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

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622 Part 4 Functional Materials

Table 4.1-56 Melting point T

of gallium compounds

Crystal Melting point T

(K) Remarks

Gallium nitride GaN 2791 GaN sublimes without decomposition around 800

◦

Gallium phosphide GaP 1749

Gallium arsenide GaAs 1511

Gallium antimonide GaSb 991 (1)

0 100 200 300 400 500

T (K)

α (10

–6

–1

)

600 700 800

GaN

Fig. 4.1-78 GaN. Coefﬁcient of linear thermal expansion

vs. temperature (curve 1, α

⊥

; curve 2, α



) [1.75]

α (10

–6

–1

)

100

300

T (K)

–2

2000

GaAs

T(K)

–3

–6

0 5 10 15 20

∆

α (10

–8

–1

)

Fig. 4.1-80 GaAs. Coefﬁcient of linear thermal expansion.

Experimental data points, and curves from an ab initio

pseudopotential calculation [1.77]

0 50 100 150 200 250 300

T (K)

GaP

α (10

–6

–1

)

Fig. 4.1-79 GaP. Linear thermal expansion coefﬁcient vs.

temperature between 0 K and 300 K [1.76]

–2

0 40 80 120 160 200 240 280 320 360

T (K)

α (10

–6

–1

)

GaSb

Fig. 4.1-81 GaSb. Linear thermal expansion coefﬁcient vs.

temperature measured with a quartz dilatometer. High-

temperature range [1.56,57]

Part 4 1.2

Semiconductors 1.2 III–V Compounds 623

Table 4.1-57 Linear thermal expansion coefﬁcient α of gallium compounds

Crystal Expansion coefﬁcient α Remarks Temperature dependence

(10

−6

−1

)

Gallium nitride GaN α

parall

5.59 X-ray diffraction Fig. 4.1-78

perpend

3.17

Gallium phosphide GaP 4.65 300 K Fig. 4.1-79

Gallium arsenide GaAs 5.87 300 K Fig. 4.1-80

Gallium antimonide GaSb 7.75 (50) Around 300 K; X-ray diffraction Fig. 4.1-81

Table 4.1-58 Heat capacity c

and Debye temperature Θ

of gallium compounds

Crystal Heat capacity (J/mol K) Temperature (K) Debye temperature (K) Remarks

Gallium nitride GaN 600 Estimated

Gallium phosphide GaP 49.65 800 445

Gallium arsenide GaAs 46.90 300 344

Gallium antimonide GaSb 266

800

700

600

500

400

300

200

100

ΓΓ

Wavenumber ν

–

(cm

–1

)

Wave vector q

KM A HL A

DOS

GaN

(LO)

(TO)

(LO)

(TO)

(LO)

(TO)

Fig. 4.1-82 GaN (wurtzite structure). Phonon dispersion curves (decomposed according to different irreducible represen-

tations along the main symmetry directions) and density of states, from a model-potential calculation. The circles show

Raman data from [1.78]. From [1.79]

Part 4 1.2

624 Part 4 Functional Materials

Table 4.1-59 Phonon wavenumbers of gallium compounds. Gallium nitride (GaN), T = 300 K, from Raman spec-

troscopy; gallium phosphide (GaP), RT, from an analysis of Raman, neutron, luminescence, and absorption data; gallium

arsenide (GaAs), T = 296 K, from coherent inelastic neutron scattering; gallium antimonide (GaSb), T = 300 K, from

second-order Raman effect

GaN GaP

Symmetry

Branch Wavenumber Symmetry Branch Frequency

point

(cm

−1

) point (THz)

low

) 145 ν(Γ) TO 10.95 (1)

ν (A

) TO 533 LO 12.06 (1)

ν (E

) TO 560 ν (X) TA 3.13 (3)

high

) 567 LA 7.46 (6)

ν (A

) LO 735 TO 10.58 (3)

ν (E

) LO 742 LO 11.09 (6)

ν (L) TA 2.58 (3)

LA 6.45 (3)

TO 10.64 (3)

LO 11.24 (3)

GaAs GaSb

Symmetry

Branch Frequency Symmetry Branch Wavenumber

point (THz) point (cm

−1

)

ν(Γ

) LO 8.55 (20)

ν (L)

TA 46

TO 8.02 (8)

ν (X)

TA 56

ν (X

) TA 2.36 (2)

ν (W)

TA 75

LO 7.22 (15)

ν (L)

LA 155

ν (X

) LA 6.80 (6) LO 204

ν (X

) TO 7.56 (8)

ν (X)

LO 210

ν (L

) LA 6.26 (10)

ν (L, X, Σ)

TO 218

LO 7.15 (7)

ν (L

) TA 1.86 (2)

TO 7.84 (12)

Table 4.1-60 Phonon dispersion curves of gallium com-

pounds

Crystal Figure

Gallium nitride GaN Fig. 4.1-82

Gallium phosphide GaP Fig. 4.1-83

Gallium arsenide GaAs Fig. 4.1-84

Gallium antimonide GaSb Fig. 4.1-85

Part 4 1.2

Semiconductors 1.2 III–V Compounds 625

400

300

200

100

ΓΓ

Wavenumber ν

–

(cm

–1

)

Wave vector q

KX L X W L

DOS

GaP

Fig. 4.1-83 GaP. Phonon dispersion curves (left panel) and density of states (right panel). Experimental data points

from [1.80] (open symbols) and from [1.81] (ﬁlled symbols) and theoretical curves from ab initio calculations [1.82].

From [1.82]

400

300

200

100

ΓΓKX L X W L

Wavenumber ν

–

(cm

–1

)

Wave vector q

DOS

GaAs

Fig. 4.1-84 GaAs. Phonon dispersion curves (left panel) and phonon density of states (right panel) [1.16]. Experimental

data points [1.83] and ab initio calculations [1.16].

Part 4 1.2

626 Part 4 Functional Materials

ΓΓ

300

200

100

Wavenumber ν

–

(cm

–1

)

Wave vector q

KX L X W L

DOS

GaSb

Fig. 4.1-85 GaSb. Phonon dispersion curves (left panel) and phonon density of states (right panel) [1.16]. Experimental

data points [1.84] and ab initio calculations [1.16]

B. Electronic Properties

Tables 4.1-61 – 4.1-66.

Band Structures of Gallium Compounds. Gallium Ni-

tride (GaN). The band structure is shown in Fig. 4.1-87.

Owing to spin–orbit interaction, the two top valence

bands (Γ

and Γ

) are split into three spin-degenerate

bands, one with quantum number J

= 3/2(Γ

) and

two with Jz = 1/2(Γ

)’. For convenience, the notation

A, B, C is used for these bands.

Gallium Phosphide (GaP). Gallium phosphide is an

indirect-gap semiconductor. The lowest set of conduc-

tion bands shows a camel’s back structure; the band

minima are located on the ∆ axes near the zone

boundary. The valence bands show the usual structure

characteristic of zinc blende semiconductors.

The spin–orbit splitting of the top of the valence

band is negligible compared with most other energy

separations in the band structure. Therefore, Fig. 4.1-86

shows the band structure calculated without inclusion of

spin–orbit splitting; the symmetry symbols of the high-

symmetry band states are symbols of the single group

of the zinc blende structure.

Gallium Arsenide (GaAs). Gallium arsenide is a direct-

gap semiconductor. The minimum of the lowest

Table 4.1-61 First Brillouin zones of gallium compounds

Crystal Figure

Gallium nitride GaN Fig. 4.1-24

Gallium phosphide GaP Fig. 4.1-23

Gallium arsenide GaAs Fig. 4.1-23

Gallium antimonide GaSb Fig. 4.1-23

conduction band is located at Γ; higher sets of min-

ima at L and near X (about 10% away from the zone

boundary) are also important for the optical and trans-

port properties. The minimum near X most probably

has a camel’s back-like structure similar to that of GaP,

AlAs, and AlSb. The valence bands have the usual struc-

ture characteristic of zinc blende crystals (Fig. 4.1-88).

Gallium Antimonide (GaSb). The conduction band of

GaSb is characterized by two types of minima, the lowest

minimum at Γ and slightly higher minima at the L points

at the surface of the Brillouin zone. A third set of minima

at the X points has been detected in optical experiments.

The valence bands show the usual structure common to

zinc blende semiconductors (Fig. 4.1-89).

Part 4 1.2

Semiconductors 1.2 III–V Compounds 627

E (eV)

–3

–6

–9

–12

Wave vector k

XΓ U,K ΓL Λ∆ Σ

GaP

Fig. 4.1-86 Band structure of gallium phosphide

–2

–4

–6

–8

–10

–12

U,KXL ΓΓΣΛ∆

E (eV)

Wave vector k

GaAs

4,5

Fig. 4.1-88 Band structure of gallium arsenide

–2

–4

–6

–8

ΓΣRL TAASHPKUM ∆Γ

E (eV)

Wave vector k

GaN

1,3

5,6

1,3

2,4

1,2

5,6

Fig. 4.1-87 Band structure of gallium nitride

E (eV)

–3

–6

–9

–12

Wave vector k

XΓ U,K ΓL Λ∆ Σ

4,5

GaSb

Fig. 4.1-89 Band structure of gallium antimonide

Part 4 1.2

628 Part 4 Functional Materials

Table 4.1-62 Energy gaps of gallium compounds

Crystal Quantity Bands Energy (eV) Temperature (K) Remarks

Gallium nitride GaN E

g, dir

3.503 (2) 1.6 Photoluminescence

A exciton 3.4751 (5) Transition from Γ

B exciton 3.4815 (10) Transition from upper Γ

C exciton 3.493 (5) Transition from lower Γ

Gallium phosphide GaP E

g, ind

to ∆

2.350 (1) 0 Extrapolated, from exciton data

2.272 300

g, ind

15v

to L

2.637 (10) 78 Electroabsorption

Gallium arsenide GaAs E

g, dir

to Γ

1.51914 0 Extrapolated, photoluminescence

1.424 (1) 300 Differentiated reﬂectivity

g, th

1.604 0 Extrapolated,

intrinsic carrier concentration

Gallium antimonide GaSb E

g, dir

to Γ

0.822 0 Extrapolated, electroreﬂectance

0.75 300

Table 4.1-63 Exciton binding energies of gallium compounds

Crystal Quantity Energy (meV) Remarks

Gallium nitride GaN E

20 A exciton, photoreﬂectance

18.5 B exciton, photoreﬂectance

22.4 C exciton, photoluminescence

Gallium antimonide GaSb E

1.6 Calculated ground-state binding energy

Table 4.1-64 Spin–orbit splitting energies of gallium compounds

Crystal Quantity Bands Energy Temperature Remarks

(meV) (K)

Gallium nitride GaN ∆

9.8 Epitaxial ﬁlms,

∆

5.6 reﬂectance measurements

∆

5.6

Gallium phosphide GaP ∆

to Γ

80 (3) 100–200 Splitting of Γ

15v

into Γ

(upper level) and

(lower level)

Gallium arsenide GaAs ∆

15v

346.4 (5) 1.7 Splitting of Γ

15v

valence band state into Γ

and Γ

, magnetoabsorption splitting of Γ

conduction band state into Γ

and Γ

∆



15c

171 (15) 4.2 electroreﬂectance

Gallium antimonide GaSb ∆

to Γ

756 (15) 10 Electroreﬂectance

∆



to Γ

213 (10)

∆

4,5v

to L

430 (10)

∆



4,5c

to L

130

Part 4 1.2

Semiconductors 1.2 III–V Compounds 629

Table 4.1-65 Effective masses of electrons (m

) and holes (m

) for gallium compounds (in units of the electron mass m

)

GaN GaP

0.22 6K Cyclotron resonance, m

n, parall

Near ∆

min

7.25 1–20K At bottom of camel’s back

polaron mass

n, parall

0.20 (6) From reﬂectance m

n, perp

Near ∆

min

0.21 Luminescence

spectrum

n, perp

0.20 (2) m

n, parall

Away 2.2 High above bottom of

from ∆

min

camel’s back

(A)

1.01 7K Estimated from m

p, heavy

Parallel 0.67 (4) 1.6 Cyclotron resonance

binding energies to [111] at 1.6K

(B)

1.1 m

p, light

Parallel 0.17 (1) Parallel

to [111] to [111]

(C)

1.6 m

p, so

0.4649 0 Spin–orbit mass,

calculated from k · p model

GaAs

(Γ) 0.0662 (2) 1.6 Cyclotron resonance

p, heavy

Parallel to [100] 0.34 10 K Electronic Raman scattering

Parallel to [111] 0.75

p, light

Parallel to [100] 0.094

Parallel to [111] 0.082

GaSb

(Γ) 0.039 (5) 2K Optically detected cyclotron resonance

) 0.11 Transverse mass

0.95 Longitudinal mass

0.226 Density-of-states mass

p, heavy

0.29 (9) 30 K B parallel to [100] Stress-modulated magnetoreﬂectance

0.36 (13) B parallel to [110]

0.40 (16) B parallel to [111]

p, light

0.042 (2)

Table 4.1-66 Electron g-factor g

of gallium antimonide

Crystal g

Temperature (K) Remarks

Gallium antimonide GaSb −9.1 (2) 4.2–50 Magnetoluminescence

C. Transport Properties

Tables 4.1-67 and 4.1-68.

Electronic Transport, General Description. Gallium

Nitride (GaN). Undoped GaN is normally an n-type

conductor. Carrier concentrations in undoped ﬁlms

can vary from 5 ×10

−3

to 5×10

−3

because

of unintentional incorporation of extrinsic impurities,

mainly silicon and oxygen. In the purest α-GaN ma-

terial (n = 10

−3

), conductivities of the order of

10 Ω cm have been found at 300 K. For the tem-

perature dependence of the electrical resistivity, see

Fig. 4.1-90.

Gallium Phosphide (GaP). Even in high-puritysamples,

intrinsic conduction occurs in gallium phosphide only

above 500

◦

C. Thus, the transport properties are in gen-

eral determined by the properties of impurities and

lattice defects. The data for the electrical conductivity σ

of pure n-type material are found to lie around 0.5 Ω cm

at 300 K.

Part 4 1.2

630 Part 4 Functional Materials

As in the case of GaAs, semi-insulating GaP can be

obtained by doping with shallow donors and acceptors

and with deep centers. Resistivities at room temperature

are of the order of 10

–10

Ω cm. Typical data at high

temperatures are shown in Fig. 4.1-91.

Gallium Arsenide (GaAs). At low ﬁelds, the electrons

are in the Γ

minima at the zone center. The domi-

nating scattering process at room temperature is polar

optical scattering, while below 60 K the most important

contribution to the lattice mobility is from piezoelectric

potential scattering. At room temperature, the hole mo-

bility of samples with p < 5×10

−3

is governed by

lattice scattering alone. An intrinsic carrier concentra-

tion of n

= 2.1×10

−3

has been found at T =300 K

(Fig. 4.1-92).

Owing to the very low intrinsic carrier concentration,

the electrical conductivity of GaAs at 300 K is deter-

–1

–2

–3

0 20406080100120

100 30 20 15 1050

–1

(10

–3

–1

)

ρ (Ω cm)

GaN

T (K)

Fig. 4.1-90 GaN. Resistivity of several undoped samples

(1 – 7) and a Zn-doped sample (8) vs. reciprocal tempera-

ture [1.85]

mined by charge carriers provided by impurities. Only

under certain doping conditions (e.g. compensation of

shallow acceptor levels by donor impurities and over-

compensation by deep Cr levels) can the Fermi level

be pinned in the middle of the energy gap and nearly

intrinsic conduction achieved (semi-insulating GaAs,

ρ =10

–10

Ω cm).

Gallium Antimonide (GaSb). Transport in n-type GaSb

is complicated by the contribution of three sets of con-

duction bands with minima located at Γ,L,andX.The

data on transport coefﬁcients can be consistently ex-

plained by a three-band model, the X bands contributing

to transport above 180

◦

C. Intrinsic carrier concentra-

tions n

of the order of 10

−3

have been estimated

for a temperature T = 365 K.

In p-type GaSb, a multiellipsoidal model has to be

used at low temperatures, taking into account the shift

of the heavy- and light-hole bands away from k = 0.

At high temperatures, a warped-sphere model (as in the

case of Si and Ge) is adequate.

1.0 2.01.5 2.5

–1

(10

–3

–1

)

n/T

3/2

(cm

–3

3/2

)

H, n

(cm

/V s)

ρ (Ω cm)

GaP

n/T

3/2

H, n

Fig. 4.1-91 GaP. Resistivity, Hall mobility, and electron

concentration (divided by T

−3/2

) vs. temperature for

a semi-insulating sample [1.86]

Part 4 1.2

Semiconductors 1.2 III–V Compounds 631

Table 4.1-67 Electron and hole mobilities µ

and µ

of gallium compounds

Crystal Temperature Carrier concentration µ

Remarks

(K) (cm

−3

) (cm

/Vs) (cm

/Vs)

Gallium nitride 300 n = 3×10

900 Hall mobility

p =5×10

350 β-GaN, Hall mobility

Gallium phosphide RT 160 135 Thin ﬁlms, Hall mobility,

maximum mobility

Gallium arsenide 300 2400–3300 Bulk GaAs, Hall mobility

300 400 Hall mobility

Gallium antimonide 300 n = 1.2×10

7620 MBE-grown on GaSb substrates

300 680 Hall mobility

See Fig. 4.1-93 for the temperature dependence of the electron mobility in GaN.

See Figs. 4.1-94a,b for the temperature dependence of the electron and hole mobilities in GaP.

See Figs. 4.1-95 and 4.1-96 for the temperature dependences of the electron and hole mobilities in GaAs.

–1

(10

–4

–1

)

(cm

–3

)

T (K)

(cm

–3

)

10 15 20 25 30 35 40 800 900 1000 1100 1200 1300 1400 1500

GaAs

Fig. 4.1-92 GaAs. Intrinsic carrier concentration vs. reciprocal temperature for the range 250–1000 K and vs. temperature

for the range 800–1500 K. The solid curves (A) are the result of a critical discussion of the literature data. The other

curves (B – G) represent data from other sources [1.87]

Part 4 1.2