Martienssen W., Warlimont H. (Eds.). Handbook of Condensed Matter and Materials Data

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582 Part 4 Functional Materials

SiC

4.384

4.380

4.376

4.372

4.368

4.364

4.360

4.356

a (Å)

T (K)

0 400 800 1200 1600

Fig. 4.1-8 SiC (3C). Lattice parameter vs. tempera-

ture [1.8]

Table 4.1-7 Linear thermal expansion coefﬁcient α of Group IV semiconductors and IV–IV compounds and its

temperature dependence

Crystal Expansion coefﬁcient α (K

−1

) Temperature (K) Remarks

Diamond C 1.0×10

−6

300 See Fig. 4.1-5 for temperature dependence

Silicon Si 2.92 (6) ×10

−6

293 See Fig. 4.1-10 for temperature dependence

Germanium Ge 5.90× 10

−6

300 See Fig. 4.1-11 for temperature dependence

Gray tin α-Sn 4.7×10

−6

293 See Fig. 4.1-12 for temperature dependence

Silicon carbide 3C-SiC 2.77 (42) ×10

−6

300 See Fig. 4.1-13 for temperature dependence

The linear thermal expansion coefﬁcient α of Si can be approximated in the temperature range 120–1500 K by

α(T ) =



3.725



1−exp[−5.88 × 10

−3

(T −124)]



+5.548× 10

−4



×10

−6

−1

) (T measured in K)

SiC

4.9085

4.9070

4.9055

a (Å)

T (K)

0 400 800 1200 1600

15.215

15.200

15.185

15.170

15.155

15.140

15.125

15.110

3.100

3.085

3.070

c (Å)

c/a

Fig. 4.1-9 SiC (6H). Lattice parameters vs. tempera-

ture [1.8]

Part 4 1.1

Semiconductors 1.1 Group IV Semiconductors and IV–IV Compounds 583

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

– 0.5

0 200 400 600 800 1000 1200 1400 1600

T (K)

α (10

–6

–1

)

Fig. 4.1-10 Si. Linear thermal expansion coefﬁcient vs.

temperature. Experimental data from [1.9] (ﬁlled circles)

and [1.10] (open circles)

–2

0 40 80 120 160 200 240

T (K)

α (10

–6

–1

)

α-Sn

Fig. 4.1-12 α-Sn. Coefﬁcient of linear thermal expansion

vs. temperature [1.12,13]

α (10

–6

–1

)

–1

50 100 150 200 250 300

T (K)

0.2

0.1

–0.1

T (K)

α (10

–6

–1

)

Fig. 4.1-11 Ge. Linear thermal expansion coefﬁcient vs.

temperature. Experimental data from various authors (sym-

bols) and theoretical results (solid line) [1.11]

–2

T (K)

α (10

–6

–1

)

0 500 1000 1500 2000 2500 3000

3C-SiC

Fig. 4.1-13 3C-SiC. Recommended linear thermal expan-

sion coefﬁcient vs. temperature according to an analysis of

data from nine references [1.5]

Part 4 1.1

584 Part 4 Functional Materials

0369121518

1.0

0.8

0.6

0.4

0.2

0.0

T (K)

(cal/K

g-atom)

(cal/K

g-atom)

T (K)

0 20406080100

α-Sn

Table 4.1-8 Heat capacities and Debye temperatures of Group IV semiconductors and IV–IV compounds

Crystal Heat capacity c

Debye temperature Θ

Temperature (K) Remarks

(J/mol K)

Diamond C 6.195 1860 (10) 300 See Fig. 4.1-14 for temperature

dependence of c

, c

Silicon Si 45.358 636 77 See Fig. 4.1-15 for temperature

45.882 298 dependence of c

Germanium Ge 51.88 374 298 See Fig. 4.1-16 for temperature

dependence of c

Gray tin α-Sn 19.09 220 100 See Fig. 4.1-17 for temperature

dependence of c

Silicon carbide 3C-SiC 28.5 1270 (20) 293 Polycrystalline sample

T (K)

5.5

5.0

4.5

4.0

3.5

3.0

2.5

2.0

1.5

1.0

, c

(cal/mol K)

900 1000 1100300 400 500 600 700 800

Diamond

Fig. 4.1-14 Diamond. Molar heat capacity vs. tempera-

ture [1.14]

(J/mol K)

100

500

T (K)

200 300 4000

02550

2.5

5.0

T (K)

(J/mol K)

Fig. 4.1-15 Si. Heat capacity at constant volume vs. tem-

perature. Figure from [1.11]

(J/mol K)

100

500

T (K)

200 300 4000

050

5.0

T (K)

(J/mol K)

Fig. 4.1-16 Ge. Heat capacity at constant volume vs. tem-

perature

Fig. 4.1-17 α-Sn. Heat capacity c

vs. temperature. Lower

curve, temperature range 0–20 K; upper curve, temperature

range 0–105 K [1.15]

Part 4 1.1

Semiconductors 1.1 Group IV Semiconductors and IV–IV Compounds 585

Table 4.1-9 Phonon frequencies at symmetry points. Diamond (C) (ν in THz, 300 K, from Raman spectroscopy); silicon

(Si) (ν in THz, 296 K, from inelastic neutron scattering); germanium (Ge) (ν in THz, 300 K, from coherent inelastic

neutron scattering); gray tin (α-Sn) (ν in THz, 90 K, from inelastic thermal neutron scattering); silicon carbide (3C-SiC)

(phonon wavenumbers

ν in cm

−1

, RT, from Raman spectroscopy); silicon carbide (6H-SiC) (ν in THz, derived from

photoluminescence data)

Diamond (C) Silicon (Si) Germanium (Ge) Gray tin (Sn) Silicon carbide (3C-SiC)

TO/LO

(Γ



) 39.9 ν

LTO

(Γ



) 15.53 (23) ν

LTO

(Γ



) 9.02 (2) ν

TO/LO

(Γ



) 6.00 (6)

(L) 266

)16.9 ν

)4.49 (6) ν

)2.38 (2) ν

) 1.00 (4)

(L) 610

)30.2 ν

LAO

)12.32(20) ν

LAO

)7.14 (2) ν

)4.15 (4)

(L) 766



)37.5 ν

)13.90 (30) ν

)8.17 (3) ν



)4.89 (8)

(L) 838



) 36.2 ν

) 3.43 (5) ν



)1.87 (2) ν



)5.74 (12)

(X) 373

) 24.2 ν

) 11.35 (30) ν

)6.63 (4) ν

)1.25 (6)

(X) 640

LA/LO

) 35.5 ν

) 12.60 (32) ν

)8.55 (3) ν

LA/LO

)4.67 (6)

(X) 761

) 32.0 ν

) 14.68 (30) ν

)7.27 (2) ν

)5.51 (8)

(X) 829

Table 4.1-9 Phonon frequencies at symmetry points, cont.

Silicon carbide (6H-SiC)

8.1 8.8 9.5 9.8 10.6 11.2 12.9

12.2 12.9 16.2 16.7 18.6

22.9 23.1 23.7

25.2 25.5 25.9

ν (THz)

0.51.0 1.0 00.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0.20.40.60.8 0.10 0.2 0.3 0.4

ξξξξ

1.0

Diamond

∆'

(0)

∆

(0)

∆

(A)

∆

(A)

[00ξ]

[1ξ0] [ξξ0] [ξξξ]

(0)

(A)

(0)

(A)

Γ'

(A)

Fig. 4.1-18 Diamond. Phonon dispersion curves. Symbols, experimental data from neutron scattering; solid curves, shell

model calculation [1.16]

Table 4.1-10 Phonon dispersion relations of Group IV

semiconductors and IV–IV compounds

Crystal Phonon dispersion curves

Diamond C Fig. 4.1-18

Silicon Si Fig. 4.1-19

Germanium Ge Fig. 4.1-20

Gray tin α-Sn Fig. 4.1-21

Silicon carbide 3C-SiC Fig. 4.1-22

6H-SiC

Part 4 1.1

586 Part 4 Functional Materials

600

400

200

ΓΓ

Wavenumber ν

–

(cm

–1

)

Wave vector q

KX L X W L

DOS

Fig. 4.1-19 Si. Phonon dispersion curves (left panel) and phonon density of states (right panel) [1.16]. Experimental data

points [1.17, 18] and ab initio calculations [1.16]

400

300

200

100

ΓΓ

Wavenumber ν

–

(cm

–1

)

Wave vector q

KX L X W L

DOS

Fig. 4.1-20 Ge. Phonon dispersion curves (left panel) and phonon density of states (right panel) [1.16]. Experimental

data points [1.19] and ab initio calculations [1.16]

Part 4 1.1

Semiconductors 1.1 Group IV Semiconductors and IV–IV Compounds 587

300

200

100

ΓΓ

Wavenumber ν

–

(cm

–1

)

Wave vector q

KX L X W L

DOS

α-Sn

Fig. 4.1-21 α-Sn. Phonon dispersion curves (left panel) and phonon density of states (right panel) [1.20]. Experimental

data points from inelastic neutron scattering at 90 K [1.21]. Solid lines, ab initio pseudopotential calculation [1.20]

Part 4 1.1

588 Part 4 Functional Materials

Wavenumber ν

(cm

–1

)

1000

750

500

250

Wave vector q

3C-SiC

Γ KML

Wavenumber ν

(cm

–1

)

1000

750

500

250

Wave vector q

6H-SiC

Γ KML

Wavenumber ν

(cm

–1

)

1000

750

500

250

Wave vector q

4H-SiC

Γ KML

Wavenumber ν

(cm

–1

)

1000

750

500

250

Wave vector q

2H-SiC

Γ KM L

Fig. 4.1-22 SiC. Phonon dispersion curves for the 3C, 2H, 4H, and 6H modiﬁcations obtained from a bond-charge model

calculation [1.22]. Symbols represent experimental data. The different types of lines indicate different polarization states

Part 4 1.1

Semiconductors 1.1 Group IV Semiconductors and IV–IV Compounds 589

B. Electronic Properties

Tables 4.1-11 – 4.1-18.

∆

Fig. 4.1-23 Brillouin zone of the diamond and of the zinc

blende lattice

Table 4.1-12 Band structures of Group IV semiconductors and IV–IV compounds

Diamond (C) (Fig. 4.1-25)

Diamond is an indirect-gap semiconductor, the lowest minima of the conduction band being located along the ∆ axes. The valence

band has the structure common to all group IV semiconductors, namely three bands degenerate at Γ. The spin–orbit splitting of these

bands is negligible.

Silicon (Si) (Fig. 4.1-26)

The conduction band is characterized by six equivalent minima along the [100] axes of the Brillouin zone located at about

= 0.85(2π/a) (symmetry ∆

). The surfaces of constant energy are ellipsoids of revolution with their major axes along [100].

Higher minima are located at Γ and along the [111] axes about 1 eV above the [100] minima.

The valence band has its maximum at the Γ point (symmetry Γ

), the light- and heavy-hole bands being degenerate at this point.

Both bands show warping. The third, spin–orbit split-off band has Γ

symmetry. The spin–orbit splitting energy at Γ is small

compared with most interband energy differences. Thus, spin–orbit interaction is mostly neglected in band structure calculations and

the notation of the single group is used to denote the symmetry of a band state.

Table 4.1-11 First Brillouin zones of Group IV semicon-

ductors and IV–IV compounds

Crystal Figures

Diamond C Fig. 4.1-23

Silicon Si Fig. 4.1-23

Germanium Ge Fig. 4.1-23

Gray tin α-Sn Fig. 4.1-23

Silicon carbide 3C-SiC Fig. 4.1-23

2H-SiC

Fig. 4.1-24

∆

Fig. 4.1-24 Brillouin zone of the Wurtzite lattice

Part 4 1.1

590 Part 4 Functional Materials

Table 4.1-12 Band structures of Group IV semiconductors and IV–IV compounds, cont.

Germanium (Ge) (Fig. 4.1-27)

The conduction band is characterized by eight equivalent minima at the end points L of the [111] axes of the Brillouin zone (symmetry L

The surfaces of constant energy are ellipsoids of revolution with their major axes along [111]. Higher energy minima of the conduction band

are located at the Γ point and (above these) on the [100] axes.

The valence band has its maximum at the Γ point (symmetry Γ

), the light- and heavy-hole bands being degenerate at this point. Both

bands are warped. The third, spin–orbit split-off band has Γ

symmetry. In contrast to silicon the spin–orbit splitting energies are considerable.

Thus, the symmetry notation of the double group of the diamond lattice is mostly used for Ge.

Gray tin (α-Sn) (Fig. 4.1-28)

The band structure of gray tin (α-Sn) is qualitatively different from those of the other Group IV elements. The s-like Γ

conduction band

edge, which decreases drastically with atomic number in the sequence C → Si →Ge, is situated below the p-like Γ

valence band edge

in α-Sn. This causes an inversion of the curvature of the Γ

light-hole band. Consequently, α-Sn is a zero-gap semiconductor, with its lowest

conduction band and its highest valence band being degenerate at Γ (symmetry Γ

). A second conduction band, with minima at L

, follows

at a slightly higher energy. This second hand determines the properties of n-type samples for n > 10

−3

(T > 77 K for intrinsic samples).

Silicon carbide

Band structure calculations show that the conduction band minima are situated along the cubic axes at the border of the Brillouin zone.

The band structure of 3C-SiC is shown in Fig. 4.1-29, and that of 2H-SiC in Fig. 4.1-30.

Silicon–germanium alloys

The band structure is characterized by a crossover in the lowest conduction band edge from Ge-like [111] symmetry to Si-like [100] symmetry

at the composition x =0.15. The shape of the band edge (and hence the effective masses) varies only slightly as a function of composition.

–5

–10

–15

–20

–25

XL ΓΣΛ∆U,K Γ

E (eV)

Wave vector k

Diamond

25'

Fig. 4.1-25 Band structure of diamond

E (eV)

–3

–6

–9

–12

Wave vector k

XΓ U,K ΓL Λ∆ Σ

25'

Fig. 4.1-26 Band structure of silicon

Part 4 1.1

Semiconductors 1.1 Group IV Semiconductors and IV–IV Compounds 591

–2

–4

–6

–8

–10

–12

U,KXL ΓΓΣΛ∆

–2

–4

–6

–8

–10

–12

XL ΓΛ∆

E (eV)

Wave vector k

4,5

E (eV)

4,5

Wave vector k

Theory

Ge (111) c(2 × 8)

Ge (111) (1 × 1)H

Ge (110) (2 × 1)

Typical error

Fig. 4.1-27 Band structure of germanium

–2

–4

–6

–8

–10

–12

U,KXL ΓΓΣΛ∆ XΓ

E (eV)

Wave vector k

4,5

Wave vector k

α-Sn

4,5

Fig. 4.1-28 Band structure of gray tin

Fig. 4.1-29 Band structure of 3C silicon carbide

–2

–4

–6

–8

–10

–12

–14

–16

ΓΣK∆ΓΛXZW Q L X

E (eV)

Wave vector k

3C-SiC

3,4

Part 4 1.1