtained removingthe top-layer from theshortterminated surfa e. Both inBe and inMg the short termination hasbeen theoreti allyshown to be the moststable and the one observed in the experiments[24,35℄.
First prin iples methods
In the previous hapter (Se . 1.1 and 1.2) we have shown that a omplete study of dy-nami al properties and of thermal expansion, within the quasi-harmoni approximation, an beperformedknowingthese ondandthirdderivativesofthetotalenergyofasystem with respe t toatomi displa ements.
Inthis hapterwegiveanoverviewofthetheoreti altoolsusedinthisworkto al ulate rystalenergyanditsderivativesuptothirdorder. Wehaveusedabinitiomethodsbased on Density Fun tional Theory (DFT)[7,8℄. These methods have been developed in the last 30years and have demonstrated tobe one ofthe most powerfultool inthe theory of the ele troni properties of solids, with results and a ura y surprising and unexpe ted [36{38℄.
In order to al ulate the ground state energy of a quantum me hani al system, in prin iple, it is not ne essary to solve dire tly S hrodinger equation, and to know the ground state wave fun tion. Within the DFT s hemethe basi quantity tobe al ulated is the ele troni harge density, and inprin iplethese ond and thirdorder derivativesof the groundstate energy an be obtained fromthe knowledge of the rst variationof the ele troni harge density thanks to anextension of the \2n+1"theorem [39℄.
On e the unperturbed problem has been solved in the framework of DFT, Density Fun tionalPerturbationTheory(DFPT) [10,11℄providesaneÆ ientmethodto al ulate the ele troni linear response toanexternal perturbationofarbitrarywavelength. DFPT has been su essfully applied to predi t vibrational properties of elemental and binary semi ondu tors [11℄ or insulators [40℄, heterostru tures [41℄, semi ondu tor alloys [42, 43℄ and surfa es [44℄, furthermore it has allowed al ulation of phonon dispersions in metalli bulk[45℄ and surfa es[46℄. DFPT has alreadyallowed third order al ulationin semi ondu ting systems[47℄,and inthis hapterwewillshowanextensionofthis s heme to metals.
In Se . 1 we give an a ount of the quite standard ab initio methods used in this workto al ulate total energy. In Se . 2,and Se . 3,afteranoverview of DFPT weshow how vibrationalproperties an be al ulated in this framework. Finally in Se . 4 we will develop a pra ti aland eÆ ient method for the al ulation of the third derivativeof the total energy formetalli systems.
The Born-Oppenheimer, or adiabati , approximation, already addressed in Se . 1.1, re-du e the full ele troni + ioni problem tothe al ulation of the ele troni ground state energyin the potentialenergy determinedby xed ions.
2.1.1 Density Fun tional Theory
The theoreti al framework of rst-prin iples al ulation used in this work is given by Density Fun tional Theory, whi h has be ome one of the most powerful tools in the theory of the ele troni properties [36{38℄. W. Kohnis one of the developer of DFTand essentiallygavearigoroustheoreti alba kgroundtoit[7,8℄. Forthishehasbeenawarded withthe 1998 NobelPrize forChemistry.
Givenamany-bodyHamiltonianofintera tingele tronsinanexternalpotentialv(r),{ that in our ase is generated by ioni ores{ one an solve, in prin iple, the S hrodinger equation and obtain the ground state many-body wavefun tion j
G
i, the ground state energyE
G
, and the ele troni density n(r).
Hohemberg-Kohn theorem [7℄ states that, for a given n(r), the v(r) generating it is unique. A ordingtothis theorem we an de neF[n(r)℄as auniversal(inthe sense that itdoes not depend onv(r)) fun tionalof n(r):
F[n(r)℄=h G jH 0 j G i=E G [n(r)℄ Z v(r)n(r)dr (2.1) whereH 0
isthe Hamiltonianof the intera ting ele trons, without the external potential. For xed externalpotential,v(r),we an de ne the energy fun tional as:
E v
[n(r)℄=F[n(r)℄+ Z
v(r)n(r)dr: (2.2) Hohemberg and Kohnalsodemonstrate [7℄that if n(r) is allowed tovary overa range of fun tionssatisfyingthe requirement
N = Z
n(r)dr (2.3)
in orresponden e of the ground-state ele tron density the following stationaryprin iple isvalid:
ÆE v
[n℄ =0: (2.4)
ItfollowsfromthistheoremthatwithoutsolvingtheS hrodingerequation{andsowithout al ulatingthe wavefun tion ofa many-body system{ one an, in prin iple, al ulatethe ground state energy and the harge density of a system. This ould be done exa tly if the formof the fun tional 2.1 were known.
Appli ationofthisvariationalprin ipleallowstodedu everysimpleone-parti le equa-tions,known asKohn andShamequations [8℄. Inordertoderive themitis ustomaryto separate F[n(r)℄ inthe followingway:
F[n(r)℄=T 0 [n(r)℄+ 1 Z n(r)n(r 0 ) 0 drdr 0 +E x [n(r)℄ (2.5)
where T 0
[n(r)℄ is de ned as the kineti energy of a system of non intera ting ele trons having n(r) as the ground state density. The two following terms are ommonly alled Hartree term (E
H
),and ex hange- orrelation energy fun tional. Eq. 2.5 is the de nition of E
x
interms of the unknown fun tional F[n℄ and of twowell de ned quantities, T 0
[n℄, and E
H [n℄.
With this de nition, the set of KS equations is the following and is to be solved self- onsistently: " p 2 2m +V KS (r) # i (r)= i i (r) (2.6) V KS (r)= Z n(r 0 ) jr r 0 j dr 0 + ÆE x [n℄ Æn(r) +v(r) (2.7) n(r)= N X i=1 j i (r)j 2 (2.8) Thus in DFT the al ulation of the ground-state properties is redu ed to a problem of nonintera ting ele trons in ane e tive potential. The groundstate energy of the system is given by: E v [n℄=E I [n℄+ N X i=1 h i j p 2 2m j i i= = N X i=1 i + E I [n℄ Z V KS (r)n(r)dr ; (2.9) where E I [n℄= 1 2 Z n(r)n(r 0 ) jr r 0 j drdr 0 +E x [n℄+ Z v(r)n(r)dr: (2.10) 2.1.2 Lo al Density Approximation
ToimplementEq.2.6inpra ti e anexpli itexpression forE x
isneeded, andinthis work wehave used the Lo al Density Approximation(LDA):
E LDA x [n(r)℄= Z n(r) hom x (n(r)) dr; (2.11) where hom x
(n) is the ex hange and orrelation energy per parti le of a uniform ele tron gas of density n. It is a very natural approximation for a system with slowly varying density andit workssurprisinglywellfor alarge varietyof systems, even better than any early expe tations. In the last de ade, many improvements to LDA were proposed [48{ 50℄, in ludinginhomogeneity orre tionsviadensity gradientexpansions ofvariouskinds,
Thea tualsolutionof KSequations anbeobtainedexpanding the KS wavefun tionson a nite basis set. One of the most widely used hoi es for this basis set is that of plane waves (PW). From the Blo h theorem it omes out that:
;k (r)= X G e i(k+G)r (k+G) (2.12)
where k belongs to the rst Brillouin Zone (BZ) of the rystal, G is a re ipro al latti e ve tor and is the band index. The PW basis set is in nite and it is usually trun ated by hoosing a kineti energy uto through the ondition:
jk+Gj 2
E ut
: (2.13)
Totreatexpli itlyalltheele tronsitwouldbene essaryto hooseaverylargenumber ofPWinordertodes ribea uratelytheirrapidos illationsnearthenu lei. Thisproblem an be avoided freezing the ore ele trons in the atomi on guration around the ions, and onsidering only the valen e ele trons. To this end one introdu es pseudopotentials able to des ribe the intera tions between valen e ele trons and the pseudoions (ions+ ore ele trons). The resultingvalen e ele trons wavefun tions are onsiderably smoother near the nu leus, but are identi al to the \true" wavefun tions outside the ore region. This method is now well-established in omputational physi s, and its results are very a urate [51{55℄.
The widely-used norm- onserving pseudopotentials a t in di erent ways on the dif-ferent angular momentum omponents, l, of the wavefun tion, and, for ea h angular momentum,usually onsistof alo al ontributionforthe radial fun tionand anon-lo al one for the angularpart:
v s (r;r 0 )=v l o s (r)Æ(r r 0 )+ l max X l =0 v s;l (r)Æ(r r 0 )P l (^r;^r 0 ); (2.14) whereP l
is the proje tor on the angular momentum l. This form of semilo al pseudopo-tentialisstillnotthemost onvenientonefroma omputationalpointofview,andforthis reasonKleinmanand Bylander (KB)introdu ed [56℄ afully non-lo alpseudopotentialin whi h alsothe radial part of the potentialis non-lo al:
v (NL) s (r;r 0 )=v l o s (r)Æ(r r 0 )+ lmax X l =0 v (NL) s;l (r;r 0 ); (2.15) where v (NL) s;l (r;r 0 )= l X m= l v s;l (r)R s;l (r)Y m l (; )Y m l ( 0 ; 0 )R s;l (r 0 )v s;l (r 0 ) hR s;l jv s;l jR s;l i : (2.16) This form of the potential allows a very onvenient simpli ation of its matrix elements
2.1.4 Smearing te hnique
Cal ulation ofmany quantitiesfor rystallinesolids involvesintegration overkve tors of Blo h fun tions inthe Brillouinzone. These integration are ommonly approximated by summation over a nite set of sampling k points, weighted with the o upation number of the ele troni states:
F = (2) 3 Z BZ ( F (k))f(k)dk' N X i=1 ( F (k i ))f(k i ) (2.17) where
(k) is the energy of an ele troni state, F
is the Fermi energy and (x) is the step fun tion: x < 0 )(x) = 0, and x >0 ) (x)= 1. In the ase of metals, due to the presen e of energy bands that ross the Fermi energy, the integrand is dis ontinuous and alarge numberof points must beused to rea h suÆ ient a ura y.
The smearingapproa h onsistsinusing a ontinuous smeared fun tion ~
(x), hara -terizedbyasmearingwidth,insteadofthe stepfun tion(x). Smoothingtheintegrand the al ulation onverges better, but the results are obviously a e ted. The only justi- ation for this ad ho pro edure is that in the limit as ! 0 one would re over the \absolutely" onverged result (at the expense of using a prohibitively ne k mesh), and that even at nite value of one an obtain a urate results. Many kind of smearing fun tions an be used: Fermi-Dira broadening, Lorentzian, Gaussian [57℄, or Gaussian ombined with polynomials [58℄, tore all only someof them.
Withinthisapproa hweshouldsubstitute,intheKohn-Shamequations,eq. 2.8with: n(r)= X i ~ ( F i )j i (r)j 2 ; (2.18) Z n(r)dr=N = X i ~ ( F i ) (2.19)
where the Fermi energy F
is de ned by eq. 2.19. The natural de nition of the total energy of the system be omes:
E v [n℄= Z F 1 n()d+ ( E I [n℄ Z ÆE I [n℄ Æn(r) n(r)dr ) = = X i f ~ 1 ( F i )+ i ~ ( F i )g+ ( E I [n℄ Z ÆE I [n℄ Æn(r) n(r)dr ) (2.20) where ~ 1 (x)= R x 1 y ~ Æ(y)dy,and ~ Æ(x)= x ~
. It anbeshown thatthe newK-S equations ome out fromthe minimization of this total energy.
2.1.5 Non Linear Core Corre tion
In their simple original form, norm- onserving pseudopotentials substitute the valen e-only harge density to the total one in all ontributions to the total energy. While this is orre t, within the frozen ore approximation, for the Hartree term, this is only
ap-the valen e harge to ompute the ex hange- orrelation energy [59℄: E x = Z V (n v (r)+n (r)) x (n v (r)+n (r))dr; (2.21) where n
is the harge density of the ore ele trons, omputed as a superposition of the atomi ore harges ofthe atomswhi hrequire NLCC,and n
v
is thevalen e harge. The ore hargeis omputedonlyon e,togetherwiththepsuedopotentialandthenitisadded tothe valen e harge to ompute the ex hange- orrelation energy.
This orre tion is parti ularly important when there is a signi ant overlap between valen eand ore harge,asinthe aseof Berylliumand Magnesiumatoms,and improves the transferability ofthe orrespondingpseudopotentials.
2.2 Density Fun tional Perturbation Theory
On ethe unperturbed problemhas been solved inthe frameworkof DFT, DFPT [10,11℄ provides an eÆ ient method to al ulate the ele troni linear response to an external perturbationof arbitrarywavelength. DFPT has been developed for semi ondu torsand su essively adapted to metalli systems [45℄. We will state DFPT for metals in a way slightly di erent and less elegant than in [45℄, be ause in the present form it is more appropriateto the purpose of this work.
When a perturbation V (1) bare
is superimposed onthe external potential a ting ona KS system, the self onsistent potential is modi ed a ordingly: V
SCF ! V SCF +V (1) SCF . If V (1) SCF
is assumed to be known, from standard rst order perturbation theory the linear variation of a wavefun tion
(1)
and thus the linear variation of ele tron density n (1) anbe al ulated. The response toaparti ularexternalperturbation anbeobtainedby iterationuptoself- onsisten y. Thesystemofequationstobesolvedinthesemi ondu tor ase isthe following:
[H KS + P v i ℄jP (1) i i= P V (1) SCF j i i (2.22) n (1) = X i fj i ihP (1) i j+ :g (2.23) V (1) SCF (r)=V (1) bare (r)+ Z n (1) (r 0 ) jr r 0 j dr 0 + dv x dn n=n 0 (n) n (1) (r) (2.24) whereirunsovertheo upiedstates,P
andP
v
are proje torson ondu tionandvalen e states, and is an energy hosen so as eq. 2.22 not to be singular. Note that, in order to solve this system, the knowledge only of the unperturbed valen e ground states is required,and thus the al ulation an be implemented eÆ iently.
To extend this approa h toa metalli system, one an onsider an energy suÆ iently high, in order to separate partially o upied states from empty ones. Chosen an energy
E >
F
+3, where isthe width ofthe smearingfun tion ~
,eq. 2.22,and 2.23 be ome: [H KS + P v i ℄jP (1) i i= P V (1) j i i (2.25)
n (1) = X i ~ Fi fjP (1) i ih i j+ g+ v X ij ~ Fi ~ Fj i j j j ih j jV (1) SCF j i ih i j+ X i ~ Æ Fi (1) F j i ih i j (2.26) where P v (P
) is a proje tor on band states with energy E(> E), P v is a summation on states with energy
E, ~ Fi = ~ ( F i ), and ~ Æ Fi = x ~ (x) F i . Eq. 2.25 should be solved only for o upied states, and is to be hosen so as eq. 2.25 not to be singular. Alsointhis ase thesolutionofthesystem requirestheknowledgeofonlya nitenumber of states of the unperturbed K-S Hamiltonian{those with energy <
E{. 2.3 Se ond order and Linear response
The ele troni linearresponse toioni displa ements onsideredasexternalperturbations givesa ess tointeratomi for e onstants. Infa t, withinthe adiabati approximation,a latti e vibration an be seen as a stati perturbation a ting on ele trons, and the linear variation of ele tron density upon appli ation of an external perturbation determines energy variationup tothird order [60℄.
Given a Kohn-Sham system in whi h the bare external potential a ting on ele trons V
ion
isafun tionofsomeparameters,theHellmann-Feynmanntheorem[27℄statesthat: E el i = Z n (r) V ion i (r)dr; (2.27) where E el
is the ele troni ground-state energy relative to some given values of the pa-rameters , and n
is the orresponding ele tron density distribution. Di erentiating this expression one obtains:
2 E el i j 0 = Z " n (r) j 0 V ion (r) i 0 +n 0 (r) 2 V ion (r) i j 0 # dr: (2.28) From this equation it results that from the knowledge of the linear response of the ele -troni harge, the se ond derivative of the total energy, and thus the dynami al matrix, an be al ulatedas: D s; s 0(q) =D el s; s 0 (q)+D ion s; s 0(q) D el s; s 0 (q)= Z n(r) u s (q) ! V ion (r) u s 0(q) dr+Æ ss 0 Z n 0 (r) 2 V ion (r) u s (q=0)u s (q=0) dr: (2.29) There are two ontributions: one is due to the ele troni harge variation, while the se ond omesfromthe dire tele trostati intera tion between ions, and isessentiallythe se ond derivative of an Ewald sum [61℄. DFPT permits to al ulate linear response to perturbation having an arbitrary wavelength q, and so every term in eq. 2.29 an be
In this se tion we develop a method to al ulate the third order derivative of the total energy for a metalli system. This method allows the al ulation of the derivatives of the dynami al matrix that will be used to study thermal expansion within the Quasi Harmoni Approximation[25℄.
Inordinary quantum me hani sRayleigh-S hrodingerperturbationtheory thereexist a well known result, alled Wigner's 2n +1 rule [39℄, whi h yields energy derivatives throughorder2n+1startingfromtheknowledgeofthen thderivativeofwavefun tions. This rule omes from a general prin iple[62℄ and itsappli ability to self onsistent eld theories has been known sin e several years [63,64℄. '2n+1' rule has been generalized to DFT by Gonze and Vigneron [60℄, and has already been applied to semi ondu ting systems [47℄ but not yetto metalli systems.
In what follows we will indi ate the n th derivative of a quantity F, with respe t toa parameter , equivalently by F (n) or d n d n
F and we will use the following notations: ~ Fi = ~ ( F i ), ~ Æ Fi = ~ Æ( F i ),and ~ Æ (1) Fi = x ~ Æ(x) F i
. Takingthethirdorderderivative withrespe t toa generi parameter of equation 2.20 we obtain:
E (3) v = X i f( d 2 d 2 ~ Fi ) (1) i +2( d d ~ Fi ) (2) i + ~ Fi (3) i g+ d 3 d 3 ( E I [n℄ Z ÆE I [n℄ Æn(r) n(r)dr ) (2.30) From ordinaryperturbationtheory appliedto the Shrodinger equation Hj
i i= i j i i, it an bedemonstrated that: (1) i = h (0) i jH (1) j (0) i i (2) i = h (0) i jH (2) j (0) i i+2h (0) i jH (1) j (1) i i (3) i = h (0) i jH (3) j (0) i i+6h (1) i jH (1) (1) i j (1) i i+ 3h (1) i jH (2) j (0) i i+3h (0) i jH (2) j (1) i i (2.31)
The rst variationof the harge density is: n (1) (r)= X i f( d d ~ Fi )j i (r)j 2 + ~ Fi [ i (r) (1) i (r)+ ℄g (2.32) Thus using 2.31 and 2.32, equation 2.30 be omes:
E (3) v = Z H (3) (r)n(r)dr+3 Z H (2) (r)n (1) (r)dr+ d 3 d 3 ( E I [n℄ Z ÆE I [n℄ Æn(r) n(r)dr ) +K (2.33) where K = X i f6 ~ Fi h (1) i jH (1) (1) i j (1) i i+( d 2 d 2 ~ Fi ) (1) i ( d d ~ Fi ) (2) i +6( d d ~ Fi )h (0) i jH (1) j (1) i ig= X i f6 ~ Fi h (1) i jH (1) (1) i j (1) i i+ ~ Æ (1) Fi ( (1) i (1) F ) 3 +6( d d ~ Fi )h (0) i jH (1) j (1) i ig
Here and in what follows H refers to the Kohn-Sham self onsistent Hamiltonian. F ol-lowing [60℄, fromthe expli it al ulation of eq. 2.33 itresults:
E (3) v = Z v (3) ext (r)n(r)dr+3 Z v (2) ext (r)n (1) (r)dr+ + Z Æ (3) E x [n℄ Æn(r)Æn(r 0 )Æn(r 00 ) n (1) (r)n (1) (r 0 )n (1) (r 00 )drdr 0 dr 00 +K: (2.35) Eventually we have expressed the third variation of the total energy in a form in whi h it depends only on the rst variation of harge density and on the rst variation of the Kohn-Sham wavefun tions.
Eq.2.35 annotbeeasilyimplementedinitspresentform: wehavealreadyshown that it is possible to al ulate n
(1)
(r) in an eÆ ient way both for metals and semi ondu tors, however sin e fromstandard rst order perturbation theory
j (1) i i= X j 6= i j j i h j jH (1) j i i i j ; (2.36)
an expli it al ulation of 2.36 requires the knowledge of both valen e and ondu tion wavefun tions, moreover the possibility that the denominator approa hes to zero may lead tonumeri alinstability.
To over ome these two diÆ ulties, in the semi ondu ting ase, it has already been shown [65℄ that eq. 2.34 redu es to:
K = 6 v X i h (1) i jH (1) (1) i j (1) i i= 6 v X i h (1) i P jH (1) jP (1) i i 6 v X ij h (1) i P jP (1) j iH (1) ji ; (2.37) whereP
istheproje tor onthe ondu tionstates, P
v
isasummationovervalen estates, and H (1) ij =h i H (1) j
i. As shown in Se . 2.2 DFPT allows dire t evaluation of jP
(1) i
i, and inthe new form eq. 2.37 an be easily al ulated.
The metalli ase is slightly di erent. Let onsider an energy E >
F
+3, where is the width of the smearing fun tion
~
, in order to separate partially lled states and empty ones, and note that:
K = 6 X i ~ Fi h (1) i P jH (1) jP (1) i i+ +6 v X ij ~ Fi h (1) i P jH (1) j j iH (1) ji ~ Fj h i jH (1) jP (1) j iH (1) ji ij + +2 v X H (1) ij H (1) jk H (1) ki ij jk ki ( ~ Fi kj + ~ Fk ji + ~ Fj ik )+
+3 F : ij ij H ij H ji +2 i ~ Æ Fi h i jH jP i i ; + +3( (1) F ) 2 ( X i ~ Æ (1) Fi H (1) ii ) ( (1) F ) 3 ( X i ~ Æ (1) Fi ); (2.38) where P
is the proje tor on the band states with energy > E, P v is a summation on states with energy
E, ij = i j . Only jP (1) i i, orresponding to i < F +3, appears in this formula, and this quantity an be al ulated within DFPT, as shown in Se .2.2. Moreoverit an be he ked that ineq.2.38everyterm havingnullorvery small denominator have a well de ned limit that is nite and expli itly omputable. As an example: lim i ! j ~ Fi h (1) i P jH (1) j j iH (1) ji ~ Fj h i jH (1) jP (1) j iH (1) ji ij = h (1) i P jP (1) j iH (1) ji f ~ Fi + i ~ Æ Fi g+h (1) i P jH (0) jP (1) j iH (1) ji ~ Æ Fi : (2.39) To summarize the third variation of the total energy has been expressed, even in the metalli ase, inaforminwhi hitdepends onlyonquantities that an beobtained from theknowledgeof thelinearresponsefor a nite numberofstatesof theK-S Hamiltonian.
Stru ture and dynami al properties
In this hapter we present our study of low temperature stru ture and dynami al prop-erties of Be(0001), Be(1010), and Mg(1010)surfa es.
Surfa e-phonondispersionsforthese surfa eshavebeen re entlyobtainedbyEle tron Energy Loss Spe tros opy (EELS) [15,16,66,67℄. Whilegood agreement between theory and experimentfor Mg wasobtained previously [67℄, earlyattempts tointerpretBe data using bulk trun ated models [15℄ are signi antly di erent from experiment [15,16,66℄, as it isshown inFig. 3.1, and 3.2. Bybulk trun ated model itis meant a model inwhi h for e onstantsare al ulatedforthe bulkand usedalsotomodeltheintera tionbetween atoms at the surfa e. These models give even a qualitative disagreement with respe t to the experiment, in parti ular, in the Be(0001) ase, the sign of the Rayleigh wave (RW) dispersion from K to M pointin the surfa e BZ is in orre tlygiven by the model al ulation. Consequently a fully ab initio approa h is needed to des ribe Be surfa es dynami al properties.
Here we show that al ulations performed within DFPT [10,11℄ are in very good agreementwith experimentsforallthe mentionedsurfa es. These resultsallowthestudy of the thermalproperties within the quasi harmoni approximation, as willbe shown in Chap. 4.
3.1 Be and Mg bulk
As a preliminary step, we have omputed the stru tural and latti e-dynami al proper-ties of the bulk metals. Our al ulations have been performed within the lo al-density approximation (LDA) using pseudopotentials and plane-wave (PW) basis sets. Atoms were des ribed by separable pseudopotentialsthat in ludenon-linear ore orre tion and have been generated so as to optimally reprodu e several atomi on gurations [68,69℄, toenhan etransferability. Ourbasisset in ludedPW'suptoakineti energy uto of22 Ry,and16Ry,forBerylliumandMagnesiumrespe tively. ForbothmaterialsBZ integra-tions were performed with the smearing te hnique of Ref. [58℄ using the Hermite-Gauss smearing fun tion of order one, a smearing width =50 mRy, and a 120-points grid in the irredu iblewedge of the bulkBZ. A urate he ks haveshown that these parameters result in a satisfa tory onvergen e of all the al ulated quantities. Values of the total
Figure 3.1: Be (0001) surfa e phonon from EELS [15,66℄. The lled (open) ir les indi ate intense(weak)featuresinthemeasureddispersion. Solidlinesindi atethe al ulateddispersion ofsurfa e modes fora bulk-terminatedslab. The shaded area orrespondsto the proje tion of bulkphonon modesonto (0001) surfa e(Fig. from [15℄).
Figure3.2: Be(1010)surfa ephononfromEELS[16℄. The lled(open) ir lesindi ateintense (weak) features in the measured dispersion. Solid lines indi ate the al ulated dispersion of surfa e modes for a bulk-terminated slab. The shaded area orresponds to the proje tion of
Table 3.1: Comparison betweenmeasured and al ulated h p latti e- onstants, a and , bulk modulus,B, and Poissonratio,
P
, ofbulkBeand Mg.
a(a.u.) =a B(Mbar) P Be(exp.) 4.33 1.568 1.1 0.02-0.05 Be(teo.) 4.25 1.572 1.25 0.04 Mg(exp.) 6.06 1.623 0.35 0.28 Mg(teo.) 5.93 1.629 0.40 0.23
energy al ulated at di erent latti e parameters a, with a xed =a, and a xed kineti -energy ut-o , have been tted to a Bir h-like equation of state [70℄. The equilibrium energy, obtained this way for di erent =a, are tted to a third order polynomial. The resulting equilibriumstru tural properties are in reasonableagreement with experiments (Tab. 3.1). Note, in the Be ase, the ontra ted value of =a and the small value of the Poisson ratio, indi ating ratherstrong and anisotropi bonding along the axis.
In Figs. 3.3, and 3.4 the al ulated phonon dispersionsof the two bulkmaterialsare displayed and ompared with neutron-di ra tion data from Refs. [71,72℄. Our al ula-tions are doneatthestati equilibrium on guration, thatiswithouttaking intoa ount latti e expansion due to nite temperature e e ts. Experimental data for Be are taken at T=80K, and the overall agreement is very good. Data for Mg are taken at T=290K. In Chap. 4it is shown that, in the ase of Magnesium, the agreement greatly improves, onsideringthermale e ts. ForBerylliumagreementisslightlyworsebutstillreasonable. Notethat ourabinitio al ulationshavenoadjustableparameterandtheonly un on-trolledapproximationsaretheadiabati oneandLDAtodealwithele troni orrelations. ThegoodagreementobtainedintheBeryllium aseisremarkable,sin eforthismaterialit is not easy toreprodu e a urately experimental data withempiri al Born-vonKarman s heme|as, for instan e, in Ref. [15℄|even after extensive tting of the experimental dispersion relations.
3.2 Be (0001) surfa e
To des ribe the surfa e we adopted a repeated slab geometry with 12-layer Be slabs separated by a 25 a.u. thi k va uum region(equivalent to8 atomi layers) tode ouple the surfa es. It has been he ked that this number of layers is enough to re over bulk properties in the middle of the slab. In the BZ integrations we used a 30-point grid, obtainedproje tingthebulkgridonthesurfa eBZ.Atomi positionsintheslabwerefully relaxedstartingfromthetrun atedbulkones,keepingthein-planelatti eparameter xed at the bulkvalue. Symmetry xes atomi in-planepositionsand relaxationinvolvesonly modi ationof the inter-layer spa ing. The three outmost layers relaxsigni antly from the bulkvalue,inagreementwithexperimentaleviden e[4,17,18℄. The al ulatedvalues fortheinterlayerspa ingvariationarereportedinTable3.2,alongwithexperimentaldata [17℄ andprevioustheoreti alresults. Ourtheoreti al al ulationagreeswellwithprevious
0
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Figure3.3: Cal ulatedphonondispersionsforbulkBe(lines)andneutrons atteringdatafrom Ref.[71℄(fulldots). The al ulateddensityofstatesisalsoshown. Experimentaldataaretaken atT=80K, whiletheoreti al al ulationsaredoneat the stati equilibriumlatti e spa ing.
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Figure3.4: Cal ulatedphonondispersionsforbulkMg(lines)andneutrons atteringdatafrom Ref.[72℄(fulldots). The al ulateddensityofstatesisalsoshown. Experimentaldataaretaken
Table 3.2: Relaxation of thethree outer layers of Be(0001) as obtainedbyLEED andseveral ele troni stru ture al ulations. The experimentaltemperature is shown,while forthe al u-lations the ex hange and orrelation fun tional used and the number of layers in the slab are indi ated. d 12 (%) d 23 (%) d 34 (%) Exp: LEED (110 K) [4℄ +3.1 +1.4 +0.9 LEED (300 K)[17℄ +5.8 -0.2 +0.2 Th: LDA-12layers (this work) +3.2 +1.0 +0.4 LDA-10layers [4℄ +2.9 +1.1 +0.4 LDA-11/13 layers [21℄ +2.7 +1.2 +0.6 LDA-9 layers [19℄ +3.9 +2.2
GGA-9layers [22℄ +2.5
determination [17℄. The use of di erent ex hange and orrelation fun tionals does not improve the omparison [22℄. Re ently the importan e, in order to get lose to LEED results, of in luding in rst-prin iples al ulations the e e t of zero-point vibrationshas been suggested fortransitionmetalsurfa es [6℄. In thepresent ase, however, al ulation shows (next hapter)that zero-pointvibrationsdonot modifysigni antly the toplayer relaxation.
Very re ently a new, low temperature (110 K), LEED-IV determinationof the stru -tural properties of Be (0001) surfa e has been obtained [4℄. The agreement between theoreti al results and this new experimental determination is very good, the value for the topmostlayerexpansionbeing3.1%,withsimilaragreementforthe innerlayers. Dis-agreement with previous experiments [17℄ seems to be due to a large extent to a very strong temperaturee e t [4℄. However, rst-prin iples al ulation of the surfa e thermal expansion givesonly arelatively small e e t, as willbe shown inthe next hapter.
Real-spa e interatomi for e onstants (IFC) of a 12-layer slab were obtained from dynami al matri es, al ulated ona 66 grid of pointsin the surfa e-BZ. Althoughthe surfa e IFC's are well onverged and re over the bulk values in the middle layers of our slab a thi ker sample is ne essary to de ouple those surfa e vibrations that penetrate deeply in the bulk. The dynami al matri es of a 30-layer slab were built mat hing the surfa e IFC's to the bulkones inthe entralregion, assket hed in Fig. 3.5.
The resulting phonon dispersions, obtained by Fourier interpolation, are reported in Fig. 3.6 together with an indi ation of the surfa e hara ter of the al ulated modes. Open dots represent modeslo alizedmorethan50% inthethreetopmostlayers (dotsize is proportional to this per entage), and full dots represent modes lo alized more than 30% inthe topmostlayer, and polarizedperpendi ularlytothesurfa e. It isevident that manylayers areinvolvedinthesurfa edynami s. Comparisonintheexperimentisshown in Fig. 3.7.
A very good agreement is found between the experimental Rayleigh Wave, the most learly revealed peaks in the experiment, and the al ulated surfa e vibration that is essentially on entrated in the rst layer and polarized perpendi ularly to the surfa e
Figure3.5: Sket h of thepro edure used to model thefor e onstant matrixC of athi kslab (onthe right). Thefor e onstants al ulated forarelaxed slab (S blo ks)are usedto des ribe the interatomi intera tion in the topmost surfa e layers (solid lines) of a thi k slab. Dashed lines indi ate bulk layers inserted in order to onstru t the thi k slab. The for e onstants des ribingbulk-bulkandbulk-surfa eintera tionsareindi atedbytheshadedblo ks,and have beenassumed to be equalto theinteratomi for e onstantsof thebulk.
However our al ulation predi ts two additional modes below the bulk-band edge that are not observed in the experiment. These modes are peaked in sub-surfa e layers and have both in-plane and out-of-plane omponents. Wesuggest that they are not observed in the experiment be ause of the strong intensity of the rather lose RW, whi h, in the regionwhere threemodesare predi ted, isfor more than70 %lo alized inthe rst layer and z-polarized.
Many of the additional surfa e vibrations in Fig. 3.7 agree qualitatively with weak features present in the experiment (open squares). In parti ular at the M point a shear horizontalmodehas beenexperimentallyobserved [66℄ intheappropriate geometry. The experimentalvalue of 50.5 meVagrees well with our result (50.0 meV).
3.3 Be(1010) surfa e
As hemati drawingofthe(1010)surfa eofanh p isshowninFig.1.4. Twotermination arepossible,intheBeryllium asethemoststableterminationistheonewiththeshortest rstinterlayerseparation[35℄. Beryllium(1010)surfa eismoreopen thanthe (0001)one andalargernumberoflayers areinvolvedintherelaxation,thusalargernumberoflayers is ne essary to des ribe this surfa e. We have used a 16-layerBe slabs separated by a 6 atomi layers equivalentva uumregiontode ouplethe surfa es. Weuseda32-pointgrid inthe irredu ible surfa e Brillouinzone (SBZ), hosen in order to give a similar density ofpointsasthe proje tion ofthe bulkgrid ontheSBZ.Atomi positionsinthe slabwere
Γ
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Figure 3.6: Cal ulated phonon dispersion for a 30-layer slab modeling the Be(0001) surfa e. Surfa e modesarerepresentedwithdots: modeslo alizedmore than50% inthethree topmost layers are shown, and the dot size is proportional to this per entage. Full dots orrespond to modes lo alized more than 30% in the topmost layer and polarized perpendi ularly to the surfa e.
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Figure 3.7: Comparison between EELS data from Ref. [15℄ and al ulated surfa e vibrations of Be(0001) surfa e. Full and open graysquares orrespond to experimental data (intense and weak features, respe tively). Dots orrespond to al ulated modes (see aption of Fig. 3.6).
Table 3.3: Relaxation of the four outer layers of Be(1010) surfa e, ompared with previous ele troni stru ture al ulation and experimental LEED results. Experimental errorbar are showninparenthesis. d 12 (%) d 23 (%) d 34 (%) d 45 (%) Theory (this work) -24.5 +6.6 -14.8 +4.7
Theory [35℄ -20 +4.4 -13 +3.8
Experiment[35℄ -25(-4/+3) +5(-3/+5) -11(-5/+8) +2(-2/+4) xed atthe bulkvalue.
The al ulated values for the interlayer spa ing variation are reported in Table 3.3, along with re ent experimental LEED stru tural data and previous theoreti al results [35℄. Goodagreement isfound withinexperimentalerrorbar. The numeri alvalues ofd
12 and d
34
relaxations seem to be very large ( 25% and 10% respe tively), but it should be kept in mind that they refers to variation of a small quantity: the short interlayer spa ing(d'1:3a.u.). DuetoBeratherlightatomi mass,apre isedeterminationofthe bulkand (1010) surfa e stru ture of Be shouldtakeintoa ount alsozero pointmotion. Inthe next hapter wewillshowthat in lusionof this termdoesnot a e t the interlayer relaxationinan appre iableway.
Surfa e-phonon dispersions of Be (1010) were al ulated by sampling the SBZ of our 16-layer slab on a 64 grid of points and Fourier interpolating dynami al matri es in betweentoobtainreal-spa einteratomi for e onstants(IFC).Thesurfa eIFC'sarewell onverged and re over the bulk values in the middle layers of the slab. The dynami al matri esof a 104-layer slab were therefore built mat hing the surfa e IFC's to the bulk ones in the entral region, with the pro edure sket hed in Fig. 3.5. A thi ker slab than the one used to des ribe (0001) surfa e is ne essary sin e vibrational modes penetrate moredeeply inthe bulk.
In Ref. [16℄ EELS spe tra have been taken in the M and A dire tions in the SBZ. Sin e in both s attering geometries the sagittal plane oin ides with a mirror plane of the rystal, vibrations polarized perpendi ularly to this plane (shear horizontal modes) annotbe ex ited[73℄. In the followingwewill allforbidden modes thosemodes that annot be dete ted in the experimental on guration used for the orresponding momentum.
In Fig. 3.9 the dispersion of a slab modeling the Be(1010) surfa e is shown together withanindi ationofthesurfa e hara terofthe al ulatedmodes. Dotsrepresentmodes lo alized more than 25% in the three topmost layers (dot size is proportional to this per entage), gray dots represent modes with a polarization forbidden in the s attering geometryused inRef.[16℄, and full dotsrepresent modes lo alizedmore than25% inthe topmostlayer, and polarizedperpendi ularlyto the surfa e.
Comparison with experiment is shown inFig. 3.10. The interpretation is straightfor-wardatzoneboundaries. Atthe Apointonlyone verypronoun edpeakismeasured(see Fig.3.8), while threesurfa e modes arepredi ted by al ulation. The twomodes at 26.4 and 32.3 meV are mainlypolarized perpendi ularly to the surfa e and are on entrated
Figure 3.8: Be(1010) surfa e: a set of typi al loss spe tra taken at the high-symmetrypoints of thesurfa e Brillouinzone (Fig. from [35℄)
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Figure 3.9: Cal ulated phonon dispersion for a slab modeling the Be(1010) surfa e. Surfa e modes of a 104-layer slab are represented with dots: modes lo alized more than 25% in the three topmost layers are shown,and the dotsize isproportionalto thisper entage. Gray dots orrespond to modes with polarizationforbidden in the s attering geometry used in Ref. [16℄, thatis: modeshavingmomentum alongthe M ( A)SBZ dire tionand polarizedalong A ( M)dire tionareforbidden. Fulldots orrespondto modeslo alizedmore than25% inthetopmostlayerand polarizedperpendi ularlyto thesurfa e.
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Figure3.10: Comparison betweenEELSdata from Ref.[16℄ and al ulated surfa evibrations ofBe(1010) surfa e. Full and open graysquares orrespond to experimental data (intense and weak features, respe tively). Dots orrespond to al ulated modes (see aption of Fig. 3.9) not forbidden in the s attering geometry used in Ref. [16℄. Thi k lines delimit the bulkband
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Figure3.11: ComparisonbetweenEELSdatafrom Ref.[16℄ and al ulatedvibrationalSurfa e Density of States(SDOS) of Be(1010) surfa e. Fulland open dots orrespond to experimental data (intense and weak features, respe tively). Shaded region represents SDOS. Modes are weighted a ording to their lo alization in the 4 outermost layers and symmetry forbidden modes are not onsidered. Di erent olor orrespond to di erent intensity a ording to the orresponden e reported intheinset.
even inside the bulk band edge. Intensity and width of this peak probably mask the othermode, as an be argued fromthe reported loss spe tra [16℄ shown inFig. 3.8. The theoreti almode at 63.7 meVis polarizedalong M and is thusforbidden. At the M pointexperimentshows twopeaks, while al ulation ndsmanymodes withdi erent po-larizations. Thereare three al ulatedmodes essentiallypolarizedperpendi ularlytothe surfa e at 41.4, 49.8 and 56.0 meV. The rst two reprodu e very well the two measured losses, the lowest one being essentially on entrated in the rst layer and orresponding tothe most intense experimentalloss.
Again the reported loss spe tra [16℄ shown in Fig. 3.8 suggest that the vi inity of the three modes and the intensity of the rst two peaks probably mask the weaker loss at56.0 meV. Cal ulation predi ts other modes with polarizationparallel to the sagittal planebetween 69.4and76.5 meV,they are lessintenseandnearone totheother,sothey probably annot be separated and are possibly masked by the tail of the Rayleigh wave. Cal ulationalso predi ts at 54.3, 61.1 and 68.7 meV three modes with very pronoun ed surfa e hara ter but polarized along the A dire tion and thus forbidden in the experimental on guration.
At zone enter the situation is less lear. Three losses are experimentally observed, measuredinthe two di erents attering geometries(see Fig. 3.8). Amongthe al ulated modes with the most pronoun ed surfa e hara ter, one (at 45.9 meV) is polarized per-pendi ulartothe surfa e and reprodu es wellthe energy andthe dispersionof the lowest measuredloss. Other al ulatedmodesare polarizedinthesagittalplanealongthe M (52.9, 73.9 meV) or A (84.78 meV) dire tions. These modes are spread on the 5-6 topmostlayers, so they probably are not easily dete table by EELS.
Ingeneral, surfa eenergy losses may derivenot onlyfrom ex itationofa singlemode lo alizedatthe surfa ebut alsofromregions of the bulkspe trum wherethe vibrational motionat the surfa e is high due to high vibrational Surfa e Density of States (SDOS), inspite of thesmall ontributionof ea hindividualmode. From the omparison between experimentaldataand al ulatedSDOS(Fig.3.11)we anasso iatetheexperimentalloss around 63.3 meV to a zone of high SDOS deriving from the high DOS in the bulk (see Fig.3.9) and not fromlarge surfa e ontributionofa singlemode. Note thatat the zone boundariesthe Rayleighwave xsisessentially on entratedinthe rst layerandgivesrise tothe most intense losses, thus we think that weak losses deriving from high SDOS an be learly dete ted onlynear zone enter.
The third experimentally observed loss at (at 55.2 meV) does not orrespond to any al ulated surfa e mode or resonan e of allowed symmetry. From Fig. 3.9 it an be seen that a forbidden mode is a tually present in that region of the spe trum. At the presentstage we an onlyspe ulate thatthis weak experimentallossmightresult froma lo albreakingofthesele tionrulefortheidealsurfa e[73℄ duetoimpuritiesorroughness probablypresent at the Be (1010) surfa e and not in luded in the al ulation.
In on lusion we nd that ab initio al ulations of the vibrational properties of Be (1010) surfa e reprodu e very well the energy and dispersion of many experimentally observedlosses. Inparti ularmodespolarizedperpendi ularlytothesurfa eandlo alized
Table 3.4: Relaxation of the four outer layers of Mg(1010) surfa e, ompared with previous ele troni stru ture al ulation and experimental LEEDresults, taken at T=120K.
d 12 (%) d 23 (%) d 34 (%) d 45 (%) Theory (this work) -19.0 +7.9 -10.8 +3.9 Theory [74℄ -14.9 +6.9 -7.5 +1.7 Experiment[24℄ -16.4 +7.8 | | at the zone boundaries.
3.4 Mg(1010) surfa e
As usual to des ribe the surfa e we adopted a repeated slab geometry with 16-layer Mg slabs separated by a 6 atomi layers equivalent va uum region to de ouple the surfa es. Using a 24-point grid in the irredu ible surfa e Brillouinzone (SBZ)all al ulated prop-erties resuted well onverged. Atomi positions in the slab were fully relaxed starting from the trun ated bulk, keeping the in-planelatti e parameters xed at the bulkvalue. Furtherdetails are in Se . 3.1.
Again many layers are involved in surfa e relaxation. The al ulated values for the interlayer spa ingvariationare reported in Table 3.4, along with experimental,low tem-perature, LEED stru tural data [24℄ and previous theoreti al results [74℄. Reasonable agreement is found. Using a slab of 22-layers and a 20 Ry plane wave ut-o results do not hange signi atively.
Surfa e-phonon dispersionsof Mg (1010)were al ulatedby samplingthe SBZ ofour 16-layer slab on a 42 grid of points and Fourier interpolating dynami al matri es in betweentoobtainreal-spa einteratomi for e onstants(IFC).Thesurfa eIFC'sarewell onverged and re over the bulk values in the middle layers of the slab. The dynami al matri es of a 96-layer slab were therefore built mat hing the surfa e IFC's to the bulk ones in the entral region (see Fig. 3.5). Dispersion is shown in Fig. 3.12 together with an indi ation of the surfa e hara ter of the al ulated modes. Dots represent modes lo alized more than 30% in the three topmost layers (dot size is proportional to this per entage). Full dots represent modes lo alized more than 25% in the topmost layer, and polarizedperpendi ularly tothe surfa e.
Goodagreement is found between measured losses and the al ulated Rayleigh wave dispersion (Fig.3.13). Comparing the two analogous Be(1010)and Mg(1010)surfa es, it is remarkable thatthe dispersion inthe A dire tion, of the mode on ned inthe rst layer has di erent sign inthe two materials.
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Figure3.12: Cal ulatedphonon dispersionfor a96-layer slab modeling theMg(1010) surfa e. Surfa emodes arerepresentedwithdots: modeslo alizedmore than30%in thethree topmost layers are shown, and the dot size is proportional to this per entage. Full dots orrespond to modes lo alized more than 25% in the topmost layer and polarized perpendi ularly to the surfa e.
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Figure3.13: ComparisonbetweenEELSdatafromRef.[67℄ (graydots) and al ulatedsurfa e vibrationsofMg(1010)surfa e. Dots orrespondto al ulatedmodes(see aptionof Fig.3.12).
Thermal expansion
In this hapterwe present a rst-prin iplesstudy, within the quasiharmoni approxima-tion, ofthe thermalbehaviour of Be(0001),Be(1010),and Mg(1010) surfa es.
Our al ulation des ribes well the thermal expansion for the bulk materialsand has been he ked against rst-prin iples mole ular dynami s simulations in the (0001) sur-fa e ase. We do not nd the large (0001) surfa e thermal expansion re ently observed experimentally [4℄ and we argue that the morphologyof the a tual surfa e ould be less ideal than assumed.
Inthe aseofBe(0001)onlyonelayeris,essentially,involvedintherelaxation,andthe study of thethermalexpansion, has beendone bydire t derivationof thevibrationalfree energy. The more omplex Be and Mg (1010) ases, where many layers are involved in the expansion, havebeen studied al ulatingthethird ordervariationofthe totalenergy, with the method des ribed in Chap. 1. In both (1010) surfa es QHA predi ts negative thermalexpansionofthe rstinterlayerspa ing,thoughsurfa esasawholeexpand. This behaviour has been re ently experimentallyobserved at Mg(1010)surfa e [24℄.
4.1 Be and Mg bulk
In this se tion the thermal expansion of bulk Be and Mg in the h p stru ture is stud-ied within the QHA s heme. The stru tural properties shown in Chap. 3, were instead obtained minimizingonly the stati energy, E
tot
(a) in eq.(1.10).
Fortheh p stru turethe QHAfreeenergy isafun tionof thetwoaxislengths,a and , and has been al ulated from rst prin iples on a grid of points in the two-parameter spa e. The vibrational ontribution to the free energy resulted to be remarkably linear in the two parameters in both materials. Due to low atomi number of Be and Mg, the ontribution of zero-point vibrationsis expe ted to bemore importantthan in other systems, espe ially for Beryllium, and in fa t it results in an in rease in both latti e parameters of 0:7% in Be and 0:3% in Mg. The agreement with experimental data is thus further improved (see Tab. 4.1).
The al ulated temperature variations of the bulk latti e parameters are reported, together with available measurements, in Figs. 4.1, and 4.2. From this omparison we on lude that QHA a ounts well for Be and Mg bulk anisotropi thermal expansion in
