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

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642 Part 4 Functional Materials

250

200

150

100

ΓΓ

Wavenumber ν

–

(cm

–1

)

Wave vector q

KX L X W L

DOS

InAs

Fig. 4.1-111 InAs. Phonon dispersion curves (left panel) and density of states (right panel) [1.82]. Experimental neutron

data (circles, T = 300 K) [1.102], thermal-diffuse-scattering X-ray data (diamonds, T = 80 K) [1.102], Raman data

(squares, T = 330 K) [1.103], and ab initio calculations (solid curves [1.82]). From [1.82]

ΓΓ

200

100

Wavenumber ν

–

(cm

–1

)

Wave vector q

KX L X W L

DOS

InSb

Fig. 4.1-112 InSb. Phonon dispersion curves (left panel) and density of states (right panel). Experimental neutron data

(circles) [1.21] and ab initio pseudopotential calculations (solid curves) [1.60]. From [1.60]

Part 4 1.2

Semiconductors 1.2 III–V Compounds 643

B. Electronic Properties

Tables 4.1-82 – 4.1-87.

Band Structures of Indium Compounds. Indium Ni-

tride (InN). The band structure (Fig. 4.1-114) shows

a direct gap at Γ very similar to that of GaN. A low-lying

many-valley conduction band leading to an indirect gap

has been suggested.

Indium Phosphide (InP). Indium phosphide is a direct-

gap semiconductor. The conduction band minimum is

situated at Γ. Higher conduction band minima at L and X

have been detected in optical experiments. The X-band

minima show no camel’s back structure, in contrast to

most other III–V compounds with the zinc blende struc-

ture. The valence band has the structure common to all

zinc blende-type semiconductors (Fig. 4.1-113).

Indium Arsenide (InAs). Indium arsenide resembles

InSb in its band structure, having only a slightly larger

energy gap and a smaller spin–orbit splitting of the top of

the valence band. The conduction band minimum (Γ

)

is situated in the center of the Brillouin zone. Near the

minimum, E(k) is isotropic but nonparabolic. The va-

lence band shows the usual structure common to all zinc

blende-type III–V compounds (Fig. 4.1-116).

–3

–6

–9

–12

XΓ U,K Γ

E (eV)

L Λ∆ Σ

Wave vector k

InP

4,5

Fig. 4.1-113 Band structure of indium phosphide

Table 4.1-82 First Brillouin zones of indium compounds

Crystal Figure

Indium nitride InN Fig. 4.1-24

Indium phosphide InP Fig. 4.1-23

Indium arsenide InAs Fig. 4.1-23

Indium antimonide InSb Fig. 4.1-23

Indium Antimonide (InSb).Indium antimonide is adirect-

gap semiconductor. The minimum of the conduction

band (Γ

) is located in the center of the Brillouin

zone. Near the minimum, E(k) is isotropic but non-

parabolic. Thus the effective mass of the electrons

is scalar and depends strongly on the electron con-

centration. Higher band minima (about 0.63 eV above

the lowest minimum) seem to have been established

by transport measurements in heavily doped n-InSb.

The valence band shows the structure common to

all zinc blende semiconductors, i. e. two subbands

degenerate at Γ

and one spin-split band at Γ

(Fig. 4.1-115).

ΓKΓ LΣ∆

–3

–6

–9

–12

TM ASH RALUMHPK

E (eV)

Wave vector k

InN

1,3

1,2

2,4

5,6

1,3

5,6

1,2

1,3

Fig. 4.1-114 Band structure of indium nitride

Part 4 1.2

644 Part 4 Functional Materials

–3

–6

–9

–12

ΓΓ X Σ∆Λ

–3

–6

ΓΣXU,KL

E (eV)

Wave vector k

InAs

Wave vector k

E (eV)

4,5

Table 4.1-83 Energy gaps of indium compounds

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

Indium nitride InN E

g, dir

1.95 300 Polycrystalline ﬁlm, absorption edge

Indium phosphide InP E

g, dir

to Γ

1.4236 (1) 1.6 From excitonic gap

1.344 300 Absorption, photoluminescence

Indium arsenide InAs E

g, dir

to Γ

0.4180 (5) 4.2 Magnetotransmission

0.354 (3) 295 Electroreﬂectance

g, x

0.41565 (1) 1.4 Excitonic band gap, photoluminescence

Indium antimonide InSb E

g, dir

to Γ

0.2352 1.7 Emission spectra, from exciton data

0.180 300 Reﬂectance, magnetoabsorption

Table 4.1-84 Spin–orbit splitting energies of indium compounds

Crystal Spin–orbit Bands Numerical Temperature Remarks

splitting value (eV ) (K)

Indium phosphide InP ∆

to Γ

0.108 (8) 4.2 Wavelength-modulated photovoltaic effect

∆

4,5v

to Λ

0.13 77 Thermoreﬂectance

0.15 (5) 300 Electroreﬂectance

∆



to Γ

0.07 300 Electroreﬂectance

Indium arsenide InAs ∆

to Γ

0.38 (1) 1.5 Magnetoelectroreﬂectance

∆

4,5v

to L

0.267 5 Wavelength-modulated reﬂectance

Indium antimonide InSb ∆

to Γ

0.81 (1) 1.5 Magnetoelectroreﬂectance

∆



to Γ

0.39 5 Wavelength-modulated reﬂectance

∆

4,5v

to Λ

0.495

–3

–6

–9

–12

XΓ U,K Γ

E (eV)

L Λ∆ Σ

Wave vector k

InSb

4,5

Fig. 4.1-115 Band structure of indium antimonide

Fig. 4.1-116 Band structure of indium arsenide

Part 4 1.2

Semiconductors 1.2 III–V Compounds 645

InAs

1.43

1.42

1.41

1.40

1.39

1.38

1.37

1.36

1.35

1.34

T (K)

E (eV)

0 50 100 150 200 250 300

Band gap

Emission

Absorption

Exciton peak

Fig. 4.1-117 InP. Energy gap and exciton peak energy vs.

temperature from absorption and emission data [1.104]

InAs

0.45

0.40

0.35

0.30

0.25

T (K)

(eV)

0 100 200 300 400 500 600

Fig. 4.1-118 InAs. Band gap vs. temperature. Solid line,

optical band gap calculated from parameters used for ﬁt-

ting various transport data; symbols, thermal band gap of

six samples, obtained from conductivity and Hall coefﬁ-

cient [1.105]

InSb

0.24

0.23

0.22

0.21

0.20

0.19

0.18

0.17

0.16

T (K)

(eV)

0 80 160 240 320

Fig. 4.1-119 InSb. Energy gap vs. temperature below RT,

measured by resonant two-photon photo-Hall effect (ﬁlled

circles); the open circles and triangles show earlier liter-

ature data for comparison. Solid curve:ﬁtbyVarshni’s

formula [1.106]

Table 4.1-85 Temperature dependence of the energy gaps

of indium compounds

Crystal Figure

Indium phosphide InP Fig. 4.1-117

Indium arsenide InAs Fig. 4.1-118

Indium antimonide InSb Fig. 4.1-119

Part 4 1.2

646 Part 4 Functional Materials

Table 4.1-86 Effective masses of electrons (m

) and holes (m

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

)

InN

0.11 300 K Optical mass, plasma edge

n, calc

0.12 Calculated effective mass

p, heavy

0.5

p, light

0.17

InP

0.0808(10) 0K Magnetophonon resonance

0.073 300 K Faraday rotation

p, heavy

0.45 (5) 4.2K Piezomodulated photovoltaic effect

p, light

0.12 (1) 4.2K

p, so

0.121 (1) 110 K B parallel to [111] and [100]

InAs

0.0265 4.2K

0.023 300 K

p, heavy

0.57 300 K

p, [100]

0.35

p, [111]

0.85

InSb

(Γ

) 0.01359 (3) 4.2K Faraday effect, n = 4.6×10

−3

0.0118 300 K Cyclotron resonance

p, heavy

0.44 20 K Parallel to [111], magnetoabsorption, Faraday rotation

0.42 Parallel to [110]

0.32 Parallel [100]

p, light

0.016 20 K Magnetoabsorption, Faraday rotation

Table 4.1-87 Electron g-factor g

of indium compounds

Crystal g

Temperature Remarks

(K)

Indium phosphide 1.48 (5) 4.2 Modulated photovoltaic effect

Indium arsenide −15.3 (2) 1.4 Magnetoluminescence

Indium antimonide −51.31 (1) 1.4 Electron spin resonance

Part 4 1.2

Semiconductors 1.2 III–V Compounds 647

C. Transport Properties

Tables 4.1-88 – 4.1-90.

Electronic Transport, General Description. Indium

Phosphide (InP). The transport properties are deter-

mined mainly by the electrons in the Γ

minimum.

Above 800 K, multivalley conduction, where the L

minimum is involved, becomes important.

Indium Arsenide (InAs). The transport properties are de-

termined mainly by the electrons in the Γ

minimum.

Pure material with intrinsic conduction down to 450 K

is available.

–1

(10

–3

–1

)

(Ω

–1

)

T (°C)

2345617

500 200 100 0 – 50 – 100

InAs

Fig. 4.1-120 InAs. Electrical conductivity vs. recipro-

cal temperature [1.107]. A, n = 1.2×10

−3

; B,

n = 1.7×10

−3

; C, n = 4×10

−3

; D, n =

7×10

−3

; E, n =2×10

−3

;1,p =2×10

−3

;

2, p =7×10

−3

Table 4.1-88 Carrier concentrations n

and electrical conductivity σ of indium compounds

Crystal Temperature Carrier concentration Conductivity Remarks

(K) n

(cm

−3

) σ(

−1

)

Indium nitride InN 300 2–3×10

Temperature coefﬁcient of resistivity

3.7×10

−3

−1

at 200–300 K

for pressed powder

Indium phosphide InP 300 6.3×10

4–5 n-type pure material

Indium arsenide InAs 300 50 See Fig. 4.1-120 for temperature dependence

Indium antimonide InSb 300 1.9×10

220 See Fig. 4.1-122 for temperature dependence

Indium Antimonide (InSb). The transport properties are

mainly determined by the extremely high mobility of the

electrons in the Γ

minimum of the conduction band.

Pure material with intrinsic conduction down to 200 K

is available. At low temperatures and in doped material,

impurity scattering limits the electron mobility.

0.45 eV

0.49 eV

–1

(10

–3

–1

)

(cm

–6

–3

)

1.0 1.4 1.8 2.2 2.6 3.0

InAs

, n

(cm

–3

)

1.2 ×10

1.7 ×10

3 ×10

7 ×10

2 ×10

Fig. 4.1-121 InAs. Square of the intrinsic carrier concentration

divided by T

vs. reciprocal temperature for several different sam-

ples [1.107]

Part 4 1.2

648 Part 4 Functional Materials

1000 500 300 200 150

123456789

–1

(10

–3

–1

)

σ (Ω

–1

) T(K)

InSb

Fig. 4.1-122 InSb. Electrical conductivity vs. recipro-

cal temperature [1.108]. Curve V , n ≈ 10

−3

;

A, n = 1.3×10

−3

; B, n = 1×10

−3

;1,p =

4×10

−3

;2,p =2.2×10

−3

;3,p =6×10

−3

;

4, p =2×10

−3

Table 4.1-89 Electron and hole mobilities µ

and µ

of indium compounds

Crystal Temperature Carrier concentration µ

Remarks Temperature

dependence

(K) (cm

−3

) (cm

/Vs) (cm

/Vs) of mobilities

Indium nitride 300 5–18×10

250 (50) Fig. 4.1-123

300 20

Indium phosphide 300 n = 0.5–1×10

4–5.4×10

Pure material, Fig. 4.1-124

77 n = 6×10

130× 10

Hall mobilities

300 190 Fig. 4.1-125

Indium arsenide 300 n ∼ 10

2–3.3×10

Pure material Fig. 4.1-121

77 0.8–1×10

300 100–450

Indium antimonide 300 70×10

Hall mobilities Fig. 4.1-126

300 850 Fig. 4.1-127

(cm

/V s)

468

262

2×10

T (K)

InN

Fig. 4.1-123 InN. Electron mobility vs. temperature

for three samples with RT carrier concentrations of

5.3×10

(1), 7.5×10

(2), and 1.8×10

−3

(3). Left

broken line, calculated ionized-impurity-scattering mobil-

ity; right broken line, empirical high-temperature mobility

(µ ∝ T

−3

) for sample 1. Solid lines, total mobility calcu-

lated for each sample [1.109]

Part 4 1.2

Semiconductors 1.2 III–V Compounds 649

(cm

/V s)

4×

2×10

648 2

T(K)

ion

=5×10

–3

InP

Ionized

impurity

Ionized

impurity

Polar

scattering

Total

ion

=5×10

–3

Fig. 4.1-124 InP. Electron mobility vs. temperature, calcu-

lated for two concentrations of ionized impurities [1.110]

(cm

/V s)

2×10

200 400 600 800

T (K)

InSb

Fig. 4.1-126 InSb. Electron mobility vs. temperature. The

experimental data are Hall mobilities. Triangles, [1.112];

ﬁlled circles [1.113]; open circles, [1.114,115]; solid line,

calculated drift mobility [1.116]

H,p

(cm

/V s)

2×10

4×10

6×

824

6×

T (K)

InP

Fig. 4.1-125 InP. Hole Hall mobility vs. temperature for

pure p-type samples, after [1.111]

H,p

(cm

/V s)

82468

2×10

T (K)

InSb

Fig. 4.1-127 InSb. Hole Hall mobility vs. tempera-

ture [1.111]. The vertical bars indicate the ranges of the

experimental values. The solid line has a slope −1.8

Part 4 1.2

650 Part 4 Functional Materials

468

2468262

T (K)

(W/cm K)

–3/2

InP

–1

Fig. 4.1-128 InP. Thermal conductivity vs. tempera-

ture [1.117,118]

Table 4.1-90 Thermal conductivity κ of indium com-

pounds

Crystal κ(W/mK) Remarks

Indium nitride InN 38.4

Indium phosphide InP See Fig. 4.1-128

Indium arsenide InAs

Indium antimonide InSb See Fig. 4.1-129

D. Electromagnetic and Optical Properties

Tables 4.1-91 – 4.1-93.

Table 4.1-91 Dielectric constant ε and refractive index n of indium compounds

Crystal Temperature ε(0) ε(∞) n at λ(µm) Remarks

(K)

Indium nitride InN 9.3 Infrared reﬂectivity, heavily doped ﬁlm

8.4

2.56 1.0 µm Interference method, n = 3×10

−3

2.93 0.82 µm

3.12 0.66 µm

Indium phosphide InP 300 12.56 (20) Table 4.1-92 Capacitance measurements

77 11.93 (20)

297 10.9 (10) Film interference

Indium arsenide InAs 300 15.15 12.37 Table 4.1-92 Infrared reﬂectance and oscillator ﬁt

Indium antimonide InSb 16.8 (2) Table 4.1-92 Gyroscopic sphere resonance, infrared

15.68 reﬂectance, and oscillator ﬁt

468

262

–1

2×

T (K)

(W/cm K)

InSb

Fig. 4.1-129 InSb. Thermal conductivity vs. temperature

for an n-type sample, and theoretical curve show-

ing the contributions of the longitudinal and transverse

phonons [1.119]

Part 4 1.2

Semiconductors 1.2 III–V Compounds 651

Table 4.1-92 Wavelength dependence of refractive index n of indium compounds at T = 300 K

InP InAs InSb

λ(µm)

n Remarks λ(µm) n Remarks λ(µm) n Remarks

14.85 3.03 Transmission 25 3.26 Interference 45 2.57 Reﬂectance and transmission,

and reﬂectance n = 6×10

−3

5 3.08 10 3.42 21.15 3.814

2 3.134 Prism method 3.74 3.52 19.98 3.826 Interference technique,

n = 2×10

−3

1 3.327 1.38 3.516 Reﬂectance and 10.06 3.953

dispersion relations 7.87 4.001

0.652 3.410 Reﬂectance and 0.517 4.558 2.07 4.03 Reﬂectance and

dispersion relations dispersion relations

0.399 4.100 0.282 3.800 0.689 5.13

0.200 1.525 0.049 1.139 0.062 1.17

0.062 0.793

Table 4.1-93 Energy dependence of optical constants of indium compounds. Real and imaginary parts of the complex dielectric

constant measured by spectroscopic ellipsometry; n, k, R, K calculated from these data

InP

hν(eV) ε

n k R K (10

−1

)

1.5 11.904 1.400 3.456 0.203 0.395 30.79

2.0 12.493 2.252 3.549 0.317 0.317 64.32

2.5 14.313 3.062 3.818 0.511 0.349 129.56

3.0 17.759 10.962 4.395 1.247 0.427 379.23

3.5 5.400 12.443 3.193 1.948 0.403 691.21

4.0 6.874 10.871 3.141 1.730 0.376 701.54

4.5 8.891 16.161 3.697 2.186 0.449 996.95

5.0 −7.678 14.896 2.131 3.495 0.613 1771.5

5.5 −4.528 7.308 1.426 2.562 0.542 1428.1

6.0 −2.681 5.644 1.336 2.113 0.461 1285.1

InAs

hν(eV) ε

n k R K (10

−1

)

1.5 13.605 3.209 3.714 0.432 0.337 65.69

2.0 15.558 5.062 3.995 0.634 0.370 128.43

2.5 15.856 15.592 4.364 1.786 0.454 452.64

3.0 6.083 13.003 3.197 2.034 0.412 618.46

3.5 5.973 10.550 3.008 1.754 0.371 622.13

4.0 7.744 11.919 3.313 1.799 0.393 729.23

4.5 −1.663 22.006 3.194 3.445 0.566 1571.2

5.0 −5.923 8.752 1.524 2.871 0.583 1455.3

5.5 −3.851 6.008 1.282 2.344 0.521 1306.6

6.0 −2.403 6.005 1.434 2.112 0.448 1284.2

InSb

hν(eV) ε

n k R K (10

−1

)

1.5 19.105 5.683 4.418 0.643 0.406 97.79

2.0 14.448 14.875 4.194 1.773 0.443 359.46

2.5 7.811 15.856 3.570 2.221 0.447 562.77

3.0 7.354 13.421 3.366 1.994 0.416 606.27

3.5 5.995 17.673 3.511 2.517 0.474 892.82

4.0 −6.722 19.443 2.632 3.694 0.608 1497.8

4.5 −6.297 8.351 1.443 2.894 0.598 1320.2

5.0 −4.250 6.378 1.307 2.441 0.537 1237.0

5.5 −4.325 4.931 1.057 2.333 0.563 1300.6

6.0 −3.835 3.681 0.861 2.139 0.572 1300.9

Part 4 1.2