Chapter 1 Introduction
1.6 Organization
In this thesis, the first chapter is the introduction consisting of the Si-based MOSFETs scaling challenges focusing on source/drain improvement mainly, the merits and drawbacks of Ge which is regarded as a potential material to replace Si, the problems for underestimation of barrier height using traditional extraction methods in the low barrier system, and finally proposing a first-principles calculations to overcome these problems and to understand the influence of dopant segregation on the NiGe/n-type Ge contact.
Chapter 2 shows some representational approximations for many-body problems before density functional theory, the frameworks of the density functional theory including the Hohenberg-Kohn theorem and Kohn-Sham formulation, and the introduction of the exchange-correlation energy functional used in the our calculations.
Simulation result is shown and discussed in Chapter 3. First, we examine the result of lattice constant and band structure for Ge bulk to find the suitable exchange-correlation functional. Then, the atomic structure of the NiGe/Ge contact consisting with the experimental simple is constructed. Finally, the behaviors of the conventional n-type dopant around the interface, the effect of the Schottky barrier height modified by dopant segregation, and the influence of the nitrogen dopant segregated at the interface are studied by the first-principles calculations. The last chapter is the conclusions and future works of this thesis.
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Table 1-1 Material characteristics of alternative channel materials [10].
Si Ge GaAs InSb
Bandgap, Eg (eV) 1.12 0.66 1.42 0.17
Hole mobility, μ h (cm2V-1s-1)
450 1900 400 1250
Electron mobility, μ e
(cm2V-1s-1)
1500 3900 8500 80000
Effective density of states in valence band, NV (cm-3)
Effective density of states in conduction band, NC (cm-3)
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Figure 1-1 Number of transistors on the CPU has been increased from 1971 to 2011; the line corresponds to exponential growth with Moore’s law. (source: Wikipedia)
Figure 1-2 N-type Metal-Oxide-Semiconductor Field-Effect-Transistor (n-MOSFET) device simulation showing series and strained Si channel resistances becoming equivalent at approximately the 32 nm logic node [2].
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Figure 1-3 Schematic representation of source/drain structure and series resistance components [3].
Figure 1-4 Relative contribution from each component of the resistance to series resistance for different technology nodes [4].
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Figure 1-5 Schematic band diagram of band bending according to the Schottky-Mott model of the metal/n-type semiconductor: (a) Before contact and (b) after contact formation [26].
Figure 1-6 Schematic band diagram of band bending according to the Bardeen model of the metal/n-type semiconductor: (a) before contact and (b) after contact formation. The Fermi-level is pinned by the high density of surface states at the semiconductor interface [26].
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Figure 1-7 Conduction mechanics for metal/semiconductor contacts, (a) thermionic emission (b) thermionic-field emission (c) field emission.
Figure 1-8 Sub-100 nm transistor technology scaling. (Source: Intel Corp.)
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Chapter 2
Calculation Methods
2.1 Representational Approaches before Density Functional Theory
For the many-body system consisted of N electrons (of coordinates r, mass m, and charge -e) and corresponding nuclei (of coordinates R, mass MI, and charge +ZIe), electrons in material are subjected both to electron-electron interactions and to external potentials in the real world. The starting point for the theoretical description of many-body system is the nonrelativistic, time-independent Hamiltonian of the coupled electron-nucleus system
Once the wavefunction is solved, any observable can be determined by the expectation value of the corresponding operator
. (2.3)
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Careful use of symmetry might help us reduce the size of the Hamiltonian, and the escalating power of modern computers growing exponentially with time allow us to consider larger system, but it is still difficult to solve exact numerical diagonalization for the problem of N electrons interacting in some external potential since the size of the Hilbert space of our N-electron problem grows much faster than exponentially with N. Therefore, proper approximations and simplifications are needed to extract useful information from such complex systems.
2.1.1 The Born-Oppenheimer approximation
The adiabatic approximation, first proposed by Born and Oppenheimer is based on the fact that the typical velocity of an electron in materials is much faster than the velocity of the nucleus, since the nuclear mass is about one thousand times heavier than electronic mass in realistic situations [27]. In other words, the time scale of the nuclear motion is several orders of magnitude larger than that of the electronic motion. Therefore, it can be assumed that at any moment the electrons will be in their instantaneous ground state determined by the interaction with nuclear distribution at a particular instant, and then the forces acting on the nuclei are determined by the instantaneous electronic distribution and the nuclear positions of the system. Based on such an assumption, the nuclear kinetic energy is considered to be negligible, and the nuclear positions are taken as fixed external parameters.
As a result, the Hamiltonian for the coupled electron-nucleus system can be reduced to the Hamiltonian for the N interacting electrons in an external potential
, (2.4)
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where the nuclear part of the Hamiltonian is represented by a fixed external potential
.
Although the Born-Oppenheimer approximation allows us to separate the degrees of freedom between electron and nucleus and to solve the nuclear part classically, the solution of the Schrodinger equation for N interacting electrons system is a formidable task since there are still the two-body mutual Coulomb interaction between these electrons. Hence, a variety of approaches have been developed to overcome this difficulty.
2.1.2 The Hartree-Fock approximation
The independent-particle approaches assume the electrons are uncorrelated except that they must obey the exclusion principle. Accordingly, the Hamiltonian of N interacting electrons (2.4) can be decoupled into N effective single-electron Hamiltonian where there is no explicit two-body term included, but we incorporated into effective potentials certain amount of the Coulomb repulsion.
Hartree in the first place proposed that the total wavefunction of N interacting electrons can be expressed as a product of N single-electron wavefunction known as the Hartree product [28-30]
, (2.5)
where denotes the spin-orbital state of one electron,
.
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Nevertheless, this simple approximation does not handle the nature of identical particles among electrons since the Hartree product fails to satisfy the Pauli principle, which demands that the N-fermions wavefunction is antisymmetric; that is, the sign of the wavefunction changes when two of its arguments are exchanged
. (2.6)
To overcome this major shortcoming of the Hartree product, his students Fock and Slater, introduced the approach to ensure that the N-electron wavefunction obeys the Pauli principle is to construct a Slater determinant of orthonormal single-electron wavefunction [27]
It obviously satisfies the Pauli principle of fermions because we exchange the coordinates of the space r and spin s with any two electrons, we will get a minus sign.
In order to obtain approximate solutions of the many-body stationary Schrödinger equation (2.2), the expectation value of the Hamiltonian for the N interacting electrons system (2.4) is calculated with respect to the Slater determinant wavefunction (2.7)
refer to the direct and exchange interactions among electrons, respectively. We have followed
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the usual practice of including the i = j self-interaction, which is spurious but such terms cancel each other in the sum of direct and exchange terms.
Equation (2.8) allows us to use the variational principle to obtain better approximate wavefunction by varying orthonormal single-electron wavefunctions until we minimize the ground state energy (2.8) within the given form (2.7), we can get the Hartree-Fock equation
. (2.9)
Equation (2.9) can be rewritten in a form analogous to many-body stationary Schrödinger equation (2.2) except that the effective one-electron Hamiltonian is an operator that depends upon the states
(2.10)
with ,
, and
,
where is the Hartree Coulomb potential and is known as the exchange potential, so the electron-electron interaction can be reduced into effective potentials that can best mimic the real interaction. Therefore, the total energy in Hartree-Fock approximation is given by
,
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where is the eigenvalue of equation (2.10).
However, the electrons get often too close to each other in the Hartree-Fock scheme since the electrostatic interaction is treated through a mean-field manner. The effective Hartree-Fock potential includes the exchange effect, but does not incorporate the instantaneous repulsion between two electrons, called correlation energy. As a consequence, the long range of the Hartree-Fock exchange interaction is overestimated and much less tractable in a large system because it neglects the collective electron correlation to screen this term and to reduce the net interaction among any two electrons. This approximation therefore would yield the wrong sign and values for the work functions of simple metals. However, the DFT calculations considering the correlation energy not only show metals to be stable, but give very good quantitative agreement with experiments.
On the other hand, considering the system including N interacting electrons in some external potential and the Slater determinant wavefunction expanded with a total of Nk basis sets, the size of the Hamiltonian matrix to be diagonalized is given by
. (2.11)
This large number makes it difficultly to solve in big system size.
2.2 Density Functional Theory (DFT)
Although we can solve the stationary Schrödinger equation for N interacting electrons in an external potential approximately, it is still a difficult problem to include electron-electron interactions in the large system size.
Density functional theory (DFT), which was conceived by Walter Kohn, et al. in the
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mid-1960s [31, 32], has provided the alternative independent-particle approach that incorporate effects of interaction and correlation among particles in realistic calculations. The main spirit of density functional theory is to allow for the systematic formulation of many-body problem in terms of its electron density instead of the many-body wavefunction as the basic variable. It will lead to enormous simplification since the basic variable of the system depends on only three rather than 3N degrees of freedom, no matter how large system is in three dimensions.
Accordingly, density functional theory is presently the most successful approach to calculate the electronic structure of many-body system, and it is increasingly important in the quantum chemistry, condensed matter physics, and even nanoscale device physics.
2.2.1 The Hohenberg-Kohn Theorem
What is truly remarkable is that all physical properties of the system can in principle be determined with the knowledge only of the ground-state density. This is precisely the statement of the Hohenberg-Kohn theorem[31]. The intuitive version of Hohenberg-Kohn theorem states that given any density n(r) as the ground-state density for some N-electron system, the Hamiltonian of that system is then uniquely determined, and so then are all the eigenstates and the expectation value of any operator.
The proof of this theorem can be accomplished of the system with non-degenerate ground states, and the Hamiltonian for the N interacting electrons in an external potential is described by (2.4). Let us assume that two external potentials and differing by more than a trivial constant can give rise to the same ground-state density n(r). Obviously, distinct Hamiltonian and due to different and will lead to distinct ground-state wavefunction and , respectively.
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An analogous argument, obtained by interchanging primed and unprimed quantities yields ground-state density. In other words, it follows that two identical ground-state densities must stem from identical external potentials, and with that our proof of the Hohenberg-Kohn theorem is complete.
There is also an important variational principle associated with the Hohenberg-Kohn theorem. Since the ground-state energy is uniquely determined by ground-state density
24 problem, it does not provide us with any useful computational scheme. This is provided by the Kohn-Sham formalism [32]. The idea here is to use a non-interacting auxiliary system and to look for an external potential such that the non-interacting system has the same ground-state density as the real interacting system.
Let us start by considering a non-interacting N-electron system in an external potential . The Hamiltonian of this system is given by
where is the kinetic energy functional of a system of N non-interacting electrons.
The ground-state density of this system is easily obtained. It is simply
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, (2.19)
where we have occupied the N single-particle states , which satisfy the single-electron Schrödinger equation
(2.20)
and have the N lowest eigenvalue .
But we are really interested in a system of N interacting electrons in an external potential
, we would determine the form that must take in order for the non-interacting system to have the same ground-state density as the interacting system in the external potential . The strategy we use is to solve for the density using the auxiliary non-interacting system, and then insert this density into an approximate expression for the total energy of the interacting system.
The first step in this process is to rewrite the energy functional of the interacting system, which was given in (2.18), as
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energy functional
. (2.22)
We have thus swept all our ignorance about electron interactions beyond the Hartree term under the rug that we call . What we gain in writing in this way is that we can eventually focus on developing reasonable approximations for .
According to the Hohenberg-Kohn theorem, the density n that minimizes the functional is the ground-state density. Thus by taking the variation of (2.21) with respect to the where we have formally defined the exchange-correlation potential as
.
We now use the auxiliary non-interacting system and its Schrödinger equation, from which we can similarly show that
. (2.24) By comparing this result with (2.23) we see that this effective potential must satisfy
. (2.25)
We are now in a position to implement the self-consistent Kohn-Sham scheme. We first choose an initial trial form of the function and substitute into (2.15) to find a trial form of . We then solve (2.20) for the single-particle wavefunction , and use (2.19) to find
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out of the next iteration. The equations are then solved again, and this process is repeated until self-consistency is obtained, i.e., until the input and output density in one iteration are sufficiently close to one another.
2.3The Exchange-correlation functional
In the Kohn-Sham formulism for DFT, the total energy written as (2.21) may be found that the exchange-correlation energy functional plays the important role of the self-consistent calculation since all components of (2.21) are known exactly with the notable exception of this term. Before we can actually implement the Kohn-Sham formalism, we have to introduce some workable approximation for the exchange-correlation energy .
Conventionally, the exchange-correlation energy is separated into exchange and correlation parts
. (2.26) Next, we are going to introduce some approximations for the exchange-correlation energy using in our calculations.
2.3.1 The local density approximation (LDA)
The first such approximation to be suggested was the Local Density Approximation (LDA) [32]. The idea behind the LDA is a uniform electron gas. In the LDA, we assume that the density of inhomogeneous system varies very slowly, so that the exchange-correlation energy is locally that of a homogeneous system at the local density. Therefore, the difficulty is avoided with the assumption that depends only on the local density , and that
can thus be written as
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, (2.27) where is the exchange-correlation energy per particle of a homogeneous system of density n.
Although no general form is known to exactly determine , the low-density and high-density limits can be calculated analytically. Usually, the density is expressed in terms of the dimensionless parameter , which is the radius of the sphere that can be assigned to each electron in average, measured in units of the Bohr radius . This is
. (2.28)
The exchange energy per particle is straightforward to calculate by Bloch and Dirac in the late 1920’s [33], and the result is
. (2.29)
No such explicit expression is known for the correlation energy per particle . However, a popular approximation for the correlation energy is based on highly accurate numerical quantum Monte-Carlo simulations of the homogeneous electron gas for certain values of by Ceperley and Alder [34]. These are then parameterized. Perdew and Zunger use a Pade approximation in for low-density case [35]:
(2.30)
with , , .
This form is then joined smoothly to the high-density form of for , which is
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(2.31)
with (all parameters given in units of )
, , , .
Practice shows that LDA yields adequate ground-state properties even for strongly inhomogeneous systems since the errors from exchange and correlation parts cancel partially.
The LDA not only yield the accurate work functions of the simple metals, but also gives bond lengths of molecules and solids typically with an astonishing accuracy of ~2%. However, the moderate accuracy that LDA delivers is certainly insufficient for most applications in chemistry.
2.3.2 The generalized gradient approximation (GGA)
An obvious way to go beyond the LDA is the gradient expansion approximation (GEA) which extends the exchange-correlation functional with terms containing gradients of the electron density.
However, it does not lead to consistent improvement over the LDA since the second-order GEA is found to violate the sum rule and the non-positivity constraint on the exchange hole, both of which are important physical conditions [36]. The question then arises of whether one can construct a gradient expansion that avoids these shortcomings, so that the resulting exchange-correlation hole satisfies the most important sum rules.
The resulting gradient corrections with exchange-correlation hole constraints restored are referred to as Generalized Gradient Approximations (GGAs). The exchange-correlation energy in the GGA is written as
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. (2.32) Perdew and Wang have developed a GGA functional based on real-space cutoff of the spurious long-range components of the second-order gradient expansion for the exchange-correlation hole [37], and then improved by the PBE (Perdew-Burke-Enzerhof) form including an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential [38].
The PBE functional for exchange energy is given by
, (2.33)
where is the Slater exchange energy density in the uniform electron gas approximation, is the local Fermi wave vector, and is the enhancement factor depending on a dimensionless density gradient s defined as .
For the linear response of uniform electron gas, LDA is an excellent approximation while GEA is not. In order to recover the LDA linear response, the enhancement factor of the exchange functional takes the form
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The great strength of the GGA lies in the dramatic improvement it gives over the LDA in calculating such properties as bond dissociation energies, which the LDA may overestimate by as much as 100%, while the GGA gives errors typically of the order of ten percent or less.
With modern GGA exchange-correlation potentials, atomic and molecular quantities can now be calculated with chemical accuracy.
2.3.3 Hybrid functionals
Unfortunately, the DFT calculations cannot correctly predict the electronic structure of
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semiconductor materials since the bandgap is considerably underestimated with the aforementioned exchange-correlation energy functional such as LDA and GGA. Recently, this problem can be solved by a new class of DFT exchange-correlation functional which is named hybrid functional.
Hybrid functionals are a class of approximations to the exchange-correlation energy functional in DFT that incorporate a portion of exact exchange from Hartree-Fock approximation. However, the Hartree-Fock exchange in hybrid DFT calculations is much less tractable in large systems arising from the slow decay of the exchange interaction with distance.
Screened Coulomb hybrid functional proposed by Heyd et al. can accelerate the decay of the Hartree-Fock exchange interaction by substituting the full 1/r Coulomb potential with a screened potential which decomposes of the exchange interaction into the short range (SR) and long range (LR) components [39].
, (2.37)
where the mixing coefficient a = 1/4 is determined by perturbation theory [40] and is the screening parameter governing the extent of short range interactions.
Numerical tests based on the particular value of screening parameter indicate that the
Numerical tests based on the particular value of screening parameter indicate that the