Supporting Information for Adatom Interactions on GaN(0001) Surface I
Supporting Information for Adatom Interactions on GaN(0001) Surface I: Coverage Dependent Adsorption Manjusha Chugh and Madhav Ranganathan* Department of Chemistry, Indian Institute of Technology Kanpur, Kanpur. India. E-mail: [email protected] Phone: +91 512 2596037. Fax: +91 512 2596806
•
To whom correspondence should be addressed. S1
The DFT results presented here have been obtained within the Generalized Gradient Approximation (GGA) parameterized by PBE exchange-correlation functional using PWscf code of Quantum Espresso package (version 5.0.2). For the optimization calculations, a kinetic energy cutoff of 70 Ry is used and the force convergence threshold was set up at 0.025 eV/Å. Sample input and output files used are provided at the end of this document. S1. k-mesh convergence: We have used (2x2), (3x3), (4x4) and (5x5) supercells for surface calculations. All these unit cells have slab geometries made up of 6 GaN bilayers and a vacuum gap of 13 Å. These unit cells differ in the lateral dimensions with respect to each other. For example, (5x5) is 5 times the unit lattice vector along the x-axis. Since all these unit cells have a slab geometry with a vacuum gap along the z-direction, we have taken single k-point for the Brillouin Zone integration along the z-direction (as is done usually). The unit cells differ in the lateral dimensions from each other, so the k-points for each unit cell should be different for the integration along lateral dimensions to have same accuracy of the calculated properties. Since the two lateral dimensions are of same magnitude for a (NxN) unit cell, k-points are also same for these two directions. We have tested the convergence of k-points by calculating the self consistent field (scf) energy for different k-points along the lateral dimensions of the unit cells. This k-mesh convergence is shown in Fig. S1 for all the unit cells considered in our work. The different number of k-points chosen for these surface supercells are as follows: surface supercell (2x2) (3x3) (4x4) (5x5)
k-points (4x4x1) (3x3x1) (2x2x1) (1x1x1)
Figure S1: Self consistent field (scf) energy (Ry/atom) as a function of number of k-points taken along the lateral dimensions for the unit cells considered in this work.
S2
S2. Relative surface energy calculations: GaN is a polar material, having alternate Ga and N layers along [0001] direction. If we cut a GaN crystal perpendicular to the [0001] direction, we can not get two equivalent surfaces. So, it is not possible to calculate the absolute surface energy for the GaN(0001) surface. Hence, relative surface energy is defined to calculate the surface energies for GaN(0001) surface configurations as shown below. To calculate the surface formation energies for the Ga and N adatom configurations, we have taken clean GaN(0001) surface as a reference. Thus, the surface formation energies would be relative to the clean GaN(0001) surface. The relative surface formation energies for the Ga and N adatom configurations for a (2x2) unit cell are plotted as a function of N chemical potential and shown in Fig. S2. From this figure, we can get an understanding of the stability of a particular configuration under growth conditions (used in experiments). The extreme right of the x-axis in Fig. S2 indicates N-rich (Ga-poor) conditions and extreme left indicates N-poor (Ga-rich) conditions. By looking at this figure, we can say that under Npoor (Ga-rich) conditions, Ga-hcp (2x2) configuration is most stable, while under the conditions of Nrich (Ga-poor), N-fcc (2x2) configuration is most stable. Thus, the relative stability of other configurations can also be predicted under different growth conditions. Our calculated relative surface formation energies of these configurations are in agreement with previous theoretical results[1].
Figure S2: Relative surface energies (in meV/Å2) as a function of N chemical potential for Ga and N adatoms configurations for a (2x2) unit cell. clean GaN means Flat Clean GaN(0001) surface. Other labels are as defined in the manuscript.
S3
S3. FCS and DCS configurations: Here, we show how Flat Clean Slab (FCS) and Distorted Clean Slab (DCS) configurations are obtained. A clean Ga-terminated slab is obtained from the bulk GaN crystal by cutting the crystal through a plane perpendicular to the z-direction. Geometry optimization of upper 3 GaN bilayers of such a slab, using the procedure mentioned in the manuscript, gives the clean slab configuration as represented in Fig. S3(a). This configuration is called a Flat Clean Slab (FCS) configuration. An adatom is adsorbed at the surface of this FCS (diagrammatically shown in Fig. S3(b)) and geometry optimization is performed. The final adsorbate/substrate configuration is shown in Fig. S3(c). The adsorption energy for an adatom A (Ga or N), calculated using FCS as the clean slab configuration, is defined as : A ,FCS A /GaN A Ead =E slab −E GaN FCS −Eiso / GaN where E Aslab is the total energy of the fully-relaxed adsorbate/substrate configuration (Fig. S3(c)), GaN A E FCS is the energy of the Flat Clean Slab (FCS) configuration (Fig. S3(a)) and Eiso is the energy of an isolated adatom. This Eq. is same as Eq. 6 of the manuscript.
We have used other clean slab configurations in our work. We take a fully-relaxed adsorbate/substrate system (Fig. S3(c)), remove the adatom from this system while keeping the substrate atoms at their relaxed positions of the adsorbate/substrate system. The configuration obtained by removing the adatom from the adsorbate/susbtrate configuration also represents a clean (no adatom) surface configuration. We call this as the Distorted Clean Slab (DCS) configuration, because the substrate atoms in this configuration are displaced from their positions as compared to the Flat Clean Slab (FCS) configuration. This DCS configuration is shown in Fig. S3 (d). We perform self consistent field (scf) calculations of this DCS configuration. The energy obtained in this way is called the energy of DCS-scf configuration in our manuscript. Note that this DCS-scf energy corresponds to energy of a clean slab configuration (i.e. one without an adatom). The adsorption energy for an adatom A, calculated using this DCS-scf energy as a reference for the clean slab energy (to see the contribution of the lattice distortions to the total adsorption energy), is defined as : A , DCS−scf / GaN A Ead =E Aslab −EGaN DCS −scf −E iso / GaN A where E Aslab and Eiso are same as defined for the previous Eq., EGaN DCS− scf is the scf energy of the DCS configuration (Fig S3(d)). This Eq. is same as Eq. 7 of the manuscript.
There is one another way to calculate the energy of a Distorted Clean Slab (DCS) configuration. Instead of just doing scf calculation, we perform geometry optimization of the DCS configuration by relaxing the upper 3 bilayers of the DCS (Fig. S3(d)). The structure obtained after geometry optimization is called as DCS-rlx, shown in Fig. S3(e). We noticed that when the DCS configuration is relaxed, the substrate atoms did not go back to the initial FCS configuration. They remain near to their positions as were in fully-relaxed adsorbate/substrate configuration. We have also tabulated the total energies of FCS, DCS-scf and DCS-rlx configurations for the Ga and N adatom for all the system sizes S4
considered in our work in Table S1. The calculated energy of the DCS-rlx configuration was found to be lower than that of the FCS configuration for all the system sizes considered for both the adatoms. We have also calculated the adsorption energy of an adatom A, using DCS-rlx energy as a reference for the clean slab energy: A , DCS−rlx A /GaN A Ead =E slab −EGaN DCS−rlx −E iso / GaN A where E Aslab and Eiso are same as defined in previous two Eqs., EGaN DCS−rlx is the energy of the DCS-rlx configuration (Fig. S3(e)).
Figure S3: Various GaN configurations : FCS represents Flat Clean Slab and DCS represents Distorted Clean Slab. Other configurations are described in the text.
S5
System Ga
N
EGaN FCS
EGaN DCS− scf
EGaN DCS−rlx
(2x2)
-132.1688
-132.1690
-132.1691
(3x3)
-132.1688
-132.1689
-132.1690
(4x4)
-132.1688
-132.1694
-132.1694
(5x5)
-132.1684
-132.1690
-132.1690
(2x2)
-132.1688
-132.1682
-132.1691
(3x3)
-132.1688
-132.1686
-132.1690
(4x4)
-132.1688
-132.1690
-132.1694
(5x5)
-132.1684
-132.1688
-132.1690
Table S1: Total energy (in Ry/Å2) for the clean slab configurations (Flat Clean Slab (FCS), Distorted Clean Slab (DCS-scf and DCS-rlx)) for all the unit cells considered in this work.
S4. Dipole moment calculations: In the manuscript, we have calculated the dipole moment according to Eq. 4 (of the manuscript). We found that the dipole moment change due to the adatom adsorption is very small. Here, we are showing other methods employed to calculate the dipole moment. We first calculate the amount of charge transfer to/from the substrate from/to the adsorbed adatoms using the Bader charge analysis method[2,3]. Then by taking the product of that charge transfer and the distance between the adsorbed species and the substrate, we calculate the dipole moment of a particular configuration. To calculate the change in dipole moment upon adatom adsorption, the dipole moment of the clean substrate is subtracted from the dipole moment of the adsorbate configuration. As an example, we calculate the dipole moment of a (2x2) N adatom configuration. The dipole moment change for this configuration according to Eq. 4 of the manuscript is -2.10 Debye, while the same calculated using the charge transfer method is -2.06 Debye. Thus, we see that both the methods give same magnitude of dipole moment change. We also calculate the change in dipole moment upon N adatom adsorption using the work function change method. In this method, first the work function of a system is calculated both with and without adatom and their difference is taken. This work function change ( Δ W ) is then used to calculate the change in dipole moment according to the following Helmholtz equation[4] : S6
Δ μ=
A ΔW 12 πθ
where A is the area of (1x1) unit cell in Å2 , θ is the coverage. The calculated change in dipole moment according to the work function method for the (2x2) Nadatom is -1.31 Debye. From all these methods, we can see that the change in dipole moment upon adsorption is very small.
S5. Sample input and output files: We show below parts of the the PWscf input and output files for (3x3) Ga adatom calculations. INPUT FILE : &CONTROL calculation = "relax", title = '3x3_UC', verbosity = 'high', restart_mode = 'from_scratch', prefix = 'Ga', pseudo_dir = '..../', outdir = '..../', tprnfor = .TRUE., etot_conv_thr = 1.0D-04, forc_conv_thr = 1.0D-03, nstep = 400 / &SYSTEM ibrav = 0 , celldm(1)= 6.088, nat = 118 , ntyp = 3 , ecutwfc = 70.0 , ecutrho = 560.0 , input_dft = 'PBE' , occupations = 'smearing', smearing = 'methfessel-paxton' , degauss = 0.03 S7
/ &ELECTRONS electron_maxstep = 500 , diagonalization='cg', conv_thr = 1.D-7, mixing_mode = 'local-TF', mixing_beta = 0.4D0 / &IONS ion_dynamics = 'bfgs' / CELL_PARAMETERS {hexagonal} 3.00 0.00 0.00 1.50 2.598 0.00 0.00 0.00 8.60 ATOMIC_SPECIES Ga 69.723 Ga.pbe-n-van.UPF N 14.006 N.pbe-van_ak.UPF H 1.0079 H.pz-vbc_075.UPF ATOMIC_POSITIONS {crystal} H 0.000000000 0.000000000 H 0.333333333 0.000000000 H 0.666666666 0.000000000 H 0.000000000 0.333333333 H 0.000000000 0.666666666 H 0.333333333 0.333333333 H 0.333333333 0.666666666 H 0.666666666 0.333333333 H 0.666666666 0.666666666 N 0.000000000 0.000000000 N 0.333333333 0.000000000 N 0.666666666 0.000000000 N 0.000000000 0.333333333 N 0.000000000 0.666666666 N 0.333333333 0.333333333 N 0.333333333 0.666666666 N 0.666666666 0.333333333 N 0.666666666 0.666666666 Ga 0.111111111 0.111111111 Ga 0.111111111 0.444444444 Ga 0.111111111 0.777777777
0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.257209186 0.257209186 0.257209186
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S8
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N
0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.111111111 0.111111111 0.111111111 0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.111111111 0.111111111 0.111111111 0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.111111111 0.111111111 0.111111111 0.444444444 0.444444444
0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444
0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.446473610 0.446473610 0.446473610 0.446473610 0.446473610 0.446473610 0.446473610 0.446473610 0.446473610 0.517853955 0.517853955 0.517853955 0.517853955 0.517853955
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
S9
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga Ga Ga Ga Ga
0.444444444 0.777777777 0.777777777 0.777777777 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.111111111 0.111111111 0.111111111 0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.111111111 0.111111111 0.111111111 0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333
0.777777777 0.111111111 0.444444444 0.777777777 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666
0.517853955 0.517853955 0.517853955 0.517853955 0.541105823 0.541105823 0.541105823 0.541105823 0.541105823 0.541105823 0.541105823 0.541105823 0.541105823 0.612486167 0.612486167 0.612486167 0.612486167 0.612486167 0.612486167 0.612486167 0.612486167 0.612486167 0.635738035 0.635738035 0.635738035 0.635738035 0.635738035 0.635738035 0.635738035 0.635738035 0.635738035 0.707118380 0.707118380 0.707118380 0.707118380 0.707118380 0.707118380 0.707118380 0.707118380 0.707118380 0.730370247 0.730370247 0.730370247 0.730370247 0.730370247 0.730370247 0.730370247 S10
Ga Ga Ga
0.666666666 0.333333333 0.730370247 0.666666666 0.666666666 0.730370247 0.444444444 0.444444444 0.780000000
K_POINTS (automatic) 3 3 1 0 0 0
OUTPUT FILE : End of BFGS Geometry Optimization Final energy = -10869.3505613211 Ry Begin final coordinates ATOMIC_POSITIONS (crystal) H 0.000000000 0.000000000 H 0.333333333 0.000000000 H 0.666666666 0.000000000 H 0.000000000 0.333333333 H 0.000000000 0.666666666 H 0.333333333 0.333333333 H 0.333333333 0.666666666 H 0.666666666 0.333333333 H 0.666666666 0.666666666 N 0.000000000 0.000000000 N 0.333333333 0.000000000 N 0.666666666 0.000000000 N 0.000000000 0.333333333 N 0.000000000 0.666666666 N 0.333333333 0.333333333 N 0.333333333 0.666666666 N 0.666666666 0.333333333 N 0.666666666 0.666666666 Ga 0.111111111 0.111111111 Ga 0.111111111 0.444444444 Ga 0.111111111 0.777777777 Ga 0.444444444 0.111111111 Ga 0.444444444 0.444444444 Ga 0.444444444 0.777777777 Ga 0.777777777 0.111111111 Ga 0.777777777 0.444444444 Ga 0.777777777 0.777777777 N 0.111111111 0.111111111
0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.328589531
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
S11
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
N N N N N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga
0.111111111 0.111111111 0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.111112271 0.111087661 0.111087543 0.444545717 0.444445637 0.444546096 0.777703472 0.777703261 0.777779133 0.111113410 0.111066056 0.111065801 0.444749881 0.444446320 0.444750398 0.777523400 0.777523206 0.777779942 0.000020703 0.333294458 0.666065318
0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.111108792 0.444542410 0.777699693 0.111084143 0.444442059 0.777699693 0.111084143 0.444542410 0.777775065 0.111106514 0.444744071 0.777517133 0.111060051 0.444440691 0.777517133 0.111060051 0.444744071 0.777773446 0.000018172 0.000018172 0.001202695
0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.446481767 0.446579427 0.446579164 0.446578975 0.446590960 0.446579164 0.446578975 0.446579427 0.446867030 0.517613787 0.517855434 0.517855177 0.517854949 0.517773623 0.517855177 0.517854949 0.517855434 0.518737196 0.541057809 0.541057809 0.541564794
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
S12
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Ga 0.000020904 0.333291525 Ga 0.001205185 0.666062624 Ga 0.333380940 0.333378291 Ga 0.333380670 0.666571992 Ga 0.666574101 0.333378291 Ga 0.666065523 0.666062624 N 0.000081952 0.000082029 N 0.333169352 0.000082029 N 0.665858251 0.001616830 N 0.000082228 0.333168878 N 0.001617841 0.665857365 N 0.333150756 0.333150458 N 0.333150837 0.667031658 N 0.667032119 0.333150458 N 0.665858126 0.665857365 Ga 0.111111633 0.111110066 Ga 0.113670007 0.439195625 Ga 0.113670686 0.780462008 Ga 0.439202597 0.113666023 Ga 0.444445602 0.444442128 Ga 0.439200638 0.780462008 Ga 0.780464711 0.113666023 Ga 0.780467700 0.439195625 Ga 0.777778621 0.777776089 N 0.111113202 0.111106929 N 0.112131689 0.440560091 N 0.112132909 0.780634588 N 0.440566050 0.112130449 N 0.444445022 0.444443288 N 0.440565836 0.780634588 N 0.780636832 0.112130449 N 0.780641553 0.440560091 N 0.777777476 0.777778380 Ga 0.004241005 0.004241842 Ga 0.324850486 0.004241842 Ga 0.663157208 0.007018917 Ga 0.004245439 0.324842456 Ga 0.007016466 0.663156320 Ga 0.334324079 0.334331197 Ga 0.334328782 0.664675769 Ga 0.664678056 0.334331197 Ga 0.663160546 0.663156320 Ga 0.444450615 0.444432101 End final coordinates
0.541058043 0.541565945 0.541106229 0.541105964 0.541106229 0.541565945 0.612005524 0.612005524 0.613986759 0.612005382 0.613988014 0.612077658 0.612077474 0.612077658 0.613988014 0.636278844 0.636394193 0.636393027 0.636392536 0.636519979 0.636393027 0.636392536 0.636394193 0.633122184 0.705856702 0.708782678 0.708782725 0.708782309 0.705011187 0.708782725 0.708782309 0.708782678 0.710581350 0.737634176 0.737634176 0.719713862 0.737649294 0.719711130 0.737080090 0.737082917 0.737080090 0.719711130 0.797782997
S13
References 1. Rosa, A. L.; Neugebauer, J. First-Principles Calculations of the Structural and Electronic Properties of Clean GaN(0001) Surfaces. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 205346. 2. Henkelman, G.; Arnaldsson, A.; Jónsson, H. A Fast and Robust Algorithm for Bader Decomposition of Charge Density. Comput. Mater. Sci. 2006, 36, 254-360. 3. Sanville, E.; Kenny, S. D.; Smith, R.; Henkelman, G. An Improved Grid-Based Algorithm for Bader Charge Allocation. J. Comp. Chem. 2007, 28, 899-908. 4. Silverstrelli, P. L.; Ambrosetti, A.; Grubisiĉ, S.; Ancilotto, F. Adsorption of Rare-Gas Atoms on Cu(111) and Pb(111) Surfaces by van der Waals Corrected Density Functional Theory. Phys. Rev. B. 2012, 85, 165405.
S14
•
To whom correspondence should be addressed. S1
The DFT results presented here have been obtained within the Generalized Gradient Approximation (GGA) parameterized by PBE exchange-correlation functional using PWscf code of Quantum Espresso package (version 5.0.2). For the optimization calculations, a kinetic energy cutoff of 70 Ry is used and the force convergence threshold was set up at 0.025 eV/Å. Sample input and output files used are provided at the end of this document. S1. k-mesh convergence: We have used (2x2), (3x3), (4x4) and (5x5) supercells for surface calculations. All these unit cells have slab geometries made up of 6 GaN bilayers and a vacuum gap of 13 Å. These unit cells differ in the lateral dimensions with respect to each other. For example, (5x5) is 5 times the unit lattice vector along the x-axis. Since all these unit cells have a slab geometry with a vacuum gap along the z-direction, we have taken single k-point for the Brillouin Zone integration along the z-direction (as is done usually). The unit cells differ in the lateral dimensions from each other, so the k-points for each unit cell should be different for the integration along lateral dimensions to have same accuracy of the calculated properties. Since the two lateral dimensions are of same magnitude for a (NxN) unit cell, k-points are also same for these two directions. We have tested the convergence of k-points by calculating the self consistent field (scf) energy for different k-points along the lateral dimensions of the unit cells. This k-mesh convergence is shown in Fig. S1 for all the unit cells considered in our work. The different number of k-points chosen for these surface supercells are as follows: surface supercell (2x2) (3x3) (4x4) (5x5)
k-points (4x4x1) (3x3x1) (2x2x1) (1x1x1)
Figure S1: Self consistent field (scf) energy (Ry/atom) as a function of number of k-points taken along the lateral dimensions for the unit cells considered in this work.
S2
S2. Relative surface energy calculations: GaN is a polar material, having alternate Ga and N layers along [0001] direction. If we cut a GaN crystal perpendicular to the [0001] direction, we can not get two equivalent surfaces. So, it is not possible to calculate the absolute surface energy for the GaN(0001) surface. Hence, relative surface energy is defined to calculate the surface energies for GaN(0001) surface configurations as shown below. To calculate the surface formation energies for the Ga and N adatom configurations, we have taken clean GaN(0001) surface as a reference. Thus, the surface formation energies would be relative to the clean GaN(0001) surface. The relative surface formation energies for the Ga and N adatom configurations for a (2x2) unit cell are plotted as a function of N chemical potential and shown in Fig. S2. From this figure, we can get an understanding of the stability of a particular configuration under growth conditions (used in experiments). The extreme right of the x-axis in Fig. S2 indicates N-rich (Ga-poor) conditions and extreme left indicates N-poor (Ga-rich) conditions. By looking at this figure, we can say that under Npoor (Ga-rich) conditions, Ga-hcp (2x2) configuration is most stable, while under the conditions of Nrich (Ga-poor), N-fcc (2x2) configuration is most stable. Thus, the relative stability of other configurations can also be predicted under different growth conditions. Our calculated relative surface formation energies of these configurations are in agreement with previous theoretical results[1].
Figure S2: Relative surface energies (in meV/Å2) as a function of N chemical potential for Ga and N adatoms configurations for a (2x2) unit cell. clean GaN means Flat Clean GaN(0001) surface. Other labels are as defined in the manuscript.
S3
S3. FCS and DCS configurations: Here, we show how Flat Clean Slab (FCS) and Distorted Clean Slab (DCS) configurations are obtained. A clean Ga-terminated slab is obtained from the bulk GaN crystal by cutting the crystal through a plane perpendicular to the z-direction. Geometry optimization of upper 3 GaN bilayers of such a slab, using the procedure mentioned in the manuscript, gives the clean slab configuration as represented in Fig. S3(a). This configuration is called a Flat Clean Slab (FCS) configuration. An adatom is adsorbed at the surface of this FCS (diagrammatically shown in Fig. S3(b)) and geometry optimization is performed. The final adsorbate/substrate configuration is shown in Fig. S3(c). The adsorption energy for an adatom A (Ga or N), calculated using FCS as the clean slab configuration, is defined as : A ,FCS A /GaN A Ead =E slab −E GaN FCS −Eiso / GaN where E Aslab is the total energy of the fully-relaxed adsorbate/substrate configuration (Fig. S3(c)), GaN A E FCS is the energy of the Flat Clean Slab (FCS) configuration (Fig. S3(a)) and Eiso is the energy of an isolated adatom. This Eq. is same as Eq. 6 of the manuscript.
We have used other clean slab configurations in our work. We take a fully-relaxed adsorbate/substrate system (Fig. S3(c)), remove the adatom from this system while keeping the substrate atoms at their relaxed positions of the adsorbate/substrate system. The configuration obtained by removing the adatom from the adsorbate/susbtrate configuration also represents a clean (no adatom) surface configuration. We call this as the Distorted Clean Slab (DCS) configuration, because the substrate atoms in this configuration are displaced from their positions as compared to the Flat Clean Slab (FCS) configuration. This DCS configuration is shown in Fig. S3 (d). We perform self consistent field (scf) calculations of this DCS configuration. The energy obtained in this way is called the energy of DCS-scf configuration in our manuscript. Note that this DCS-scf energy corresponds to energy of a clean slab configuration (i.e. one without an adatom). The adsorption energy for an adatom A, calculated using this DCS-scf energy as a reference for the clean slab energy (to see the contribution of the lattice distortions to the total adsorption energy), is defined as : A , DCS−scf / GaN A Ead =E Aslab −EGaN DCS −scf −E iso / GaN A where E Aslab and Eiso are same as defined for the previous Eq., EGaN DCS− scf is the scf energy of the DCS configuration (Fig S3(d)). This Eq. is same as Eq. 7 of the manuscript.
There is one another way to calculate the energy of a Distorted Clean Slab (DCS) configuration. Instead of just doing scf calculation, we perform geometry optimization of the DCS configuration by relaxing the upper 3 bilayers of the DCS (Fig. S3(d)). The structure obtained after geometry optimization is called as DCS-rlx, shown in Fig. S3(e). We noticed that when the DCS configuration is relaxed, the substrate atoms did not go back to the initial FCS configuration. They remain near to their positions as were in fully-relaxed adsorbate/substrate configuration. We have also tabulated the total energies of FCS, DCS-scf and DCS-rlx configurations for the Ga and N adatom for all the system sizes S4
considered in our work in Table S1. The calculated energy of the DCS-rlx configuration was found to be lower than that of the FCS configuration for all the system sizes considered for both the adatoms. We have also calculated the adsorption energy of an adatom A, using DCS-rlx energy as a reference for the clean slab energy: A , DCS−rlx A /GaN A Ead =E slab −EGaN DCS−rlx −E iso / GaN A where E Aslab and Eiso are same as defined in previous two Eqs., EGaN DCS−rlx is the energy of the DCS-rlx configuration (Fig. S3(e)).
Figure S3: Various GaN configurations : FCS represents Flat Clean Slab and DCS represents Distorted Clean Slab. Other configurations are described in the text.
S5
System Ga
N
EGaN FCS
EGaN DCS− scf
EGaN DCS−rlx
(2x2)
-132.1688
-132.1690
-132.1691
(3x3)
-132.1688
-132.1689
-132.1690
(4x4)
-132.1688
-132.1694
-132.1694
(5x5)
-132.1684
-132.1690
-132.1690
(2x2)
-132.1688
-132.1682
-132.1691
(3x3)
-132.1688
-132.1686
-132.1690
(4x4)
-132.1688
-132.1690
-132.1694
(5x5)
-132.1684
-132.1688
-132.1690
Table S1: Total energy (in Ry/Å2) for the clean slab configurations (Flat Clean Slab (FCS), Distorted Clean Slab (DCS-scf and DCS-rlx)) for all the unit cells considered in this work.
S4. Dipole moment calculations: In the manuscript, we have calculated the dipole moment according to Eq. 4 (of the manuscript). We found that the dipole moment change due to the adatom adsorption is very small. Here, we are showing other methods employed to calculate the dipole moment. We first calculate the amount of charge transfer to/from the substrate from/to the adsorbed adatoms using the Bader charge analysis method[2,3]. Then by taking the product of that charge transfer and the distance between the adsorbed species and the substrate, we calculate the dipole moment of a particular configuration. To calculate the change in dipole moment upon adatom adsorption, the dipole moment of the clean substrate is subtracted from the dipole moment of the adsorbate configuration. As an example, we calculate the dipole moment of a (2x2) N adatom configuration. The dipole moment change for this configuration according to Eq. 4 of the manuscript is -2.10 Debye, while the same calculated using the charge transfer method is -2.06 Debye. Thus, we see that both the methods give same magnitude of dipole moment change. We also calculate the change in dipole moment upon N adatom adsorption using the work function change method. In this method, first the work function of a system is calculated both with and without adatom and their difference is taken. This work function change ( Δ W ) is then used to calculate the change in dipole moment according to the following Helmholtz equation[4] : S6
Δ μ=
A ΔW 12 πθ
where A is the area of (1x1) unit cell in Å2 , θ is the coverage. The calculated change in dipole moment according to the work function method for the (2x2) Nadatom is -1.31 Debye. From all these methods, we can see that the change in dipole moment upon adsorption is very small.
S5. Sample input and output files: We show below parts of the the PWscf input and output files for (3x3) Ga adatom calculations. INPUT FILE : &CONTROL calculation = "relax", title = '3x3_UC', verbosity = 'high', restart_mode = 'from_scratch', prefix = 'Ga', pseudo_dir = '..../', outdir = '..../', tprnfor = .TRUE., etot_conv_thr = 1.0D-04, forc_conv_thr = 1.0D-03, nstep = 400 / &SYSTEM ibrav = 0 , celldm(1)= 6.088, nat = 118 , ntyp = 3 , ecutwfc = 70.0 , ecutrho = 560.0 , input_dft = 'PBE' , occupations = 'smearing', smearing = 'methfessel-paxton' , degauss = 0.03 S7
/ &ELECTRONS electron_maxstep = 500 , diagonalization='cg', conv_thr = 1.D-7, mixing_mode = 'local-TF', mixing_beta = 0.4D0 / &IONS ion_dynamics = 'bfgs' / CELL_PARAMETERS {hexagonal} 3.00 0.00 0.00 1.50 2.598 0.00 0.00 0.00 8.60 ATOMIC_SPECIES Ga 69.723 Ga.pbe-n-van.UPF N 14.006 N.pbe-van_ak.UPF H 1.0079 H.pz-vbc_075.UPF ATOMIC_POSITIONS {crystal} H 0.000000000 0.000000000 H 0.333333333 0.000000000 H 0.666666666 0.000000000 H 0.000000000 0.333333333 H 0.000000000 0.666666666 H 0.333333333 0.333333333 H 0.333333333 0.666666666 H 0.666666666 0.333333333 H 0.666666666 0.666666666 N 0.000000000 0.000000000 N 0.333333333 0.000000000 N 0.666666666 0.000000000 N 0.000000000 0.333333333 N 0.000000000 0.666666666 N 0.333333333 0.333333333 N 0.333333333 0.666666666 N 0.666666666 0.333333333 N 0.666666666 0.666666666 Ga 0.111111111 0.111111111 Ga 0.111111111 0.444444444 Ga 0.111111111 0.777777777
0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.257209186 0.257209186 0.257209186
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 S8
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N
0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.111111111 0.111111111 0.111111111 0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.111111111 0.111111111 0.111111111 0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.111111111 0.111111111 0.111111111 0.444444444 0.444444444
0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444
0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.446473610 0.446473610 0.446473610 0.446473610 0.446473610 0.446473610 0.446473610 0.446473610 0.446473610 0.517853955 0.517853955 0.517853955 0.517853955 0.517853955
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
S9
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga Ga Ga Ga Ga
0.444444444 0.777777777 0.777777777 0.777777777 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.111111111 0.111111111 0.111111111 0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.111111111 0.111111111 0.111111111 0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333
0.777777777 0.111111111 0.444444444 0.777777777 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666
0.517853955 0.517853955 0.517853955 0.517853955 0.541105823 0.541105823 0.541105823 0.541105823 0.541105823 0.541105823 0.541105823 0.541105823 0.541105823 0.612486167 0.612486167 0.612486167 0.612486167 0.612486167 0.612486167 0.612486167 0.612486167 0.612486167 0.635738035 0.635738035 0.635738035 0.635738035 0.635738035 0.635738035 0.635738035 0.635738035 0.635738035 0.707118380 0.707118380 0.707118380 0.707118380 0.707118380 0.707118380 0.707118380 0.707118380 0.707118380 0.730370247 0.730370247 0.730370247 0.730370247 0.730370247 0.730370247 0.730370247 S10
Ga Ga Ga
0.666666666 0.333333333 0.730370247 0.666666666 0.666666666 0.730370247 0.444444444 0.444444444 0.780000000
K_POINTS (automatic) 3 3 1 0 0 0
OUTPUT FILE : End of BFGS Geometry Optimization Final energy = -10869.3505613211 Ry Begin final coordinates ATOMIC_POSITIONS (crystal) H 0.000000000 0.000000000 H 0.333333333 0.000000000 H 0.666666666 0.000000000 H 0.000000000 0.333333333 H 0.000000000 0.666666666 H 0.333333333 0.333333333 H 0.333333333 0.666666666 H 0.666666666 0.333333333 H 0.666666666 0.666666666 N 0.000000000 0.000000000 N 0.333333333 0.000000000 N 0.666666666 0.000000000 N 0.000000000 0.333333333 N 0.000000000 0.666666666 N 0.333333333 0.333333333 N 0.333333333 0.666666666 N 0.666666666 0.333333333 N 0.666666666 0.666666666 Ga 0.111111111 0.111111111 Ga 0.111111111 0.444444444 Ga 0.111111111 0.777777777 Ga 0.444444444 0.111111111 Ga 0.444444444 0.444444444 Ga 0.444444444 0.777777777 Ga 0.777777777 0.111111111 Ga 0.777777777 0.444444444 Ga 0.777777777 0.777777777 N 0.111111111 0.111111111
0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.196479930 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.233957319 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.257209186 0.328589531
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
S11
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
N N N N N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga Ga Ga Ga Ga Ga Ga N N N N N N N N N Ga Ga Ga
0.111111111 0.111111111 0.444444444 0.444444444 0.444444444 0.777777777 0.777777777 0.777777777 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.000000000 0.333333333 0.666666666 0.000000000 0.000000000 0.333333333 0.333333333 0.666666666 0.666666666 0.111112271 0.111087661 0.111087543 0.444545717 0.444445637 0.444546096 0.777703472 0.777703261 0.777779133 0.111113410 0.111066056 0.111065801 0.444749881 0.444446320 0.444750398 0.777523400 0.777523206 0.777779942 0.000020703 0.333294458 0.666065318
0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.111111111 0.444444444 0.777777777 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.000000000 0.000000000 0.000000000 0.333333333 0.666666666 0.333333333 0.666666666 0.333333333 0.666666666 0.111108792 0.444542410 0.777699693 0.111084143 0.444442059 0.777699693 0.111084143 0.444542410 0.777775065 0.111106514 0.444744071 0.777517133 0.111060051 0.444440691 0.777517133 0.111060051 0.444744071 0.777773446 0.000018172 0.000018172 0.001202695
0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.328589531 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.351841398 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.423221743 0.446481767 0.446579427 0.446579164 0.446578975 0.446590960 0.446579164 0.446578975 0.446579427 0.446867030 0.517613787 0.517855434 0.517855177 0.517854949 0.517773623 0.517855177 0.517854949 0.517855434 0.518737196 0.541057809 0.541057809 0.541564794
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
S12
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Ga 0.000020904 0.333291525 Ga 0.001205185 0.666062624 Ga 0.333380940 0.333378291 Ga 0.333380670 0.666571992 Ga 0.666574101 0.333378291 Ga 0.666065523 0.666062624 N 0.000081952 0.000082029 N 0.333169352 0.000082029 N 0.665858251 0.001616830 N 0.000082228 0.333168878 N 0.001617841 0.665857365 N 0.333150756 0.333150458 N 0.333150837 0.667031658 N 0.667032119 0.333150458 N 0.665858126 0.665857365 Ga 0.111111633 0.111110066 Ga 0.113670007 0.439195625 Ga 0.113670686 0.780462008 Ga 0.439202597 0.113666023 Ga 0.444445602 0.444442128 Ga 0.439200638 0.780462008 Ga 0.780464711 0.113666023 Ga 0.780467700 0.439195625 Ga 0.777778621 0.777776089 N 0.111113202 0.111106929 N 0.112131689 0.440560091 N 0.112132909 0.780634588 N 0.440566050 0.112130449 N 0.444445022 0.444443288 N 0.440565836 0.780634588 N 0.780636832 0.112130449 N 0.780641553 0.440560091 N 0.777777476 0.777778380 Ga 0.004241005 0.004241842 Ga 0.324850486 0.004241842 Ga 0.663157208 0.007018917 Ga 0.004245439 0.324842456 Ga 0.007016466 0.663156320 Ga 0.334324079 0.334331197 Ga 0.334328782 0.664675769 Ga 0.664678056 0.334331197 Ga 0.663160546 0.663156320 Ga 0.444450615 0.444432101 End final coordinates
0.541058043 0.541565945 0.541106229 0.541105964 0.541106229 0.541565945 0.612005524 0.612005524 0.613986759 0.612005382 0.613988014 0.612077658 0.612077474 0.612077658 0.613988014 0.636278844 0.636394193 0.636393027 0.636392536 0.636519979 0.636393027 0.636392536 0.636394193 0.633122184 0.705856702 0.708782678 0.708782725 0.708782309 0.705011187 0.708782725 0.708782309 0.708782678 0.710581350 0.737634176 0.737634176 0.719713862 0.737649294 0.719711130 0.737080090 0.737082917 0.737080090 0.719711130 0.797782997
S13
References 1. Rosa, A. L.; Neugebauer, J. First-Principles Calculations of the Structural and Electronic Properties of Clean GaN(0001) Surfaces. Phys. Rev. B: Condens. Matter Mater. Phys. 2006, 73, 205346. 2. Henkelman, G.; Arnaldsson, A.; Jónsson, H. A Fast and Robust Algorithm for Bader Decomposition of Charge Density. Comput. Mater. Sci. 2006, 36, 254-360. 3. Sanville, E.; Kenny, S. D.; Smith, R.; Henkelman, G. An Improved Grid-Based Algorithm for Bader Charge Allocation. J. Comp. Chem. 2007, 28, 899-908. 4. Silverstrelli, P. L.; Ambrosetti, A.; Grubisiĉ, S.; Ancilotto, F. Adsorption of Rare-Gas Atoms on Cu(111) and Pb(111) Surfaces by van der Waals Corrected Density Functional Theory. Phys. Rev. B. 2012, 85, 165405.
S14
