Theoretical Investigation of Ultrathin Gate Dielectrics

We describe a theoretical methodology for screening potential gate dielectric materials. A recently proposed method for constructing realistic structural models of the Sidielectric interface is used to generate the Si-SiO2-Si and Si-SiON-SiO2-Si model metaloxide-semiconductor (MOS) structures. We discuss methods to estimate the valence band discontinuity at the corresponding interface. We use Landauer’s ballistic transport approach to investigate the low bias leakage through these ultrathin dielectric layers.


I. INTRODUCTION
The International Technology Roadmap for Semi- conductors (ITRS) indicates that the strategy of scaling complimentary metal-oxide-semiconductor (CMOS) devices will come to an abrupt end around the year 2012 [1].The main reason for this will not be the lithography resolution problems as was anticipated, but rather the leakage current through the silicon dioxide gate with a thickness below 2 nm.
In other words, the insulation of the oxide layer breaks down for a very thin layer [2].The end of SiO2 as a gate insulator spurred an active search for an alternative gate dielectric.Finding such a material has proven to be far from trivial.
There are four principle requirements that a successor gate material should meet.(1) The leakage problem is rooted in the continuous reduction of the gate dielectric's thickness neces- sary to maintain the capacitance of the MOS structure as its lateral dimensions are being scaled.Therefore a material with a dielectric constant higher than that of SiO2 (approximately four) is sought.
(2) The new material should be stable in contact with Si (that is, not to form silicides or silicates) at temperatures up to 900C.This is re- quired to maintain the integrity of a sharp interface during the CMOS processing, such as, e.g., the im- plant anneal.(3) The density of interfacial traps should be below 1012 cm -2 to be on par with SiO2.
(4) The conduction band offset should be at least eV (3.2 eV at the Si-SiO interface) to provide a sufficient energy barrier necessary for the gate action at the operational bias.
*Corresponding author.Tel.: 480-755-5016, Fax: 480-755-5055, e-mail: alex.demkov@motorola.comWithin the adiabatic approximation the dielec- tric constant of a solid has two major contri- butions: the electronic and lattice susceptibilities.The orientational term important in polar liquids and molecular solids may be neglected for a covalent or ionic material.The electronic susceptibility scales approximately as a square of the ratio of the plasma frequency Op (measures the electron density) over the so called Penn gap E,G (measures the average energy gap of the electronic spectrum) [3]: With EpG being about 4-6eV for most semicon- ductors, and hap being in the range of 15-17 eV, it is clear that it is unlikely that the electronic component alone allows to achieve a dielectric constant much higher than approximately 15.That is why the vast majority of novel gate dielectrics proposed to this day, e.g., SrTiO3 [4][5][6], ZrO2 [7], HfO2 [8], TiO2 [9], or Ta205 [10], are insulators that display the high lattice polarizability in the crystalline form due to a soft phonon mode.These materials, however, present a different set of challenges.First and foremost, one needs to suppress the oxidation of Si.Secondly, the thermo- dynamic stability in contact with Si is a problem [10].Third, in most transition metal oxides the bottom of the conduction band is derived from the d-states of the transition metal which are close in energy to the conduction band of Si.This results in a small or negative conduction band offset [11].In addition, the dielectric constant is typically reduced in the oxide thin films, and there are indications that this reduction is caused by the soft- mode hardening [12].This may have severe im- plications for the feasibility of achieving a high dielectric constant, particularly in amorphous films.
The empirical search for a new gate dielectric is rather costly, and theoretical modeling could prove to be very helpful.In particular, theory provides the information on the chemical bond- ing, structure, and stability of the Si-dielectric interface, and band offsets.An estimate of the low bias leakage current through the gate, and how it is affected by the atomic structure would be of great interest in this ongoing investigation.In this paper we present a possible methodology that may be used to guide the experimental effort.Using a theoretical atomic level Si-SiO2 model system, we demonstrate how the band offset may be com- puted.We study the role of nitrogen at the inter- face.We investigate the leakage current through an ultra-thin MOS structure using the quantum transmission probability that when intergrated produces an I-V characteristics.A combination of the density functional quantum molecular dynamics (QMD) and ballistic transport theory are employed.To our knowledge, this is the first entirely first principles theoretical analysis of this problem.

II. PRELIMINARY THEORETICAL CONSIDERATIONS
We now discuss our approach to the leakage current calculations in nanoscale devices such as ultrathin MOS structures.As the size of the electronic devices continues to shrink, and the computational power of the electronic structure methods steadily increases, we now can compute quantum mechanically both the atomic structure of a very small device and its transport character- istics such as an I-V curve.The electronic structure methods allow to perform predictive calcula- tions for systems containing hundreds and even thousands of atoms.This translates into a few nanometer length scale, and we could take ad- vantage of a simple ballistic transport theory [13].
In this theory a current calculation is reduced to an effective one-dimensional scattering problem.For example, the leakage through a MOS structure is described as tunneling through a potential barrier [14].The use of a ballistic de- scription is justified for systems with dimensions below the physically important scattering lengths such as the Fermi length, mean free path, and the phase relaxation length which in a typical semi- conductor are 10nm, 30tm, tm, respectively.
Transport through SiO2 relevant to our case has been studied theoretically [15,16] and experimen- tally [17], and the mean free path is reported to be between 0.6 and 1.5nm, with the LO-phonon emission being a dominant energy-loss mechan- ism.Therefore, for a nm thick oxide layer one can use Landauer's formalism.The current is given by: ILR -dET(E)dE. (2)  where the #1 and /z2 are the chemical potentials of the left and right leads, respectively.Thus the central problem is to calculate the barrier trans- mission function T(E).The transmission prob- ability is often calculated within the WKB approximation.To do so, one needs to know the barrier thickness, height, and the carrier's effective mass m* [18].Choosing a proper effective mass for the barrier region appears somewhat difficult.A wide range of values for the "tunneling" mass is used in the literature [19,20].The small parameter in the Luttinger-Kohn theory that allows the use of the envelope function and the effective mass itself is of the order of (a/ai)2, where a is the lattice constant, and the ai is the extent of the "defect" state [21], which in our case is the oxide thickness.
When the thickness of the gate dielectric (the scatter) is only 1 nm the effective mass approxima- tion does not hold (the lattice constant of Si is 0.535nm).Therefore, the band structure-derived effective mass may be used only with extreme care at this length scale.We will describe two complementary methods of estimating the band offset at the interface, that gives the barrier height.For the transport calcula- tions we follow the approach of Fisher and Lee which relates the barrier transmission to the Green's function for the scatering region [22].We generate an atomistic model of the oxide layer, which will be referred to as "defect" to represent a scattering region.Two semi-infinite perfect Si regions with the same cross section as the "defect", representing the leads are considered attached on both sides of the "defect" structure.The problem is infinite in three dimensions.Applying periodic boundary conditions in two transverse directions we reduce it to a 1D propagation in the direction normal to the interface.The local orbital basis (see below) allows us to use the L6wdin representation of the Hamiltonian, and the geometry of the effective 1D problem is reflected in the matrix.The semi-infinite leads are treated via corresponding self-energy operators related to the lead's Green's function.This is calculated using a slab-recursion method [23].Due to the finite range of the basis orbitals the slab-slab coupling in the leads and the defect-lead coupling are treated exactly.In our formulation neither the effective mass nor the electron velocity appear explicitly in the formal- ism.Instead, a spectral density operator, which can be defined regardless of the system size, plays their role.
Structural calculations have been performed by using a local orbital first principles quantum molecular dynamics (QMD) method designed for applications to large systems and implemented in the Fireball package [24].The method employs density functional theory within the local density approximation and hard norm-conserving pseudopotentials.The simplified energy functional is a self-consistent generalization of that due to Harris.The total electron density is approximated by a sum of densities of fragments, e.g., atoms that are not necessarily neutral.The method is entirely real space (except in a simple Ewald summation).A short-range non-orthogonal local orbital basis of Fireball orbitals of Sankey and Niklweski offers a good variational flexibility combined with a significant computational advantage [25].Integrals over the Brillouin zone are evaluated using the special k-points of Monkhorst and Pack.If the local energy minimum is sought a fictitious friction force is introduced to guide the system to the minimum energy configuration.Numerous recent applications of the technique to a vareity of materials problems are reviewed in Ref. [26].

III. STRUCTURE FROM QUANTUM MOLECULAR DYNAMICS
We have theoretically built several model MOS structures.As a starting point we use theoretical structures of the Si-SiO2 interface generated by the "direct oxidation" method described in Ref. [27].
Briefly, the structure consists of a silicon slab with an oxide layer "grown" on it in a QMD simula- tion.Si layers are separated from the stoichio- metric oxide by approximately 4 ]k of sub-oxide.
Structural models of that type are used in calculations of the band offset at the interface discussed in Section IV.We constructed several cells of various dimensions and thicknesses, with and without dangling bonds at the interface.Then, in the simplest case, a mirror image of the cell is generated, and the two are fused together.That   procedure results in a Si-SiOx-SiO2-SiOx-Si struc- ture.Some special care needs to be taken of the SiO2 bonding pattern in the plane of contact.
Mixing and matching the initial Si-SiO2 structures both symmetric and asymmetric MOS models could be built.The structure then is annelated in a high-temperature QMD run followed by a quench to guide the system to a local energy minimum.Models thus generated are used as scattering regions or "defects" in transport calculations of Section V.In particular we will describe the tunneling through a "defect" with the dimen- sions of 0.543 nm 0.543 nm 3 nm.It consists of two monolayers of perfect (stoichiometric) SiO2, which are sandwiched between two SiOx transition layers 0.3 nm each.This structure is em- bedded between two nm thick regions of perfect silicon, and relaxed.The tunneling region is about 1.1 nm.There are dangling bonds in one of the interfacial layers, while the other represents a perfect interface.Once we have a "defect" model, two perfect Si regions with the same cross section as the "defect" representing the leads are consider- ed attached on both sides of the tunneling struc- ture.Thus the total length of the system is taken to infinity.The left Si region represents the channel, and the right one a grain of poly-Si (gate electrode).The metallic nature of the gate elec- trode is taken into account by adjusting the Fermi level at the appropriate energy in the conduction band.
Silicon oxynitrides are considered leading can- didates for an interim medium-k gate dielectric [28].However, the properties of these materials are not very well understood.We investigate an oxynitride layer at the Si-SiO2 interface and the effect of nitrogen on the band offset and leakage.In order to build a nitrided "defect" we first introduce nitrogen as nitric oxide NO at the Si- SiO2 interface with the dangling bonds, and relax the structure.The details of these calculations, and the experimental verification of the proposed structures are reported in Ref. [29].The total nitrogen concentration in the sample discussed in the following is 6.78 1014cm -.T.he Si-SiON- SiO2 structure is shown in Figure 1.Si-N bonds are ranging from 0.165 to 0.172 nm, and there are no N-O bonds in the sample.All N atoms are three fold coordinated by Si just as in stoichiometric Si3N4 or Si2N20.The Si-N-Si bond angles are close to 120.To build a tunneling "defect" structure we fuse this cell with a nitrogen free one with a perfect Si-SiO interface and relax the N1 D (Si-N1) 0. 174 nm D (Si-N2)" 0. 173 nm 0. 165 nm 0. 182 nm 0. 170 nm 0. 167 nm

FIGURE
The Si-SiON-SiO2 structure built in a Quantum Molecular Dynamics simulation.Nitrogen is at the inter- face where it is easier to realize the preferred three-fold co- ordination.
local strain via a QMD anneal.The resulting asymmetric Si-SiON-SiO2-SiOx-Si structure has no dangling bonds, and the thickness of the tunneling region is similar to that of a nitrogen free sample described above.
To summarize, we have described a method to produce atomistic level models of MOS structures that could be used to calculate band offsets and transport properties.

IV. BAND OFFSET ESTIMATES
We now briefly describe the "reference energy method" introduced in Ref. [27], and used here to estimate the valence band discontinuity at the interface.The technique is a local orbital formulation of the method due to Van de Walle and Martin [30].We use the average of the expectation value of the Hamiltonian calculated for the valence (3s or 3p) states of Si, es(i)= H(s)aver and ep(i)--H(p)aver (H(s)= (s(i)[H[fS(i)) and H(p)= (hp(i)[HlfSp(i)) respectively) as a reference energy against the valence band top of the bulk material.This is similar to a technique adopted in all-elec- tron calculations for core level shifts [31].The spa- cial dependence of H(s) and H(p) across a 20 slab of silicon with two 2 reconstructed sur- faces is shown in Figure 2. It is remarkable how fast both curves converge to their bulk values.
In the following, we will use e as the reference energy.
The Si 3s valence state matrix element H(s) plotted across the model heterojunction along the line perpendicular to the plane of the interface (Si left, SiO2 right) is shown in Figure 3. Crosses cor- respond to the "as grown" sample, and diamonds correspond to the "annealed" sample.The inter- facial region is clearly seen between -0.2 and 0.2 nm.In Si (left), the average value of H(s) is near 13 eV while in the oxide (right) it is about 20 eV.
The valence band offset, Awe, is obtained from Awe AER(Si, SiO2) + AERv(Si) AEv(SiO2), ( Distance (rim) FIGURE 3 The Si 3s valence state matrix element H(s) plotted across the model heterojunction.Crosses correspond to the "as grown" sample, and diamonds correspond to the "annealed" sample.The interfacial region is clearly seen between 0.2 and 0.2 nm.
where AER(Si, SiO2) is the energy difference in the reference levels in Si and SiO respectively, and AERv(i) is the energy difference between e(i) and the top of the valence band edge in bulk material i. AEv(Si) is 8.31 eV.The value of AERv(SiO2) (which determines where to place the top of the valence band in the oxide layer) depends on Si-O bond lengths and Si-O-Si bond angles.We estimate this dependence using/3-cristobalite, and find that to first order the dependence on Si-O-Si angle is weak for angles in the range from 130 to 180 (where nearly all bond angles in silica polymorphs lie), and we shall neglect it in the analysis.In Figure 4 we show AERv(SiO2) as a function of the bond length dsio.The dependence is well described by a linear equation: 11.4dsio eV/ + 28.03 eV.(4) For the thin oxide layer we obtain a valence band offset of 4.35 eV, and for a thicker oxide layer we obtain 4.65 eV in good agreement with experiment.
In this procedure two quantities define the val- ence band offset.First, it is AER(Si, SiO2), the difference in the Si reference energy level between the silicon and oxide layers.This difference takes into account the interfacial dipole.Secondly, it is AERv(SiO2) that is sensitive to the local strain in the oxide layer.
A more direct theoretical technique to estimate the valence band offset is the density of states analysis.We use the Green's function method within the local-orbital formalism to calculate the total and partial densities of states.The density of states is computed as: U(z) --Tr(Im(G(z)S-1)), (<slHIs>-EV) as a function of the Si-O bond length Si-O bond length (/) 1.63 1.65 FIGURE 4 AERv(SiO2) as a function of of the bond length dsio calculated for fl-cristobalite exhibits a strong linear dependence.AERv(SiO2) is therefore sensitive to the local strain in the oxide layer.
where G(z) is the matrix of the resolvant operator G (z-H)-1 in the local orbital basis not to be mixed with the representation of the resolvant in this basis.S-1 is the inverse of the overlap matrix.
The presence of the overlap is due to the non- orthogonality of the basis.The partial density of states is computed for the atoms in the different layers of the interfacial model and provides the energy-position correlation required for this analysis.The details of the calculation could be found in Ref. [29].The valence band offset of 4.25eV is found in close agreement with the "reference energy method" analysis.
We repeated the analysis described above for the N containing sample.The valence band offset of 4.67 eV is computed using the "reference energy method", and 4.7 eV is obtained using the direct DOS analysis.The 0.32 eV increase of the valence band offset after the nitridataion is mainly due to the strain relaxation caused by nitrogen in the interfacial layer (AERv(SiO2)), only 0.12eV are coming from a change in AE(Si, SiO2) that describes the interfacial dipole.Another potentially important result is a significant difference in the densities of states of two samples just below the Si conduction band edge.The partial density of states analysis indicates that the states in this energy region are due to the interface.The interface band is more pronounced in the nitrogen- containing sample, and is shifted down in energy by 0.3 eV.This possibly suggests that a very high nitrogen concentration at the Si-SiO2 interface may result in the additional scattering due to the states just below the conduction band of Si.
V. TRANSPORT CALCULATIONS A detailed discription of the ballistic conductance calculation within a local orbital formalism will be reported separately [32].Here we briefly outline the salient features of our approach.We identify the low-bias leakage through the ultrathin MOS structure with the carrier tunneling through a potential barrier caused by the band discontinuity at the silicon-dielectric interface.Therefore, we need to calculate the transmission probability for the electron traveling from the left lead to appear in the right lead after being "scattered" by a thin layer of the oxide or "defect".Mathematically, the problem is most conveniently dealt with within the framework of the sca_ttering theory.In the eigen- vector representation, the barrier transmission function is given by: T(E) I<Zltlr>l=6(g gl)6(g gr), (6) lr where ]/)and Ir) represent the unperturbed states of the semi-infinite leads without the coupling to the "defeact" (the choice of the "defect" layer will be discussed later), 5 functions ensure the energy conservation, and represents the so-called t- matrix which is defined as: t= v + vuv.(7) Here V describes the interaction between the "defect" and the leads.This interaction could be written as v + va , where Vai is the coupling between the "defect" and the leads.We use the short range basis orbitals, the cutoff radii for the O and Si orbitals are chosen to be rc 3.6, and 5.0 Bohr, respectively, that results in a third nearest neighbor model for Si.We define the "defect" as the oxide layer sandwiched be- tween two sub-oxide regions and four atomic layers of Si on each side.With this choice of a "defect" structure, it can be shown [32] that the transmission is given by: r Tr[rGFrG+], where F is defined as: Fi Z Vaili)(i[6(E Ei)Wffi WdimiWffii, (10)   and we used the following identity: 6(E-Ei)li)(il-6(E H) m i. (11) Note, that Eq. ( 9) is an operator expression independent of the basis.A very similar expression can be derived for the non-orthogonal basis [32].One needs to keep in mind, however, that the trace in this case is defined as TriO] (alOS-11).
Fortunately, because the "defect"-lead coupling matrix is so sparse, we are only interested in the sub-matrix Ga of the total Green's function, describing the "defect".This portion of the Green's function can be written as: where Ha is the unperturbed "defect" Hamilto- nian, and Ei is the self energy describing the coupling of the "defect" to the leads: Y] VGi s Vai (14) From Eq. ( 14) it follows that we only need to know the "surface" block G/ of the Green's function matrix of a semi-infinite lead to compute the self-energy.In the non-orthogonal basis employed here, the Hamiltonian is a band matrix.Defining a slab as four atomic layers of Si we ensure that there is no interaction between the slabs beyond the nearest neighbour interaction.This results in an effective "tridiagonal" matrix similar to that for a 1D chain.Now the "surface" block is easily obtained by recursion.Also note, that by including a sufficient number of Si layers into the "defect" structure the coupling of the lead to the "defect" Vdi, is made essentially the same as the slab-slab coupling within the lead.The actual silicon-oxide coupling is "absorbed" in the "de- fect" Hamiltonian, and is treated exactly within the present formalism.first entirely quantum mechanical approach to this problem.We believe that theoretical modeling could prove to be a useful tool in reducing the cost of the empirical search for a new gate dielectric.
FIGURE 5 The transmission probability through two MOS structures.One is a nitrogen free sample with dangling bonds at the interface, and the other contains 6.78 x 1014 cm -2 nitrogen at the silicon-silicon dioxide interface.
We have applied the described formalism to the gate leakage problem using our ultrathin MOS models as "defect" structures.In Figure 5 we show the transmission probability through two previously described MOS structures.One is a nitrogen free sample with dangling bonds at the interface, and the other contains 6.78 x 1014 cm -2 nitrogen at the silicon-silicon dioxide interface.
Note that the electron leakage is higher for the nitrogen-containing sample due to the increase in the density of states at the conduction band edge described in the previous section.That suggests that nitridation of the oxide layer may result in a higher leakage due to the segregation of nitrogen to the oxide-silicon interface in the case of high the nitrogen concentration.

VI. CONCLUSIONS
We have described a theoretical approach suitable for screening of potential gate dielectrics.Using a combination of the density functional quantum molecular dynamics and ballistic transport theory we investigate the leakage current through ultra- thin MOS structures.To our knowledge, this is the

FIGURE 2
FIGURE2 The spacial dependence of the 3s, and 3p expectation values of the Hamiltonian H(s)=(fs(i)lH]fs(i) and H(p)= (p(i)lH Ibp(i)) across and 20]k slab of silicon with two 2 reconstructed surfaces.