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in the complementary metal oxide semiconductor ( cmos ) devices , sio@xmath3 has been the first and only choice of gate oxide material for a long time . however , with the continued decrease in the feature size of the cmos devices , sio@xmath3 is no longer reliable as a gate oxide due to the high tunneling leakage current through it at small thickness . major research efforts are continuously being carried out in search for a suitable replacement of the sio@xmath3 as a gate material.@xcite several promising candidates are metal oxides such as hfo@xmath3,@xcite zro@xmath3@xcite and al@xmath3o@xmath5,@xcite all of which have high value of dielectric constant @xmath0 . high-@xmath0 metal oxides in their amorphous ( a- ) form are more preferable as a gate oxide over their crystalline form due to several important advantages they provide : ( i ) isotropic physical properties ; ( ii ) no crystalline domain boundary which leads to less defects at the interface with the si substrate ; and ( iii ) good compatibility with the conventional cmos fabrication process . some alloy structures of these high-@xmath0 metal oxides are also being studied extensively@xcite and , in fact , a hf based alloy material has already been in its third generation of production as a gate oxide in the semiconductor industry and further improvements of thermal stability , dielectric constant and material preparations are underway.@xcite    in clear contrary to the abundance of experimental results in the literature , theoretical studies of the structure and dielectric properties of amorphous metal oxides and their alloys are quite limited . one of the reasons is the difficulties in generating reasonable and reliable amorphous structures with the available theoretical methods . experimentally , the high-@xmath0 materials are deposited on silicon substrate by vapor deposition followed by annealing processes at around @xmath6 k.@xcite simulating the deposition process and the resulting amorphous high-@xmath0 material structures by molecular dynamics ( md ) is extremely time consuming because . the as - deposited a - hfo@xmath3 and a - al@xmath3o@xmath5 films have been generated by kinetic monte carlo ( kmc ) methods@xcite with certain assumptions such as the atoms are kept fixed on pre - determined crystal sites and less important dynamics processes of atomic motion are omitted . both md and kmc are useful for the simulation of the initial atomic layer deposition processes of the amorphous film growth . to generate bulk amorphous structures , _ ab inito _ md simulations can be used in a melt and quench scheme , a - hfo@xmath3,@xcite a - zro@xmath3@xcite and their silicates@xcite have been successfully simulated this way . the activation - relation technique is another method to generate structures of continuous disordered systems@xcite and has been applied to generate a - zro@xmath3 structures.@xcite in this method , one optimizes the structures many times to determine a local energy minima ( in configurational space ) using a force field which can be calculated from quantum mechanical first principles or described by an empirical potential function . for the former the calculation cost is high while for the latter a reliable potential function is required . recently , several extended - tersoff potentials@xcite were proposed for certain metal oxides and applied to generate their amorphous structures from classical md.@xcite    amorphous structures are long range disordered systems which typically requires much larger than @xmath7 atoms to simulate . another issue , perhaps more difficult to deal with , is the simulation time scale that is needed for the atoms to evolve into the amorphous structure . while _ ab initio _ methods have simulated up to picoseconds time scale , one certainly wishes to investigate the structures and electronic properties of metal oxides at much larger time scales and much larger sizes in order to reveal the long time limit of the structural evolution and the physical properties of the resulting amorphous material . it is the purpose of this work to report such an investigation .     generated by classical md starting from the cubic crystal form of hfo@xmath3 with different lattice constants , corresponding to different densities . if the density is high , the system recrystallizes to its cubic form ( a ) or any other crystal form ( b ) ; if the density is low , the system tends to form disordered structure with big holes ( d ) , whereas with moderate density of @xmath8 g/@xmath9 which corresponds to lattice constant of @xmath10  for crystalline hfo@xmath3 , a reasonable amorphous structure is generated ( c ) . blue spheres represent hf atoms , red spheres are for o atoms.,title="fig : " ] +    in particular , we calculate the lattice dielectric constants of bulk amorphous metal oxides by a combined approach of classical md - for structure evolution , and quantum mechanical first principles dft - for electronic structure analysis . we overcome the size and time scale difficulty by classical md simulations in a melt and quench scheme and generate amorphous structures of hfo@xmath3 , zro@xmath3 and their alloys in the form of hf@xmath1zr@xmath2o@xmath3 with different values of the concentration x ( @xmath11 ) . the classical md is based on the interatomic born - mayer - buckingham potential function.@xcite this potential has been developed over many years and extensively applied to investigate the growth processes of y@xmath3o@xmath5 , zro@xmath3 and mg - al - o thin films.@xcite detailed analysis of the amorphous structures are carried out including the coordination number , the radial distribution function and the potential energy per atom of the structures . using the atomic structures generated this way , we carry out first principles density functional theory ( dft ) calculations of the lattice dielectric constant and the results are averaged over ten independent samples for each material . the sample averaged dielectric constants show very good agreement with the corresponding experimental data . the combined approach of classical md and dft is an excellent way to deal with the problem of large sizes and large time scales required by the materials physics of metal oxides . several advantages can be summarized : ( i ) classical md is inexpensive for simulating systems containing large number of atoms which is important for a proper characterization of the amorphous structures having long range disorders , here we have gone up to 768 atoms although several thousand atoms can be easily handled ; ( ii ) in the melt and quench scheme a much longer relaxation time scale , here to several nano seconds , can be evolved after the material is heated above the melting point . such a long time scale is important for erasing the memory of the initial structure thereby the final outcome becomes independent of the initial conditions . here we show that the same kind of amorphous structure is obtained by evolving from different initial structures including from a completely random structure . ( iii ) since the calculated physical properties of amorphous material should be averaged over an ensemble of samples due to the randomness of the structure , the combined classical - quantum approach allows such configurational average to be done . here we show that the calculated dielectric constants of the metal oxides become spatially isotropic after configurational average , but it is typically anisotropic if only one sample is calculated . the paper is organized as follows . in sec . ii we present the classical md method as employed in this work . in sec . iii we discuss the structural properties of our simulated amorphous structures , and also present our results for the lattice dielectric constants of these structures . section iv concludes the paper . structure obtained by the classical md simulations . green , red and blue lines in ( b ) represent hf - o , o - o and hf - hf rdf , respectively . the results are obtained from structures having 768 atoms.,title="fig : " ] + in this section we discuss the potential function used in our classical md simulations along with a short description of the melt and quench scheme . in our md method the interactions between the atoms are described by the born - mayer - buckingham potential function@xcite as shown below : @xmath12 the cation - cation interactions are assumed to be purely coulombic , the parameters for @xmath13 interactions and cation - anion interactions are presented in table . all the parameters are obtained from ref . we employed wolf _ et . s method @xcite for the summation of the coulomb interaction term . at the initial stage of the simulation , all the atoms are assigned velocities taken from a maxwell s distribution at the room temperature @xmath14 . the system is then heated up to a temperature @xmath15 k , followed by an evolution of @xmath16 ns at the same temperature ; and then quenched back to the room temperature @xmath14 . the temperature of the system is adjusted by resetting the velocity of a randomly selected atom at the interval of @xmath17 according to the following expression:@xcite @xmath18 where @xmath19 is a parameter between @xmath20 and @xmath21 , @xmath22 is a velocity selected randomly from the maxwell s distribution at temperature @xmath23 or @xmath14 via a random number @xmath24 . after the system is quenched to @xmath14 , it is allowed to evolve for @xmath25 ps to calculate the radial distribution function ( rdf ) . finally , the system is cooled down to @xmath20 k by a damping method,@xcite and thus concludes the procedure for generating one amorphous structure of a particular metal oxides .    in the simulations , the initial structure is modeled with a @xmath26 super cell of hf@xmath1zr@xmath2o@xmath3 containing 96 atoms ; or @xmath27 containing 768 atoms . a periodic boundary condition is imposed on the supercell . the initial atomic configuration is the cubic crystalline form or in the form of a random structure by _ randomly _ placing the constituents atoms ( with the correct ratio ) inside the simulation box . these drastically different initial conditions verify that the md simulation time scales given above are large enough since the same kind of final amorphous structures are obtained . ccccc elements & charge & a(ev ) & @xmath28( ) & c + hf & + 4 & 1454.6 & 0.350 & 0 + zr & + 4 & 1453.8 & 0.350 & 0 + o & -2 & 22764.3 & 0.149 & 20.4 +    [ tab1 ] to generate reliable amorphous structures it is important to find the optimal density to use for the initial structure . hence , we first perform a series of independent simulations on the 96-atom super cell , starting from initial structures of different lattice constants ( _ i.e. _ different densities ) of cubic hfo@xmath3 and zro@xmath3 in the range from @xmath29  to @xmath30  with an interval of @xmath31  . the density of @xmath8 g/@xmath9 for hfo@xmath3 and @xmath32 g/@xmath9 for zro@xmath3 . if we start with a lattice constant less than @xmath33  , the systems recrystallize to a cubic form ( fig.1a ) or some other crystal form ( fig.1b ) ; while with a lattice constant greater than @xmath34  , the systems tend to form disordered structures with big holes inside ( fig.1d ) . our optimized density values are close to those obtained previously from _ ab initio _ calculations which were @xmath35 g/@xmath9 for hfo@xmath3 @xcite and @xmath36 g/@xmath9 for zro@xmath3.@xcite hence , the lattice constant of 5.6  is used as initial condition ( cubic form ) in subsequent melt and quench evolution to the amorphous structures of hf@xmath1zr@xmath2o@xmath3 .    to verify the generated amorphous structures we calculate the coordination numbers and the radial distribution functions . results are presented in fig . [ fig2 ] for a - hfo@xmath3 obtained from the 768-atom supercell . from fig . [ fig2]a we note that the three - coordinated o atoms and six - coordinated hf atoms dominate the a - hfo@xmath3 structures which agree with the known _ ab initio _ results.@xcite from fig . [ fig2]b we find that the first peaks in the hf - o , o - o and hf - hf rdfs are at 2.1  , 2.8  , and 3.6  , respectively , again compare well with _ _ studies.@xcite these comparisons indicates that the classical md can reliably generate amorphous structures of the metal oxides . importantly , the efficiency of classical md allows us to generate ensembles of amorphous samples of hf@xmath1zr@xmath2o@xmath3 for each given concentration @xmath4 , which is essential for statistical analysis of the dielectric properties ( see below ) . zr@xmath2o@xmath3 averaged over @xmath25 samples for each value of @xmath4 ( solid red line ) . the potential energies per atom for each individual sample are also shown in the figure ( circles ) . the potential energies of these samples reside in the range of @xmath37 ev to @xmath38 ev , which are about 0.3 ev higher than that of the cubic hfo@xmath3 . the very small differences in the potential energies among the samples with the same value of @xmath4 confirms the reliability of the structures generated.,title="fig : " ] +    for further analysis , we calculate the potential energy per atom of the a - hf@xmath1zr@xmath2o@xmath3 structures for different values of @xmath4 ranging from @xmath20 to @xmath21 with an interval of @xmath39 . for each value of @xmath4 , we generated @xmath25 different amorphous samples by varying the initial conditions , and then calculated the potential energy per atom for each sample which are presented in fig . the results show that the potential energies of a - hf@xmath1zr@xmath2o@xmath3 structures as generated are located in the range from @xmath37 ev to @xmath38 ev which is a reasonable and small spread . we also generate a structure of a - hfo@xmath3 starting from an initial system of @xmath40 hf atoms and @xmath41 o atoms placed _ randomly _ in a cubic simulation box . the calculated potential energy of this particular a - hfo@xmath3 structure comes to @xmath42 ev which is in energy range of the a - hfo@xmath3 structures that were generated from the cubic hfo@xmath3 initial conditions . it is very satisfactory that the generated amorphous structures are not dependent on the initial condition , indicating that presented above are adequate .    ) for the amorphous structures of hf@xmath1zr@xmath2o@xmath3 . all the black symbols are for the values in each direction of @xmath43 , @xmath44 and @xmath45 averaged over @xmath25 samples , while the square symbols connected with a solid red line represent the values which are averaged over all the three directions for each value of @xmath4 . , title="fig : " ] +      a most important property of the high-@xmath0 metal oxides is their dielectric constant . in this section we present our calculations of the lattice dielectric constant ( static ) of hf@xmath1zr@xmath2o@xmath3 amorphous structures generated in the last section . we found that the electronic contribution to the dielectric constant is substantially smaller than the lattice contribution and does not change very much for different structures , hence it will not be discussed until when we calculate the total dielectric constant to compare with the measured data ( see below ) . ccccc oxide & amorphous & cubic & monoclinic & tetragonal + [ 0.5ex ] hfo@xmath3 & 24.52 & 31.08 & 14.49 & 125.95 + zro@xmath3 & 35.76 & 33.80 & 16.14 & 26.72 +    [ tab2 ]    for each amorphous structure the dielectric constant is calculated for an ensemble of @xmath25 different samples ( the 96-atom super cell ) . the sample averaged values are presented in table  [ tab2 ] for a - hfo@xmath3 and a - zro@xmath3 along with the dielectric constants for all @xmath46 of their crystal forms . we notice that the lattice dielectric constants of a - hfo@xmath3 and a - zro@xmath3 are close to that of their cubic form , suggesting that some cubic - like short range order content may dominate the amorphous structures . indeed , recently experimental measurements@xcite on sputtering samples of a - hfo@xmath3 discovered a cubic - like short range coordination in the amorphous microstructure which led to high measured dielectric constant @xmath47 . our calculated lattice @xmath0 ( see table  [ tab2 ] ) is 24.52 , adding the electronic contribution which we found to be around 4.4 , the total @xmath48 which agrees quite well with the measured data . to confirm that the @xmath0 of a - hfo@xmath3 being close to that of the cubic hfo@xmath3 , is not due to the cubic initial condition of the structure simulation , we have evolved a single a - hfo@xmath3 structure from a completely random initial condition ( see previous section ) : its lattice dielectric constant is found to be 21.785 , 25.43 and 25.163 in the three spatial directions respectively ; its electronic contribution is found to be 4.45 . these values are totally consistent with that obtained from the cubic initial conditions .    in comparison to previous theoretical first principles investigations,@xcite our @xmath0 values are higher . this difference may be explained by noting that in ref . [ ] the amorphous structures were obtained from a relaxed monoclinic structure with cubic lattice vectors which tends to have lower @xmath0 values . we should mention that direct comparison with experimental measurement is often difficult due to scarcity of data for dielectric constants of amorphous bulk structures . the ensemble averaged lattice dielectric constants of all the amorphous structures generated for different values of the concentration @xmath4 are presented in fig . we observe a gradual increase ( solid red line ) in the values of dielectric constant from a - hfo@xmath3 , through hf@xmath1zr@xmath2o@xmath3 , to a - zro@xmath3 . we also note that the spatial resolved dielectric constants in @xmath43 , @xmath44 and @xmath45 directions ( represented by the black symbols in fig . [ fig4 ] ) are quite _ isotropic _ which is expected for amorphous structures . it is very important to note that an isotropic dielectric constant is obtained only when the values are averaged over the ensemble of samples of the same structure ( same @xmath4 ) . for a single sample , we always observe large spreads in the values of dielectric constant in different directions . this finding underpins the major advantage of employing a rather inexpensive classical md to efficiently generate amorphous structures for the statistical average of the physical properties . using classical md simulations within a melt and quench scheme , the amorphous structures of hfo@xmath3 and zro@xmath3 , along with their alloy structures of the form of hf@xmath1zr@xmath2o@xmath3 , have been generated . a suitable potential function , namely the born - mayer - buckingham potential , has been adopted for these simulations . the calculated coordination numbers and radial distribution functions of the generated structures agree well with the previous studies . the values of the ensemble averaged lattice dielectric constants of these amorphous structures are nearly isotropic , close to that of their cubic form , and compare well with available experimental data.@xcite on the other hand , a single sample always produces anisotropic @xmath0 values . our results indicate that the combined approach of classical md - for structure evolution , and quantum mechanical first principles dft - for electronic structure analysis , is a reliable technique for investigating high-@xmath0 metal oxides . we would like to thank h.z . shao of fudan university , c. yin of dec and j.n . zhuang of the university of hong kong for useful discussions . this work is supported by the university grant council ( contract no . aoe / p-04/08 ) of the government of hksar , the small project funding of hku ( yw ) , nserc of canada ( hg ) , and cifar ( hg ) . x. zhao and d. vanderbilt , phys . b * 65 * , 233106 ( 2002 ) . d. ceresoli and d. vanderbilt , phys . rev . b * 74 * , 125108 ( 2006 ) . x. zhao and d. vanderbilt , phys . b * 65 * , 075105 ( 2002 ) . x. zhao , d. ceresoli , and d. vanderbilt , phys . b * 71 * , 085107 ( 2005 ) . g. mazaleyrat , a. estve , l. jeloaica , m.d . rouhani , comput . , * 33 * , 74 ( 2005 ) . scopel , a.j.r . da silva , and a. fazzio , phys . b * 77 * , 172101 ( 2008 ) zhu , t. tamagawa , m. gibson , t. furukawa , and t.p . ma , ieee electron device lett . * 23 * , 649 ( 2002 ) . m.y . ho , h. gong , g.d . wilk , b.w . busch , m.l . green , w.h . lin , a. see , s.k . lahiri , m.e . loomans , p.i . risne , and t. gustafsson , appl . phys . lett . * 81 * , 4218 ( 2002 ) . y. xiong , h. tu , j. du , m. ji , x. zhang , l. wang , appl . phys . * 97 * , 012901 ( 2010 ) . h. momida , t. hamada , y. takagi , t. yamamoto , t. uda , and t. ohno , phys . b * 75 * , 195105 ( 2007 ) ; h. momida , t. hamada , and t. ohno , jpn . j. appl 46 * , 3255 ( 2007 ) . p. broqvist and a. pasquarello , microelectron . eng . * 84 * , 2416 ( 2007 ) ; p. broqvist and a. pasquarello , appl . . lett . * 90 * , 082907 ( 2007 ) . perevalov , v.a . gritsenko , s.b . erenburg , a.m. badalyan , h. wong , and c.w . kim , j. appl . * 101 * , 053704 ( 2007 ) . bayer , s. hofmann , j.d . dutson , s.j . wakeham , m.j . thwaites , w.i . milne , and a.j . flewitt , appl . lett . * 98 * , 252903 ( 2011 ) . a. dkhissi , a. estve . c. mastail , s. olivier , g. mazaleyrat , l. jeloaica , and m.d . rouhani , j. chem . comput . , * 4 * , 1915 ( 2008 ) . chen , z.f . hou , and x.g . gong , appl . . lett . * 95 * , 102905 ( 2009 ) . d. vanderbilt , x. zhao , d. ceresoli , thin solid films * 486 * , 125 ( 2005 ) . barkema , n. mousseau , phys . lett . * 77 * , 4358 ( 1996 ) ; n. mousseau , g.t . barkema , phys . b * 61 * , 1898 ( 2000 ) .        t.p . perumal , v. sridhar , k.p.n . murthy , k.s . easwarakumar , s. ramasamy , phys . a , * 309 * , 35 ( 2002 ) . t. arima , k. fukuyo , k. idemitsu , y. inagaki , j. mol . liq . * 113 * , 67 ( 2004 ) . m. kilo , m.a . taylor , c. argirusis , g. borchardt , r.a . jackson , o. schulz , m. martin , m. weller , solid state ionics * 175 * , 823 ( 2004 ) . v. georgieva , i.t . todorov , a. bogaerts , chem . phys . lett . * 485 * , 315 ( 2010 ) . v. georgieva , m. saraiva , n. jehanathan , o.i . levelev , d. depla , and a. bogaerts , j. phys . phys . , * 42 * , 065107 ( 2009 ) .
high-@xmath0 metal oxides are a class of materials playing an increasingly important role in modern device physics and technology . here 
 we report theoretical investigations of the properties of structural and lattice dielectric constants of bulk amorphous metal oxides by a combined approach of classical molecular dynamics ( md ) - for structure evolution , and quantum mechanical first principles density function theory ( dft ) - for electronic structure analysis . using classical md based on the born - mayer - buckingham potential function within a melt and 
 quench scheme , amorphous structures of high-@xmath0 metal oxides hf@xmath1zr@xmath2o@xmath3 with different values of the concentration @xmath4 , are generated . 
 the coordination numbers and the radial distribution functions of the structures are in good agreement with the corresponding experimental data . 
 we then calculate the lattice dielectric constants of the materials from quantum mechanical first principles , and the values averaged over an ensemble of samples agree well with the available experimental data , and are very close to the dielectric constants of their cubic form .