n n n n n n
R R H S H S (3.3.3) , and possesses a quadratic unrestricted minimum at the each eigenfunction n( )r . If a good starting guess for the eigenfunction it is possible to use the algorithm without the knowledge of other wavefunctions, and therefore without explicit orthogonalization ofthe preconditioned residual vector.
3.4 Ab-initio molecular dynamics (AIMD)
A natural consequence of the Car-Parrinello formalism61, 62 is that electronic and ionic relaxation can take place simultaneously, the time–dependence of the electronic structure is a consequence of nuclear motion. For this to be performed, the total forces on the ions must be found to update the ionic positions. In order to process this part, Born–Oppenheimer Molecular Dynamics was employed in VASP program63. The force on an ion, I, at positionRI can be obtained from the full derivative of the total
42 be altered; therefore the full derivative has to be expressed in terms of changes in the wavefunction: Therefore after some algebraic manipulation the last two terms can be written as:
i i i
i i
R . This shows that when each electronic orbital is an eigenstate of the Hamiltonian, then the partial derivative of the total energy with respect to the ionic positions is the force felt by the ions. According to the Hellmann-Feynman theorem64,it can be generalized to any order of derivative of the total energy. In practice, the wavefunctions are only calculated to a given tolerance therefore they are not exact eigenstates of the Kohn-Sham Hamiltonian. As a result the forces calculated using the Hellmann-Feynman theorem incurs error. This can be seen when the expression for the force is written formally as:( ) ( ) the variational force, vanishes when the wavefunctions are completely converged so that the conditions of the Hellmann-Feynman theorem are satisfied. It is in this term that the errors occurring from unconverged wavefunctions or incomplete basis sets are incurred. The error on the force is first order with respect to error in the wavefunction.
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It follows that the Hellmann-Feynman theorem can only be implemented when the wave functions are very close to self consistency. Only then can the ionic equations of motion be integrated and the ionic positions updated. This leads to the use of the Born-Oppenheimer approximation. This is based on the fact that typical electronic velocities are much greater than that of the ions. It can therefore be assumed that the ions move so slowly relative to the electrons that at any point in time, the electrons will be in their ground state for that instantaneous ionic configuration (see Fig.
3.4.1).The ground state wavefunction is first calculated self-consistently for a fixed set of ionic coordinates. In this method an initial wavefunction has to be assumed this is given by initializing the expansion coefficients of the wavefunction by random numbers. Although this initial state is far from the Kohn-Sham eigenstates it does not assume any initial symmetry which, if incorrect, could lead to prohibitively long calculation to find the correct wavefunctions. The total energy is reduced by the residual minimization scheme(RMS) until the wavefunctions are sufficiently converged to give correct forces. New ionic positions are then calculated under the influence of these forces either by integration of the equations of motion or by direct minimization techniques. After each change in the position of the ions, the wavefunctions will no longer be eigenstates of the new ionic positions. In order to find the wavefunctions for this better set of ionic positions it is not necessary to randomize the wavefunctions. The motion of the ions will be sufficiently small that the old wavefunctions will be a good place to initialize the iterative procedure again.
After electronic self-consist iteration, a molecular dynamic iterative procedure as shown in Fig. 3.4.2. The verlet algorithm is employed for time iteration in VASP program which running as follow step. Consider expansion of coordinate forward and backward in time scale we have:
44 As the equation shows, we only need to know current acceleration from the potential gradient of current position: ( ) ( )t U[ ( )]t
schematic view of iterative procedure as shown in Fig. 3.4.2.
Figure 3.4.1 Schematic diagram of an MD trajectory on BO surface
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Figure 3.4.2 Flow chart of iterative procedure of ab initio molecular dynamics simulations (AIMD).
Chapter 4 Experimental results
In chapter we will first introduce the principle of scanning tunnel microscope (STM) and X-ray photoelectron spectroscopic (XPS); and then, discuss experimental results. Two measurement experiments have same sample preparation procedure as shown in following. The sample was placed in a grounded -metal chamber with ultrahigh-vacuum (UHV) environment (1~3×10-10 torr) at room temperature (~300K).
The Si(100) samples used in our experiment were sliced into pieces of size 1 mm×8 mm .After loading the samples to the chamber; the samples were then being degassed for over 12 hours at ~900 K. After degassing, the sample was flashed at ~1450 K for a few seconds (~9s) to remove the surface oxide layer. After these processes the surface would form a Si(100)-2×1 dimerized surface. More detailed sample cleaning procedure please refer this article65. And then, chlorine gas was induced and controlled through a leak valve. The amount of dosage of chlorine gas was about 2 10 82 minutes (2.4 Langmuir) .After reaction; the surface would form a Cl-terminated Si (100)-2×1 structure. After that, we employed the DC current controlled “alkali metal disperser” to deposit Sodium atom under varying exposure time. The deposition rate was calibrated by thickness monitor; and to converter the unit into “ML” by referring atom density of surface silicon. About one minute afterwards, the surface would reach thermal equilibrium. After that time, we just start
46
to take data.