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IOP Concise Physics

Radiative Properties of Semiconductors

N M Ravindra, Sita Rajyalaxmi Marthi and Asahel Banobre

Chapter 1

Introduction to radiative properties

1.1 IntroductionRadiative properties are fundamental physical properties that describe the inter-action of electromagnetic waves, ranging from ultraviolet (UV) to deep infrared(IR) spectral regions, with matter. They can be broadly classified into opticalradiative properties and thermal radiative properties. Understanding the radiativeproperties of materials is critical for the proper selection and interpretation oftemperature measurement techniques in various industrial processes. Radiativeproperty measurements enable us to understand the physics of solids and otherstates of matter. A schematic of the electromagnetic radiation spectrum ispresented in figure 1.1.

Heat transfer requires a medium, either a solid or liquid. Heat transfer takesplace mainly by two methods, conduction and convection. However, when heat is

Figure 1.1. Electromagnetic radiation wave spectrum. Reproduced with permission from [1].

doi:10.1088/978-1-6817-4112-3ch1 1-1 ª Morgan & Claypool Publishers 2017

transferred by radiation, no medium is required. Thermal radiation coversthe range of wavelengths from 0.1 to 100 microns. Radiation travels throughvacuum with speed of light (c) in the form of electromagnetic waves. Thespeed of propagation is related to its wavelength (λ) and frequency (v) by theequation:

λ=c v. (1.1)

Frequency is independent of the medium of propagation, but velocity is dependenton the medium of propagation.

1.2 Properties1.2.1 Emissivity

Radiation emitted by a surface is determined by introducing emissivity. Emissivity of asurface is its potential to emit energy in the form of radiation in comparison toblackbody radiation at the same given temperature. Emissivity ranges between 0 and 1.

Emissivity of a surface is given by the relation:

ε = qq

(1.2)b

(e)

(e)

where,q(e) – energy emitted per unit area by a real bodyqb

(e) – energy emitted per unit area by blackbodyEmissivity of black body is 1 and for real body, emissivity is <1.Emissivity is dependent on wavelength, temperature and angle of emission of

radiation. The spectral emissivity is given by:

ε λ θ ϕλ θ ϕ

λ≡λ θ

λ

λT

I T

I T( , , , )

( , , , )

( , ). (1.3)

e

b,

,

,

The spectral hemispherical emissivity (directional average) is:

∫ ∫

∫ ∫ε λ λ

λ

λ θ ϕ θ θ θ ϕ

λ θ θ θ ϕ≡ =λ

λ

λ

π πλ

π πλ

TE TE T

I T d d

I T d d( , )

( , )( , )

( , , , ) cos sin

( , ) cos sin. (1.4)

b

e

b,

0

2

0

/2,

0

2

0

/2,

The total hemispherical emissivity (a directional and spectral average) is:

∫ε

ε λ λ λ≡ =

λ λ∞

TE TE T

T E T d

E T( )

( )( )

( , ) ( , )

( ). (1.5)

b

b

b

0 ,

Emissivity of oxidized metal is greater than that of polished metals. Emissivity ofnon-conductors is relatively high. A table of emissivity of a few common materials ispresented in table 1.1.

Radiative Properties of Semiconductors

1-2

There are three responses to incident radiation (Gλ) on a surface:• Reflection from surface (Gλ,ref)• Transmission through surface (Gλ,tr)• Absorption by surface (Gλ,abs)

= + +λ λ λ λG G G G . (1.6),ref ,tr ,abs

1.2.2 Spectral absorptivity

Thermal radiation that is incident on the surface of an opaque solid is eitherabsorbed or reflected. The absorptivity is defined as the fraction of the incidentradiation that is absorbed. For semi-transparent solids, thermal radiation impingingon the surface is absorbed, reflected or transmitted, as shown in figure 1.2.

Assuming temperature independence, spectral absorptivity is:

α λ θ ϕλ θ ϕ

λ θ ϕ≡λ θ

λ

λ

I

I( , , )

( , , )

( , , ). (1.7),

,i,abs

,i

Spectral hemispherical absorptivity is:

∫ ∫

∫ ∫α λ

λλ

α λ θ ϕ λ θ ϕ θ θ θ ϕ

λ θ ϕ θ θ θ ϕ≡ =λ

λ

λ

π πλ θ λ

π πλ

G

G

I d d

I d d( )

( )

( )

( , , ) ( , , ) cos sin

( , , ) cos sin. (1.8)

i

i

,abs 0

2

0

/2, ,

0

2

0

/2,

Table 1.1. Emissivity of a few materials [2].

Material Temperature (°C) Emissivity

Aluminium (polished) 100 0.095Aluminium (oxidized) 200 0.11Aluminium oxide 500–827 0.42–0.26Bismuth, unoxidized 25 0.048Bismuth, unoxidized 100 0.061Graphite 0–3600 0.70–0.80Earthenware Ceramic 20 0.90Smooth glass 0–200 0.95Smooth glass 250–1000 0.87–0.72Iron, unoxidized 100 0.21Iron, oxidized 200–600 0.64–0.78Iron oxide 500–1200 0.85–0.89Plaster 0–200 0.91Zinc, highly polished 200–300 0.04–0.05Zinc, oxidized 24 0.280Zirconium silicate 238–500 0.920–0.800Zirconium silicate 500–832 0.800–0.520

Radiative Properties of Semiconductors

1-3

The total absorptivity is:

∫∫

αα λ λ λ

λ λ≡ =

λ λ

λ

∞

∞GG

G d

G d

( ) ( )

( ). (1.9)oabs

0

1.2.3 Spectral reflectivity

Spectral directional reflectivity of a material is:

ρ λ θ ϕλ θ ϕ

λ θ ϕ≡λ θ

λ

λ

I

I( , , )

( , , )

( , , ). (1.10),

,i,ref

,i

The spectral hemispherical reflectivity is:

∫ ∫ρ

λλ

ρ λ θ ϕ λ θ ϕ θ θ θ ϕ

λ θ ϕ≡ =λ

λ

λ

π πλ θ λ

λ

G

G

I d d

I

( )

( )

( , , ) ( , , ) cos sin

( , , ). (1.11)E,ref 0

2

0

/2

, ,

,i

The total spectral reflectivity is:

∫∫

ρρ λ λ λ

λ λ≡ =

λ λ

λ

∞

∞GG

G d

G d

( ) ( )

( ). (1.12)abs 0

0

1.2.4 Spectral transmissivity

Assuming negligible temperature dependence, spectral, directional transmissivity isgiven by:

τλ

λ≡λ

λ

λ

G

G( )

( ). (1.13),tr

Figure 1.2. Various responses to surface irradiation. Reprinted from [3] with permission from JohnWiley& Sons.

Radiative Properties of Semiconductors

1-4

The total hemispherical transmissivity is:

∫∫

τλ λ

λ λ≡ =

λ

λ

∞

∞GG

G d

G d

( )

( ). (1.14)tr 0 ,tr

0

For a semi-transparent medium,

τρ + + α =Reflectivity ( ) Transmissivity ( ) Absorptivity ( ) 1. (1.15)

Kirchhoff’s law states that, for a material in thermodynamic equilibrium, the totalhemispherical absorptivity is equal to the total hemispherical emissivity from thesurface of the material, where irradiation of the surface corresponds to emissionfrom a blackbody at the same temperature as the surface.

ε α= . (1.16)

1.2.5 Thermal radiation properties

1.2.5.1 RadianceFor a blackbody, radiance is defined as the radiant power dq from a pencil cone,from surface area dA in the direction ŝ, per unit wavelength interval per unit solidangle and per unit area projected onto the direction ŝ. Spectral radiance ismathematically given by:

λ θ φθ λ

=ΩλL

dqdA d d

( , , )cos

(1.17)

where, dΩ is solid angle

λ θ θ φ=d d dsin .

The total radiance is given by the integral over all wavelengths:

∫ λ λ= λ

∞L L d( ) . (1.18)

0

The concept of radiance and solid angle are illustrated in figure 1.3. The radiativeheat flux is the rate of energy flow per unit area across an element area dA, withunits W m−2.

∫ ∫ θ θ θ φ′ = =φ

π

θ

π

= =q

dqdA

L d dcos sin (1.19)0

2

0

/2

where, dA is the area of the surface.The radiative heat flux leaving the body solely by emission is also called emissive

power.An ideal blackbody absorbs all the incident radiation and does not reflect any

radiation. At thermodynamic equilibrium, blackbody emits all the radiation at any

Radiative Properties of Semiconductors

1-5

given temperature. Blackbody is used as a reference in thermal radiation to measurethe absorption of a real body.

Stefan Boltzmann law provides a relationship between the energy radiated byblackbody and its temperature. The law states that the energy radiated byblack body ( j) per unit area is directly proportional to the fourth power of itstemperature (T ).

σ=j T (1.20)4

where, σ, the Stefan Boltzmann constant is equal to 5.67 × 10−8 W m−2 K−4.This law implies that the energy radiated by objects at higher temperatures is

greater than the energy radiated by those at lower temperatures.Planck’s radiation law describes the relationship between the spectral density of

the electromagnetic radiation emitted by blackbody at thermal equilibrium andtemperature T. The spectral density (Bλ), in terms of wavelength, is given by therelation:

λλ

=−λ

B Thc

e( , )

2 1

1. (1.21)

2

5 hckT

Spectral density (Bv), in terms of frequency, is given by:

ν ν=−νB T

hc e

( , )2 1

1. (1.22)

5

3 hkT

The Planck’s law describes the radiation distribution that peaks at a certainwavelength or frequency. The peak shifts to shorter wavelengths or higherfrequencies for higher temperatures, and the area under the curve increases withincreasing temperature.

Planck’s distribution curves are shown in figure 1.4. Wein’s law states that theblackbody radiation curve peaks at different temperatures for a given wavelength

Figure 1.3. Illustration of radiance and solid angle [4].

Radiative Properties of Semiconductors

1-6

and is inversely proportional to the temperature. This is as a result of Planck’s law.Mathematically, Wein’s law is given by the relation:

λ = bT

(1.23)max

where, b is the Wein’s displacement constant equal to 2.898 × 10−3 mK.

1.2.6 Radiative properties of semiconductors

Study of the radiative properties in semiconductors is very important and useful formaterials and device processing. For real-time process monitoring and control, thesemiconductor industry requires studies on the radiative properties of wafers. In thefabrication of advanced semiconductor devices, techniques such as rapid thermalprocessing (RTP) are often employed. This technique requires the understandingand modeling of thermal radiative properties of semiconductor wafers [2]. Spectralemissivity of wafer affects the amount of radiation emitted. At any given temper-ature, the hemispherical emissivity affects the heat loss by radiation from the wafer.

Semiconductorwafers are laminatedwithdifferent typesof coatings.Understandingthe radiative properties of these coatings is critical for the successful implementa-tion of pyrometers. In applications of pyrometry, the knowledge of emissivity ofsemiconductors, as a function of temperature and wavelength, is required. Severalother factors such as surface roughness, doping, presence of coatings also influencethe emissivity of the semiconductor wafer. Understanding the radiative propertiesunder varying surface conditions and temperatures of wafers is critical in semi-conductor process monitoring and control.

Semiconductors, when heated to the desired operating temperature, emit radia-tion. The free carriers present in the semiconductor alter the infrared absorptionwithin the material and hence the resulting emissivity.

Spectral emissivity is a complicated function of:a) intrinsic emissivity of the semiconductor,b) extrinsic emissivity of the layers or thin films on the semiconductor,c) optical properties of the reflective chamber walls and lamp radiation that

might transmit through the wafer and get detected by the pyrometer.

A combination of all these is defined as the effective emissivity.

Figure 1.4. Planck’s distribution [5].

Radiative Properties of Semiconductors

1-7

References[1] http://www.mpoweruk.com/radio.htm (accessed 27 September 2016)[2] Table of emissivity of various surfaces, Mikron Instrument company, Inc.[3] Incropera F P, DeWitt D P, Bergman T L and Lavine A S Fundamentals of Heat and Mass

Transfer (New York: Wiley)[4] http://nptel.ac.in/courses/105104100/lectureD_23/D_23_4.htm (accessed 6 November 2016)[5] https://physics.stackexchange.com/questions/279561/radiometric-quantity-radiance-along-

the-ray

Radiative Properties of Semiconductors

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