RESULT AND DISCUSSION

4-1 Magnesium Silicide (Mg

2

Si)

4-1-1 Paper work

Following the procedures in Ref. 9 for Mg2Si in which B3LYP method and 6-311G basis set are used. We have now added the other another three basis sets ( 6-311G (d), 6-311+G and 6-311+G (d) ) in comparison with 6-311G, respectively.

The total electronic energy is obtained at various electric fields: 0.00, 0.01, 0.02, 0.03, 0.04, 0.05 and 0.06 atomic unit ( a.u. ). Then energies are fit with respect to electric fields for each method and each basis set. (See Table 10)

Table 10: Energies (E) are fit with electric fields (F) and polarizability volumes are obtained from deriving the fitting equations at the B3LYP level.

Mg2Si (B3LYP) Fitting equation ^αtot (Hartree/a.u.²) αtot(cm3)

6-311G E = -88.245F² - 1.1318F - 689.63 177.6218 2.63E-23

6-311G(d) E = -87.955F² - 1.0027F - 689.64 176.9127 2.62E-23

6-311+G E = -177.52F² + 0.2477F - 689.63 354.7923 5.26E-23

6-311+G(d) E = -176.01F² + 0.2143F - 689.64 351.8057 5.21E-23

Firstly, the fitting equation (E = -88.245F² - 1.1318F - 689.63) is taken as an illustration in the Table 10, due to the definition of dipole moment ^E

µ_{= −⎜}^⎛∂F

⎝∂ ⎠

⎞⎟, the

equation can be written as µ=176.49F+1.1318. When F=1 (a.u.), µ=177.6218 (Hartree/a.u.). Then the polarizability α is assumed as being in the unit electric field, α=177.6218 (Hartree/a.u.²). Finally we must convert unit (Hartree/a.u.²) of the polarizability to polarizability volume (cm³), See Table 10. Because the polarizability is from total energy, the polarizability volume is called total polarizability volume (αtot ).

Secondly, dipole moments are obtained at various electric fields (0 ~0.06 a.u.), and are fitted with fields, see Table 11. Because of

F α = ^∂µ

∂ and the base of a unit electric field, distortion polarizability (αd) is obtained easily.

Table 11: Dipole moments (μ) are fit with fields (F) and distortion polarizability volumes are obtained from deriving fitting equations at the B3LYP level.

Mg2Si(B3LYP) Fitting equation αd(debye/a.u.) αd(cm3)

6-311G μ =- 102.46F² +453.33F + 2.8494 248.41 1.45E-23

6-311G(d) μ = 15.238F² + 445.17F + 2.5882 475.646 2.77E-23

6-311+G μ = 1887.5F² + 758.3F + 1.5688 4533.3 2.64E-22

6-311+G(d) μ = 1371.2F² + 782.54F + 1.28 3524.94 2.06E-22

4-1-2 Correction for electric field and polarizability

From Table 11, it can be clearly seen that the distortion polarizability is sensitive to the choice of the basis sets. In order to find out why distortion polarizability is so sensitive to the basis sets, we compare total electronic energy at a varying field with ionization energy from a minus of electronic HOMO energy without electric field in Table 12.

Table 12: In B3LYP/6-311+G (d), total electronic energy with a varying electric field and the absolute value of the energy difference between electric fields: one exists the field and the other dose not. It also shows the HOMO, LUMO energy and ionization energy without electric field.

B3LYP/6-311+G(d)

electric field (z-axis, a.u.)

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0

Ee -690.2525 -690.0270 -689.8504 -689.7385 -689.6616 -689.6376 -689.6385

| Ee-Ee(F=0) | 0.6140 0.3885 0.2118 0.1000 0.0230 0.0009 0.0000

HOMO -0.1537

LUMO -0.0982

ionization energy 0.1537

B3LYP/6-311+G(d)

electric field (z-axis, a.u.) 0.01 0.02 0.03 0.04 0.05 0.06

Ee -689.6586 -689.7014 -689.7887 -689.9164 -690.0692 -690.2588

| Ee-Ee(F=0) | 0.0200 0.0629 0.1502 0.2778 0.4306 0.6203

HOMO

LUMO

ionization energy

In Table 12, it shows that when the electric field is outside [-0.04, 0.04] a.u., the absolute value of energy difference between electric fields: one exists the field and the other does not is larger than ionization energy (0.1537 hartree). It also means that the field strength outside of [-0.04, 0.04] a.u. is too large, and then leads the electrons to ionize and electronic structure is changed severely. But in the present work, we focus on the variation of energy and dipole moment in the smaller electric field. So we must choose smaller electric fields. Moreover, we make a plot of dipole moments against fields from -0.06 to 0.06 a.u. in Fig.15 with four basis sets.

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08

Fig. 17 Fitting plot of dipole moment versus electric field by four basis sets in B3LYP level

In Fig. 17, all four fitting lines coincide when electric field is in between -0.01 and 0.01 a.u. and are dispersive when field is outside of [-0.01, 0.01] a.u.. This is due to effect of the stronger field, and results in obtaining dispersive fitting lines and different distortion polarizability by four basis sets in Table 11. When the field strength is larger, the convergence of the computation is out of control by the perturbation method, and also the molecule is going to undergo multiple ionizations at that strong field. Energy calculation is meaningless at |F| > 0.01 a.u.. On the other hand, we want to obtain the convergent polarizability with different basis sets, so the range of the field from -0.01 to 0.01 a.u. is chosen in the following investigations. The results of the renewed fitting are listed in Table 13.

Table 13: Dipole moments (µ) are fit with electric field (F) and distortion polarizability volume. Fitting field is from -0.01 to 0.01 a.u..

Mg2Si (B3LYP) Fitting equation αd (debye/a.u.) αd(cm3)

6-311G μ = 595.77F² - 454.13F - 2.8765 737.41 4.30E-23

6-311G(d) μ = 628.68F² - 449.92F - 2.6034 807.44 4.71E-23

6-311+G μ = 456.99F² - 489.37F - 2.9321 424.61 2.48E-23

6-311+G(d) μ = 454.91F² - 491.02F - 2.6724 418.8 2.44E-23

In Table 13, it is quite obvious that distortion polarizability volume is still sensitive for using different basis set, especially for additional diffuse functions. In order to decrease the inconsistency in polarizability, we check that previously mentioned about the base of one a.u. of the electric field again. It is found that total energy from the Taylor expansion at F = 0 is more reasonable than at F=1 following the published paper to evaluate. (See eq. (4.1) and (4.2) )

2 3

2 3

Now, total polarizability from the fitting of total energy and electric field and distortion polarizability from the fitting of dipole moment and field are evaluated in Table 14 and 15 again.

Table 14: Energy (E) is fit with fields (F) and derives polarizability. Fitting field is from -0.01 to 0.01 a.u. and polarizability is evaluated from Taylor expansion at F=0.

Mg2Si (B3LYP) Fitting equation αtot (Hartree/a.u.²) αtot(cm3)

6-311G E = -89.259F² - 1.1259F - 689.63 178.518 2.65E-23

6-311G(d) E = -88.426F² - 1.0182F - 689.64 176.852 2.62E-23

6-311+G E = -95.367F² - 1.1492F - 689.63 190.734 2.83E-23

6-311+G(d) E = -95.613F² - 1.047F - 689.64 191.226 2.83E-23

Table 15: Dipole moments ( μ ) are fit with fields and derive distortion polarizability. Fitting field is from -0.01 to 0.01 a.u. and polarizability is evaluated from Taylor expansion at F=0.

The total polarizability αtot is the sum of the distortion polarizability αd and orientation polarizability αo, but by comparing Table 14 with Table 15 it shows that the polarizabilities from the fitting of energy or dipole moment is the same.

Fitting results of dipole moment cannot be taken as distortion polarizability directly, because dipole moment in the calculation is derived from the quantum mechanical dipole operator.

ˆ _{i A}

i A

r Z

µ = −

∑ ∑

+ R_A (4.5)

ri and RA are the vectors of the electrons and nucleus respectively, ZA represents nuclear charges.

The mean value of dipole moment is evaluated in eq. (4.6) directly.

It shows clearly that the dipole moment as the function of electric field should represent total dipole moment including electron part and nucleus part. From Table 14 and 15 we can see that polarizability from the fitting results of the dipole moment and the electric field should be assumed as total polarizability, is the same as the result of the fitting from the total energy and the electric field.

4-1-3 Dielectric constant for solid Mg

2

Si

For solid Mg2Si, there are no translational and rotational motions in degrees of the freedom, so orientation polarizability has no contribution to total polarizability (αo = 0). Now, we must remove orientation polarizability from total polarizability.

Because of the effect from Boltzmann distribution, the orientation polarizability can be written as

Applied electric field is 0.01 a.u. and temperature is 300K in our case. When B3LYP, with 6-311G basis set are used, the permanent dipole moment is 2.8765 Debye. But µFloc ( = 4.93E-20 J ) is greater than kT ( = 4.14E-21 J ) in fact, so Boltzmann distribution isn’t used in obtaining distortion polarizability.

We would like to understand is how to calculate the distortion polarizability directly. Because distortion polarizability is the sum of the electronic polarizability

and atomic polarizability, we can obtain electronic part from charge population directly. Moreover, for atomic polarizability, because it is about 5 % ～ 10 % of the electronic polarizability, it can be almost neglected.

Starting from charge population, we adopt the Mulliken partial charge and APT partial charge to derive the Mulliken and APT dipole moment. Firstly, there is an optimized structure of Mg2Si at F = 0 in Fig. 18.

Fig. 18 The optimized structure has two bond length “r1” and “r2”, angle “θ” and the partial charge of Mg “q1” and “q2”. Here are r1 = r2

= r and q1 = q2 = q.

From Fig. 18 the dipole moment is derived in eq. (4.8)

1 1 2 2

(

1 2

)

2 cos

2

i i i

q r q r q r q r r q r

θ

µ

=

∑

= ^K + ^K = ^{K K}+ = ⎜^⎛⎝ ^⎞⎠^⎟ (4.8)

Here r1, r2 and θ is calculated at electric field equal to zero and q is APT charge or Mulliken charge calculated with varying electric field. Because when perturbation of electric field is added, the output of the Mg2Si geometry from G03 calculation isn’t changed, only partial charges are changed. Therefore, we think that

the APT dipole and Mulliken dipole is only from electronic part. So we call the polarizability from APT and Mulliken dipole as electronic polarizability.

Mulliken and APT charge are chosen respectively and we can obtain two kinds of dipole moment, Mulliken dipole and APT dipole. Now, we compare all dipoles : dipole moment (direct output from G03), APT dipole, Mulliken dipole without electric field in Table 16. It is very clear that APT dipole is contributed to electronic polarizability very well, because the APT dipole is much smaller than dipole moment. But Mulliken dipole is larger than dipole moment at 6-311G, 6-311+G(2d), 6-311+G(3df), 6-311+G(3d2f) and LANL2DZ, it cannot represent electronic part well at those basis sets.

Table 16: The comparison of dipole moment, APT dipole and Mulliken dipole without electric field at the B3LYP level.

no electric field (B3LYP) Dipole moment APT dipole Mulliken dipole

6-311G 2.8765 0.159514 3.373304

6-311G(d) 2.6034 0.185058 1.660079

6-311+G 2.9311 0.182463 2.419225

6-311+G(d) 2.6716 0.243587 0.93931

6-311+G(2d) 2.5595 0.185182 3.0296904

6-311+G(2df) 2.5452 0.247732 1.1804818

6-311+G(3df) 2.5357 0.209746 3.1015206

6-311+G(3d2f) 2.5376 0.194249 3.1624394

LANL2DZ 2.9674 0.044939 3.7838176

CEP-31G 2.8633 0.524192 1.348365

CEP-121G 2.9241 0.56176 1.900735

Then it is as the same as the previous fitting of dipole moment and electric field, polarizability can be obtained easily. Because two kinds of dipole contribute to electronic part only, the obtained polarizability is called as the electronic

Table 17: Mulliken dipole is fit with electric field (from -0.01 to 0.01 a.u.) and derives electronic polarizability at the B3LYP level.

Mulliken dipole Fitting equation αe(debye/a.u.) αe (cm3)

6-311G μ = 797.56F² + 215.77F + 3.3734 215.77 1.26E-23

6-311G(d) μ = 709.74F² + 222.06F + 1.6602 222.06 1.29E-23

6-311+G μ = 801.95F² + 168.16F + 2.4304 168.16 9.80E-24

6-311+G(d) μ = 336.26F² + 170.61F + 0.9527 170.61 9.95E-24

6-311+G(2d) μ = 639.61F² + 190.57F + 3.0422 190.57 1.11E-23

6-311+G(2df) μ = 555.59F² + 196.23F + 1.1929 196.23 1.14E-23

6-311+G(3df) μ = -1606.9F² + 167.54F + 3.123 167.54 9.77E-24

6-311+G(3d2f) μ = -1404.5F² + 172.79F + 3.1857 172.79 1.01E-23

LANL2DZ μ = 357.76F² + 164.93F+ 3.7838 164.93 9.62E-24

CEP-31G μ = 7.4405F² + 185.44F + 1.3484 185.44 1.08E-23

CEP-121G μ = 23.877F² + 204.23F + 1.9008 204.23 1.19E-23

Table 18: APT dipole is fit with electric field (from -0.01 to 0.01 a.u.) and derives electronic polarizability at the B3LYP level.

APT dipole Fitting equation αe (debye/a.u.) αe (cm3)

6-311G μ = 245.14F² + 189.49F + 0.1592 189.49 1.10E-23

6-311G(d) μ = 389.28F² + 177.78F + 0.1849 177.78 1.04E-23

6-311+G μ = 1017.8F² + 205.41F + 0.1774 205.41 1.20E-23

6-311+G(d) μ = 1580.7F² + 193.93F + 0.2377 193.93 1.13E-23

6-311+G(2d) μ = 1766F² + 189.15F + 0.1798 189.15 1.10E-23

6-311+G(2df) μ = 1747.2F² + 187.65F + 0.2423 187.65 1.09E-23

6-311+G(3df) μ = 2012.7F² + 185.85F + 0.2041 185.85 1.08E-23

6-311+G(3d2f) μ = 1998.7F² + 185.94F + 0.1885 185.94 1.08E-23

LANL2DZ μ = -1132.8F² + 148.82F - 0.0451 148.82 8.68E-24

CEP-31G μ = -454.06F² + 171.9F - 0.5254 171.9 1.00E-23

CEP-121G μ = -643.99F² + 200.23F - 0.563 200.23 1.17E-23

Compare Table 17 to Table 18, it shows that the spread of electronic polarizability from Mulliken dipole is more dispersive than APT dipole by Pople’s basis sets and effective core potential, because Mulliken charge is sensitive to the choice of basis sets and it usually is reasonable as small- or moderate-basis sets are used. So if we adopt the polarizability from APT dipole to calculate the dielectric constant, the larger basis sets should be used. And if we adopt the polarizability from Mulliken dipole to calculate the dielectric constant, the smaller basis sets should be used.

By applying Clausius-Mossotti equation we can combine the macroscopic amounts of dielectric constant with the microscopic amounts of polarizability.

r

Dielectric constant is obtained in eq. (4.10) by rearranging eq. (4.9). ρ is the density and M is the molecular weight.

r

Dielectric constants with respect to different basis sets are calculated and summarized in Table 19 and 20. The density is 1.99 g/cm³ and Molecular weight is 76.6955 g/mol for Mg2Si which are taken from experiment results.

Table 19: Use APT charges to obtain electronic polarizability and dielectric constant at the B3LYP level.

APT αe (debye/a.u.) αe(cm3) dielectric constant

6-311G 189.49 1.10E-23 8.82

6-311G(d) 177.78 1.04E-23 7.32

6-311+G 205.41 1.20E-23 11.86

6-311+G(d) 193.93 1.13E-23 9.53

6-311+G(2d) 189.15 1.10E-23 8.77

6-311+G(2df) 187.65 1.09E-23 8.56

6-311+G(3df) 185.85 1.08E-23 8.31

6-311+G(3d2f) 185.94 1.08E-23 8.32

LANL2DZ 148.82 8.68E-24 4.94

CEP-31G 171.9 1.00E-23 6.71

CEP-121G 200.23 1.17E-23 10.70

Table 20: Use Mulliken charges to obtain electronic polarizability and dielectric constant at the B3LYP level.

Mulliken αe (debye/a.u.) αe(cm3) dielectric constant

6-311G 215.77 1.26E-23 14.95

6-311G(d) 222.06 1.29E-23 17.61

6-311+G 168.16 9.80E-24 6.37

6-311+G(d) 170.61 9.95E-24 6.59

6-311+G(2d) 190.57 1.11E-23 8.99

6-311+G(2df) 196.23 1.14E-23 9.93

6-311+G(3df) 167.54 9.77E-24 6.31

6-311+G(3d2f) 172.79 1.01E-23 6.80

LANL2DZ 164.93 9.62E-24 6.09

CEP-31G 185.44 1.08E-23 8.25

CEP-121G 204.23 1.19E-23 11.58

The high frequency dielectric constant ε∞ is 13.3* by experiment. The contribution of the high frequency dielectric constant is totally from electronic polarizability. From Table 19 no matter how to choose Pople’s basis sets, the obtained dielectric constants are close to the experimental value when the larger basis sets are used ( 6-311+G, 6-311+G(d)…etc ), and for effective core potential it shows the good result only at a larger basis set ( CEP-121G ), but LANL2DZ is unavailable here.

However, in Table 20 for Pople basis sets it shows good results corresponding to experimental value only when the smaller basis sets ( 6-311G, 6-311G(d) ) are used.

For ECP, it also shows that the available values are from CEP-series basis sets, but for LANL2DZ it shows a bad result. The results of Table 19 and 20 are vey correspondent to the theory mentioned earlier (in 2-2 charge population). The theory shows below:

1. Mulliken population analysis is affected largely by basis functions, and it usually is most useful for comparing trends in charge distributions, when small- or medium-size basis sets are used.

2. APT charge isn’t directly related to the choice of a particular basis set, its basis set dependence stems only from the fact that the basis set can be incomplete. So the basis-set dependence is modest

4-1-4 Other methods (BLYP, M05 and M05-2X)

Now, we try other methods: BLYP, M05 and M05-2X. The process is the same as the previous B3LYP level.

In Table 21, for the Pople’s basis set, we show that the dielectric constant obtained from APT charge is in good agreement with experimental value (ε^∞ = 13.3 ) as the basis set is large enough ( see larger bold-faced words ) at the BLYP level. And for effective core potential it also shows that the dielectric constant is in good agreement with experimental value at a larger basis set (CEP-121G).

Table 21: Use APT charges to obtain electronic polarizability and dielectric constant at BLYP level.

APT (BLYP) αe (Debye / a.u. ) αe (cm3 ) dielectric constant

6-311G 189.97 1.11E-23 8.89

6-311G(d) 181.66 1.06E-23 7.77

6-311+G 209.81 1.22E-23 13.03

6-311+G(2df) 197.76 1.15E-23 10.21

6-311+G(3d2f) 196.33 1.14E-23 9.95

LANL2DZ 138.64 8.08E-24 4.37

CEP-31G 163.09 9.51E-24 5.94

CEP-121G 184.83 1.08E-23 8.17

In Table 22 for Pople basis sets, the dielectric constant is fit well for a small basis set. However, for 6-311+G (2df) the dielectric constant is a good fit ( see asterisk ), it is contradictory to the unexpected results of Mulliken charge at a larger basis set. So we try the largest basis set ( 6-311+G (3d2f) ) in order to check this contradiction. However, the dielectric constant at 6-311+G (3d2f) isn’t a good fit.

So we consider that it is a coincidence for the good result of 6-311+G (2df). And for ECP, it corresponds to the experimental value only at a larger basis set (CEP-121G).

Table 22: Use Mulliken charges to obtain electronic polarizability and dielectric constant at the BLYP level.

Mulliken (BLYP) αe (Debye / a.u. ) αe (cm3 ) dielectric constant

6-311G 217.72 1.27E-23 15.70

6-311G(d) 222.99 1.30E-23 18.08

6-311+G 176.44 1.03E-23 7.17

6-311+G(2df) 202.41 1.18E-23 11.16*

6-311+G(3d2f) 184.22 1.07E-23 8.09

LANL2DZ 162.86 9.49E-24 5.92

CEP-31G 185.26 1.08E-23 8.23

CEP-121G 203.11 1.18E-23 11.32

In Table 23, for Pople basis sets it shows a good fit only at 6-311G and 6-311+G, but at a larger basis set the dielectric constant is much far from the experimental value. So Pople basis sets in the M05 level are unavailable for the APT derivation. There is no similar trend like B3LYP or BLYP in the M05 level.

For the new M05 method it doesn’t include enough parameters, so it isn’t suited to use in our case. But for ECP, the dielectric constant is correspondent to the experimental value at larger basis sets ( CEP-31G and CEP-121G ).

Table 23: Use APT charges to obtain electronic polarizability and dielectric constant at the M05 level.

APT (M05) αd (Debye / a.u. ) αd (cm3 ) dielectric constant

6-311G 191.1129 1.1142E-23 9.0701*

6-311G(d) 174.7550 1.0188E-23 6.9982

6-311+G 192.6017 1.1229E-23 9.3070*

6-311+G(2df) 56.7148 3.3065E-24 1.8282

6-311+G(3d2f) 125.7389 7.3306E-24 3.7651

LANL2DZ 1274.8705 7.4325E-23 4.7766

CEP-31G 208.4698 1.2154E-23 12.6489

CEP-121G 200.1852 1.1671E-23 10.6906

In Table 24 for Pople basis sets, the dielectric constant is still not correspondent to experimental value, so those results are not also suited to analyses in our case. For ECP, it shows a good results only at CEP-31G..

Table 24: Use Mulliken charges to obtain electronic polarizability and dielectric constant at the M05 level.

Mulliken (M05) αe (Debye / a.u. ) αe (cm3 ) dielectric constant

6-311G 235.0190 1.3702E-23 26.9794

6-311G(d) 238.2464 1.3890E-23 30.8908

6-311+G 171.2971 9.9866E-24 6.6558

6-311+G(2df) 250.8102 1.4622E-23 67.3060

6-311+G(3d2f) 143.2398 8.3509E-24 4.6136

LANL2DZ 185.2729 1.0801E-23 8.2292

CEP-31G 208.2056 1.2138E-23 12.5772

CEP-121G 233.1315 1.3592E-23 25.0950

In Table 25 and 26 based on the M052X level, no matter APT charge or Mulliken charge both show bad results by Pople basis sets and ECP, although for APT charge the dielectric constant is similar at each basis set. Those bad results of M052X are very reasonable, because the new method – M052X is parameterized only for non-metals. But Mg atom is metal.

Table 25: Use APT charges to obtain electronic polarizability and dielectric constant at the M052X level.

APT (M052X) αe (Debye / a.u. ) αe (cm3 ) dielectric constant

6-311G 163.5600 9.5355E-24 5.9766

6-311G(d) 175.3025 1.0220E-23 7.0550

6-311+G 171.4508 9.9956E-24 6.6705

6-311+G(2df) 181.7957 1.0599E-23 7.7866

6-311+G(3d2f) 178.4675 1.0405E-23 7.3974

LANL2DZ 250.5412 1.4607E-23 65.7015

CEP-31G 129.6915 7.5610E-24 3.9372

CEP-121G 7.4667 4.3531E-25 1.0880

Table 26: Use Mulliken charges to obtain electronic polarizability and dielectric constant at the M052X level.

Mulliken (M052X) αe (Debye / a.u. ) αe (cm3 ) dielectric constant

6-311G 224.2322 1.3073E-23 18.7371

6-311G(d) 229.1981 1.3362E-23 21.8616

6-311+G 178.5347 1.0409E-23 7.4050

6-311+G(2df) 194.3835 1.1333E-23 9.6043

6-311+G(3d2f) 150.9721 8.8017E-24 5.0735

LANL2DZ 166.0045 9.6781E-24 6.1794

CEP-31G 195.0367 1.1371E-23 9.7172

CEP-121G 219.9386 1.2822E-23 16.6282

Briefly, For Pople basis sets, there are good results from APT dipole by larger basis sets in the B3LYP and BLYP levels, and there are good results from Mulliken dipole by small basis sets in the B3LYP and BLYP levels. For ECP, no matter APT or Mulliken partial charge, there are good results by CEP-series basis sets at the B3LYP, BLYP and M05 levels .But for M052X it isn’t suited to calculate in our case ,due to the fact that M052X is parameterized only for non-metals.

We conclude that in the present calculation for dielectric constant that M05 and M05-2X are not really better functionals than their original forms of B3LYP and

4-1-5 Seebeck coefficient

Because the definition of Seebeck coefficient is

e T 0 Se is Seebeck coefficient, Vp is potential and T is absolute temperature.

From thermodynamic relation

If we consider electric work, Helmholtz free energy can be written as

dF = - PdV -SdT + V dq

_p (4.12) In order to obtain the Seebeck coefficient, we combine the relation of the eq. (4.13) with the definition of the eq. (4.11). Then it leads to

V ,T 6-311+G basis set at the B3LYP level to calculate Seebeck coefficient. In first step, Helmholtz free energy at zero electric field is taken, and its unit is changed from

atomic unit to electron volt. Because we want to obtain the relation of the potential and the temperature, the electron charge would be based on one atomic unit.

Therefore Helmholtz free energy with unit of electron volt is divided by unit electron charge (See Table 27). Then the obtained result is fit with the temperature from 300K to 800K. (See eq. 4.15)

Table 27: Helmholtz free energy from B3LYP/6-311+G and the potential at one unit electron charge (V).

B3LYP/6-311+G 300K 400K 500K 600K 700K 800K

Helmholtz energy (Hartree) -689.656798 -689.669068 -689.681888 -689.695150 -689.7087777 -689.7227217

Helmholtz energy (eV) -18766.7160 -18767.0499 -18767.3988 -18767.7596 -18768.13048 -18768.50992

potential (at unit charge,V) -18766.7160 -18767.0499 -18767.3988 -18767.7596 -18768.13048 -18768.50992

The fitting equation via potential and temperature from Table 27 is listed below.

V = 6 10 T + 0.003T + 18766_p × ^{-07 2} (4.15) represented as gas phase. But the Mg2Si material is a polar solid so that the electric potential inside this dielectric should be divided by the calculated dielectric constant of 11.86 from gas phase⁹, and then actually the Seebeck coefficient of Solid state is

given as

S = 1 10 T + 2.52 10_e × ^-7 × ^-4 (4.17)

In Fig. 19 it shows the fitting profiles from solid-state, gas-phase and Material studio 4.0 of Mg2Si and in Fig. 18³² it shows the experimental results.

Compare Fig. 19 to Fig. 20, we find that our solid-state case agrees with experimental results of non-BN coated, and also find our fitting result is linear even at higher temperature. But in Fig. 20 the Seebeck coefficient of the BN coated fitting decreases as temperature increasing, which may originate in the effect thermal excitation of carriers across the gap from the conduction band³³. However, in our case because we don’t consider energy band structure theory but directly based on thermodynamic method, it doesn’t show that Seebeck coefficient decrease at high temperature.

Fig. 19 The fitting profiles from solid-state, gas phase and Material studio 4.0

Fig. 20. Seebeck coefficient of the grown ingot over the range is from 345K to 840 K³². BN coated means encapsulated sample with a boron nitride (BN)-based anti-adhesion coating.

A summary of Seebeck coefficient is listed in Table 28.

Table 28: Seebeck coefficients are obtained from gas phase, solid state and Material studio 4.0⁹ respectively, and compare to experiment value.

B3LYP/6-311+G fitting equation Se (µV/K) 300K Se (µV/K) 800K

gas phase S_e = 1.2*E-06 T + 0.003 3360 3960

solid state S_e = 1*E-07 T +2.52*E-04 284 334

material studio 4.0^{9 S}e = 6*E-06 T + 0.0018 3600 6600

experimental value³² 180 280

In Table 28 it shows clearly that Seebeck coefficient agrees with experiment value by our treatment of solid state very well. But for Material studio 4.0 based on the energy band structure theory, the Seebeck coefficient that is similar to our gas phase case isn’t fitted to experimental value. So we can conveniently obtain more accurate Seebeck coefficient from the thermodynamic method with the present new scheme.

4-2 Iron Disilicide (FeSi

₂

)

4-2-1 Polarizability and dielectric constant

Because Fe atom is a transition metal which belongs to the system of the heavy atoms, we would consider the relativistic effect for FeSi2 molecule, and then we adopt the effective core potential (ECP). Here, LANL2DZ, CEP-31G and CEP-121G basis sets and 6-311+G(2d), 6-311+G(2df), 6-311+G(3df) and 6-311+G(3d2f) are used, besides 6-311G, 6-311G(d), 6-311+G and 6-311+G(d) basis sets. Furthermore, we try other methods such as BLYP, M05 and M05-2X, in order to compare with B3LYP mentioned previously.

There are polarizabilities and dielectric constants calculated with different basis sets from APT charge at B3LYP level in Table 29. The experimental value of high-frequency dielectric constant is 27.6³⁴. We compare this experimental value with Table 29. Basis sets are in two parts: one is Pople style basis set, and the other is effective core potential (ECP). The dielectric constants from Pople style basis sets are in good agreement with experimental value, when basis sets are larger ( 6-311+G ~ 6-311+G (3d2f) ). But when basis sets aren’t enough large, the dielectric constants aren’t good fits ( 6-311G and 6-311G(d) ). Those results of the derivation of APT charge match the phenomenon that the basis-set dependence is modest for these APT charges, and those APT charges usually converge at the larger basis set. Expectably, for Pople basis sets of APT part, the dielectric constants of FeSi2 show the same trend as of Mg2Si at the B3LYP level. The second part of

There are polarizabilities and dielectric constants calculated with different basis sets from APT charge at B3LYP level in Table 29. The experimental value of high-frequency dielectric constant is 27.6³⁴. We compare this experimental value with Table 29. Basis sets are in two parts: one is Pople style basis set, and the other is effective core potential (ECP). The dielectric constants from Pople style basis sets are in good agreement with experimental value, when basis sets are larger ( 6-311+G ~ 6-311+G (3d2f) ). But when basis sets aren’t enough large, the dielectric constants aren’t good fits ( 6-311G and 6-311G(d) ). Those results of the derivation of APT charge match the phenomenon that the basis-set dependence is modest for these APT charges, and those APT charges usually converge at the larger basis set. Expectably, for Pople basis sets of APT part, the dielectric constants of FeSi2 show the same trend as of Mg2Si at the B3LYP level. The second part of

在文檔中 利用熱力學方法探討半導體材料的熱電效應 (頁 63-125)