2 Optical Theory for Semiconductor Characterization

2.1 Electromagnetic waves in free space

Before presenting semiconductor optical properties, I will discuss the basic properties of electromagnetic waves. The relatively simple behavior of these waves in free space underlies their far more complicated interaction with solids. M y development here shows how to relate the electromagnetic behavior in the semiconductor to quantities characteristic of the microscopic nature of the solid, such as the frequency-dependent dielectric function ε(ω) and absorption coefficient α(ω). These can in turn be con­nected to the quantities actually measured spectroscopically: the reflectance, transmit-tance, and absorptance determined in infrared measurements; the scattered and fre­quency-shifted intensity seen in Raman experiments; and the luminescent intensity observed in photoluminescence work.

Maxwell's equations define basic electromagnetic behavior. Their solutions in free space describe undamped transverse waves, composed of perpendicular oscillating electric and magnetic fields, propagating at the fixed speed of light. In the complex environment inside a solid, however, the electromagnetic field interacts with the constituent charges, to produce new phenomena which do not exist in space; the waves are damped as they impart energy to the solid, the speed of propagation depends on frequency, and longitudinal electromagnetic waves may occur.

In this chapter I review Maxwell's equations in free space, then focus on their solutions in a solid, which require the constitutive equations. These relate current and charge to electric field, hence specifying the physics of the electromagnetic-solid interaction. The presentation is general, except that I ignore magnetic effects. After this discussion, I derive expressions for the spectroscopic quantities reflectance, trans-mittance, and absorptance, and other optical parameters important for characteriz­ation measurements. The next chapter fleshes out the formal approach by inserting into the equations specific microscopic processes in semiconductors.

Maxwell's equations, the complete classical description of light, are written using different systems of units. However , the internationally accepted convention is the SI or rationalized M K S A (meter, kilogram, second, ampere) system, which I use in this chapter. M y other conventions for the electromagnetic quantities are as given by Holm (1991), who has made a thorough study of this often muddled area.

Maxwell's equations in SI units are:

V Χ H = j + -

V x . - f

V D = ο (2.1)

V D = ρ V B = 0

8 Optical theory for semiconductor characterization

where H is magnetic field strength, j is current density, D is the displacement vector, Ε is electric field, Β is magnetic induction, ρ is the free carrier charge density, t is time, and V is the vector gradient operator, which in cartesian coordinates is λ d θ θ i— + j — + k—. These equations present a remarkably concise description of electro-magnetism, giving them impressive formal power. However, they are not in solvable form until the exact relations between D and E, and Β and H, are known. The basic definitions are:

D = ε0Ε + Ρ = εε0Ε (2.2) Β = μ 0 Η + Μ = μμ0Β (2.3)

where ε 0 is the permittivity of free space (8.85 x 10~ 1 2 C 2 N - 1 m - 2 ) , Ρ is electric dipole per unit volume, μ 0 is the permeability of free space (4ΊΓ Χ 10~ 7 Ν S 2 C~ 2), M is magnetic dipole per unit volume, ε is the electrical permittivity of a medium other than vacuum, and μ is the magnetic permeability of a medium other than vacuum. These formal constitutive equations cannot be written in detail without knowing the details of the medium; when they are so written, they contain all the physics of the microscopic electromagnetic-solid interaction. In all further analysis I exclude mag­netic effects. This eliminates some of the more exotic semiconductors such as Cdi-^Mn^Te, where the addition of manganese to CdTe imparts decided (and inter­esting) magnetic properties; but these exotics are rare enough to justify their omis­sion.

I consider first the simplest medium, vacuum or free space, where there is no free charge density or current flow, so j = 0 and ρ = 0. There is also no electrostatic or magnetic polarization, so the constitutive equations become

D = ε 0Ε (2.4) Β = μ 0 Η (2.5)

With this information inserted into equations (2.1), appropriate manipulation gives for the electric and magnetic fields

V E = * 7 (2.6)

and

V B = * ¥ (2.7)

which have the standard form of wave equations. These indicate a certain parallelism between the Ε and Β fields, and every electromagnetic wave includes both. Because I have limited our discussion to non-magnetic media, I concentrate on E, which causes the important physical effects in a semiconductor.

The standard solution for a wave equation of the form equation (2.6) is an oscillat­ory disturbance which propagates at a speed determined by the medium. The oscillat­ory form is conveniently represented by the complex term e ^ q r ~ ^ , where r is the displacement vector; / is time; the wave vector q points in the direction of wave propagation and has magnitude 2 π/λ = 2rf where λ is wavelength and / = 1/Λ is the wavenumber (expressed in number of waves per centimeter, or c m - 1 ) ; and ω is the radian frequency 2ττυ, with υ the frequency of wave oscillation in hertz. (The sign

Electromagnetic waves in semiconductors 9

convention in the expression e ' ( ' q ' r ~ f t ^ can take on four different configurations, as discussed by Holm (1991). I use the one favored by physicists.) In this solution, the velocity of wave propagation is ω/q.

It is easier to visualize the wave behavior, and the solution is still completely general, if the wave is taken as propagating along the positive ζ axis. Then the assumed solution is

E(z,0 = Ene'"'*2-^ (2.8)

where the vector E 0 defines the direction and maximum strength of the electric field. Substituting this into equation (2.6) yields

q2 = ε0μ0ω2 (2.9)

which gives for the electric field

Ε = E 0 e i £ u ( z V ^ ^ _ i ) (2.10)

Equation (2.9) is the first dispersion relation I derive between q and ω, or from the quantum viewpoint, between photon momentum and energy. This yields a constant wave propagation speed (independent of ω) c = ω/q = ω/q = l/Ve 0^o? which works out to 3 x 10 8 m s - 1 , the speed of light in vacuum. It was this result by Maxwell that first showed that light is an electromagnetic wave. But more complex dispersion relations arise in semiconductors, giving frequency-dependent velocities.

If Maxwell's equations are solved for B, the result is that the magnetic field accom­panies the electric field, propagating at the same speed, but oscillating perpendicular to E. Both Ε and Β oscillate perpendicular to the direction of propagation, making electromagnetic waves of the transverse type. The fact that they are transverse comes directly from the third of Maxwell's equations, which gives VO = e0V-E = 0 when ρ = 0; this gives immediately q-E = 0, that is, the Ε field is perpendicular to the direction of propagation. In a semiconductor, where VD is not necessarily zero, there is a different outcome. Note also that nothing in the solution for E(z,t), equation (2.10), causes loss of energy; a wave propagating in free space, and encountering nothing but more vacuum, never loses amplitude.

2.2 Electromagnetic waves in semiconductors

Unlike free space, a solid contains both mobile free charges and bound charges. The free charges in a semiconductor are the conduction electrons and valence band holes. The bound charges are cemented into the lattice structure; the atomic charges com­prising the lattice itself, and the inner electrons tightly localized at the atomic cores. The current and charge density due to the free charges, and the polarizability of the bound charges, all effect Maxwell's equations.

To see this, first consider only the bound charges. They produce a dipole moment per unit volume, a polarization P. In the first approximation, the magnitude of Ρ is proportional to the magnitude of E, and Ρ lies in the same direction, so the relation between Ρ and Ε becomes

Ρ = χε0Ε (2.11)

where χ is called the 'susceptibility'. The full relation is more complex, with Ρ depend-

10 Optical theory for semiconductor characterization

ing on higher orders of E. Considering only the linear approximation, the constitutive equation (2.2) becomes

D = ε 0 Ε + Ρ = ε 0(1 + * ) E = ε ΐ 3 ί ( ω ) ε 0 Ε (2.12)

where e]at = 1 + χ is the dielectric response function that describes the polarization. It is called e l a t as a reminder that it comes from electrons bound into the lattice and the lattice atoms themselves. Their polarization depends on the frequency of the electric field, as will be shown in the next chapter, which accounts for the frequency depen­dence of S{at((u).

The effects due to the current of free electrons and holes could also be put into equation (2.12), but it is enlightening to consider them from a different perspective. The charge current can be related to the electric field by the microscopic version of Ohm's law:

j = σ(ω)Ε (2.13)

where σ(ω) is the conductivity, in general a frequency-dependent or a.c. quantity. A s in the relation between Ρ and E, more complicated non-linear relationships, and even spatially non-local, relationships are possible, but I will not treat them. Substituting the linear constitutive relations (2.12) and (2.13) into Maxwell's equations (2 .1) , and manipulating to eliminate B, gives the wave equation for the electric field Ε in a solid:

V χ V x Ε = V 2 E - V (V · E) = ε1άίε0μ0 -jp: + ο-μ0— ( 2 1 4 )

where the term V 2 E - V ( V - E ) is a standard expansion of the double cross-product V x V x E.

This equation for Ε differs from the result in free space (equation (2 .6 ) ) , in two important ways. One is in the term V-E , which was zero in free space but may be finite here, suggesting that longitudinal waves exist. The second is the last term on the right, which includes the current of free charges. T o understand these new features, it is again useful to assume a plane wave solution Ε = Eoe 1 ^ 2 "*^, the substitution of which in equation (2.14) gives:

q2E - q(q -E) = ελΆϊε0μ0ω2Ε + ο~μ0ΐωΕ = ω2ε(ω)ε0μ0Ε (2.15)

In the term on the right, the newly defined quantity ε(ω) is the total dielectric response function, given by

/ \ / \ ι ίσ(ω) ε ( ω ) = £ , a t (a>) + ( 2 1 6 )

It combines the bound charge polarization effects and the free carrier current effects in the semiconductor, and is frequency dependent because of είαί(ω), σ-(ω), and the denominator in the second term. It turns out to be the central quantity that describes the electromagnetic-semiconductor interaction, in the linear response limit.

Unlike equation (2.6) in free space, equation (2.15) cannot be solved immediately for q versus ω because of the longitudinal term q ( q - E ) . But the solution becomes clear if Ε is expressed as the sum of transverse and longitudinal components:

Ε = Eti + Eqq (2.17)

where t is a unit vector in the x-y plane perpendicular to the direction of propagation,

Electromagnetic waves in semiconductors 11

and q is a unit vector along q. With this decomposition, equation (2.15) becomes

- ε(ω) - qz Ett + | " j e(co)Eqq = 0 ( 2 1 8 )

Because the two terms in this result are linearly independent, equation (2.18) is satisfied only if the coefficients of t and q are each zero. Hence,

q =-?*(*>) ( 2 1 9 )

is the generalized dispersion relation for transverse waves in a solid of total dielectric function ε ( ω ) , and

ε ( ω ) = 0 (2.20)

is the condition for longitudinal modes to exist. Both these abstract statements will take on more meaning when I derive explicit forms for ε(ω) in Chapter 3. The less familiar condition (2.20) for longitudinal waves turns out to be important for the infrared and Raman analysis of semiconductor phonons and free carriers.

Before obtaining the explicit results, however, the dispersion relation for the trans­verse waves (equation (2.19)) can be put into a form with more physical meaning. First it is crucial to recognize that ε(ω) is a complex quantity e R + isi by its fundamen­tal definition (equation (2.16)) , even if s l a t and σ are themselves real. In fact, e l a t and σ each also include imaginary parts if the polarization and conduction processes in the semiconductor are lossy, which is always the case. This means the magnitude q of the wave vector is also a complex quantity q = g R + iq\. If this is inserted into the plane wave Ε = Ε 0 ^ ζ _ ω ί ) used to solve equation (2.14), the result is:

Ε = E o e - ^ V ^ 2 - ^ (2.21)

This makes it clear that the imaginary part of the wavevector derived from equation (2.19) is related to the damping of the electric field as it penetrates the medium, whereas the real part describes the propagation of the electromagnetic wave in the medium.

These observations can be put into a form more familiar for optical usage by defining a new quantity called the complex index of refraction, h = η + ik, as:

h= n + ik = VEJÙJ) (2.22 )

Then, combinin g equation s (2.19) , (2.21) , an d (2.22 ) give s

Ε = E o e - ^ V ^ - " ' ) (2.23)

where q0 = ω/c is the wavevector in vacuum. Equation (2.23) gives immediate physi­cal meaning to the complex refractive index and the complex wavevector. It shows that the real part of h is related to the propagation of the wave through the definition of propagation velocity ν = œ/(nq 0). Henc e η = civ, the ratio of the speed of light in vacuum to that in the solid, the familiar elementary definition for refractive index. The imaginary part of h is related to a new effect which did not appear in free space; the decrease in electric field as the wave penetrates the medium, expressed by the damping term e ~ ^ ° z . This accounts for the name given to k, which is the 'extinction coefficient'. It will become more meaningful when I consider the transmittance of a semiconductor in the next section.

12 Optical theory for semiconductor characterization

2.3 Quantities for semiconductor spectroscopy

2.3.1 Reflectance, transmittance, and absorbance

It should be clear by now that if one knows the frequency-dependent dielectric function of a solid, one knows everything about its microscopic optical response. Actual measurements, however, are made under macroscopic conditions. Light shines on the surface of a sample, usually a slab with flat faces. Standard spectroscopy, such as is carried out in the infrared, measures the amount of light reflected at the front surface, or transmitted through the back surface, which depend on the geometry of the sample. Reflectance and transmittance are not what is measured in photolumines­cence and Raman spectroscopy, but the behavior of the light at the air-semiconductor interfaces is still important to determine the conditions and meaning of the measure­ment. For any kind of spectroscopy, if the sample is not a slab of a single material, but consists of layers of semiconductors, each interface between different materials affects the impinging electromagnetic waves.

The behavior of electromagnetic waves at interfaces is determined by the boundary conditions which accompany Maxwell's partial differential equations. From these, it is possible to derive the fraction of the incident light which is reflected at an interface between two materials, the reflectance; and the fraction transmitted through the interface, the transmittance. Consider a typical sample, a slab of thickness d formed of a single type of semiconductor, as shown in Fig. 2.1.

The reflectance and transmittance depend on the angle of incidence of the radi­ation. A t any angle other than normal incidence, the behavior at the interface also depends on the polarization of the light. I consider only normal incidence, but the full equations can be found in many sources, such as Palik and Holm (1979). A t normal incidence, however, the polarization is not relevant. I take the interface as perpendi­cular to the direction of wave propagation, the + z direction. A t the front surface, the

Ί 'R

/ n2+ ik2

\

\ d

I

n3+ /7c3,

Fig. 2.1 A semiconductor sample of thickness d (medium 2) between media 1 and 3, showing incident, reflected, and transmitted intensities. Each medium is described by its complex refrac­tive index rij + ikj (j = 1, 2, 3). These determine the reflectance and transmittance of the sample.

Quantities for semiconductor spectroscopy 13

Fresnel amplitude coefficients for reflectance and transmittance, derived from Max­well's boundary conditions, are:

— ELL — ^ 1 ~ ^2

r ~ Ex " nx + n2 (2.24)

t = E± = 2nx

where E{, ET, and Et are the incident, reflected, and transmitted electric field ampli­tudes respectively, and hi and n2 are the complex refractive indices of the two media. What is measured in an actual experiment is the fraction of the incident intensity which is reflected, denoted by R\ or the fraction of the incident intensity which is transmitted, denoted by T. Intensity is proportional to amplitude squared, so that R = A T * = \r\2 and Τ = tt* = \t\2. T o put this in familiar form, consider medium 1 to be air, with Πχ = 1 and kx = 0, and the slab to be a semiconductor with η = η + ik. Then the reflectance is:

R = = (n- I)2 + k2

(n + If + kz (2.25)

This gives only the first surface reflectance, which is also the complete reflectance for an ideal sample which is semi-infinite along the ζ axis, or when the sample behaves as if its back surface were infinitely distant. This is approximated by a finite sample which absorbs light heavily, so that little light reaches the back surface, which then hardly affects the front surface reflection. Another approximation which often holds, at least over part of a spectral range, is that η > 1 and k « 1. Then equation (2.25) becomes:

Ρ = (n ~ If (n + If (2.26)

In this case, the reflectance behavior represents the real part of the refractive index and not the absorptive part. But it is helpful to remember that the reflectance can be total, with R = 1, under two very different conditions: whenever k » n; and near a longitudinal mode, where ε(ω) = 0 and hence η = 0.

There are cases, however, where the finite slab has virtually no absorption, so its extinction coefficient k is small. This can be a good approximation for a pure, high resistivity semiconductor at wavelengths far from its gap and its lattice absorptions. Then the behavior at the rear interface must also be included. Light is reflected and transmitted at the back surface, as determined by equation (2.24) with the appropriate refractive indices. Some returns through the front surface, to change the reflectance. A striking new effect comes when waves reflected from the front combine with those reflected from the back to give interference fringes. For this situation, the reflectance R' and transmittance Τ for a semiconductor slab in air become:

D , 4R sin 2 θ K ~ ~ - ο 2 Λ 2 _l AU2^2

Γ =

(1 - Rzf + 4f l 2 s in 2 6 (2.27)

( i - * ) 2

(1 - Rzf + 4f l 2 s in 2 9

where θ = ωηά/c is the phase angle for the beam travelling through the film of thickness d, and R is the front surface reflectance from equation (2.25). These ex-

14 Optical theory for semiconductor characterization

pressions clearly show the effects of interference in the sin θ terms, which produce periodic structure in the spectrum. The reflectance has a minimum and the transmit-tance a maximum when sin0 = 0. For spectral data as a function of wavenumber/, reflectance minima and transmittance maxima occur at the values of / = mllnd, where m = 0, 1, 2 , . . . . The spacing between two such successive features is Af = \llnd. Hence, the observation of such interference fringes gives a simple way to determine layer thickness if η is known.

Equations (2.27) become inordinately complex for a slab with significant absorp­tion. There is, however, a simplifying case which often occurs in actual measurements. If the resolution of the spectrometer is larger than the spacing between the inter­ference fringes, then the equations can be averaged over the phase angle. This yields, for the transmittance of an absorbing slab

[(1 - R)2 + 4 / ? s in 2 ^ ]e" t t r f

1 - R2e~2ad (2.28)

where two new quantities have been defined: φ = tan - 1 [2A:/( l - n2 - k2)] and the absorption coefficient

2u)k . r t

α = — = 4φ ( 2 2 9 )

where the term on the right is convenient when wavenumber fis used. The term sin φ in equation (2.28) is often small enough to ignore, leading to a yet simpler form which is often quoted:

(1 - R)2Q-ad

1 - R2e~2«d (2.30)

Equation (2.30) shows that the intensity transmitted through the slab is affected both by the front surface reflectance and by exponential loss, tracing back to equation (2.23) which includes the extinction coefficient. The meaning of the term e~ad be­comes clear if I consider equation (2.30) in the limit where there is no front surface reflection (R = 0 ) , so the transmitted intensity is not reduced by reflection losses. Then the intensity leaving the back surface It is related to the incident intensity Ix by

It = IiQ-ad (2.31)

which means the absorption coefficient a is the fractional change in intensity per unit length of penetration, due solely to absorption in the semiconductor. This direct connection to physical processes makes a a more useful spectroscopic quantity than the transmittance. It is also often true that the absorption is proportional to the concentration TV of the absorbing entities, which can be useful for quantitative charac­terization. It is then sometimes helpful to define an absorption cross-section σ = al Ν. If the transmittance is measured under appropriate conditions, a can be found from equation (2.30) when front surface reflectance R and thickness d are known.

Equations (2.25) to (2.30) cover the most important cases seen in practice for a slab of bulk semiconductor in air. There are analogous closed form expressions for struc­tures with two or three layers, the algebraic complexity of which dictates that they be evaluated by computer. In most cases, it is more productive to use a computer program which traces through the Fresnel coefficients (equation (2.24)) at each inter­face, eventually arriving at a total reflectance or transmittance for the structure. This

Quantities for semiconductor spectroscopy 15

'interface-by-interface ' approach has the advantage that it can be applied to as many layers as necessary, making it possible to analyze layered semiconductor microstruc­tures of any complexity. But, as the next section shows, depending on the wavelength, it is not always necessary to analyze the entire structure.

2.3.2 Penetration depth

Equation (2.31), which defines the absorption coefficient a, can be cast in another form useful for considering what optical radiation actually probes in a semiconductor structure. A quantity called the skin depth or penetration depth can be defined as δ = l /α , to give at depth ζ within the material:

A = he'* (2.32)

so that δ is the distance over which the incident intensity falls to 1/e of its value at the surface of the semiconductor. This characteristic length is useful to estimate how far light penetrates into a semiconductor, if it is kept firmly in mind that δ depends on frequency and on the semiconductor parameters. In the visible light region, δ is typically 0.1 — 1 μηι in silicon and 0.1 μιη in G a A s , according to Nakashima and Hangyo (1989). In the middle and far infrared, the strongest absorption likely to be encountered is about 1 0 4 c m _ 1 , giving δ > 1 μηι. Hence infrared radiation generally penetrates most semiconductor epitaxial layers and microstructures, the thickness of which rarely exceeds a few micrometers. Raman and photoluminescence spectroscopy are more nearly limited to surface regions. However , the frequency dependence of δ means that in a Raman experiment a change in the exciting wavelength changes the region which is characterized. I give values of δ for several materials in Table 3.1.

2.3.3 Spatial resolution

In applications where spectral information is obtained over the surface of a semicon­ductor wafer, it is important to know the spatial resolution of the map. The limiting factor is diffraction, the smearing out of sharp boundaries due to the wave nature of light. Here standard optical theory is helpful. For light of wavelength λ passing through a lens of focal length F and diameter Z) , the diameter d of the minimum size spot which can be formed is (Hecht and Zajac, 1979):

FA D (2.33)

which can also be expressed in terms of the/-number a s / /# = FID. Typical / / # values for a photoluminescence or Raman system range from 1 to 8. For an experiment excited by the 488 nm line from an A r h laser, the spot size could therefore be as small as 0.6 μπι. Although this ideal value will probably not be attained in practice, the resolution in the visible is adequate to see spatial detail on a scale appropriate for device geometries. Such resolution is not possible at longer infrared wavelengths, but infrared maps can still give useful information on the scale of tens of micrometers to millimeters.

16 Optical theory for semiconductor characterization

2.3.4 Frequency, wavelength, wavenumber, and energy

Each region of the electromagnetic spectrum has its own conventions for frequency and related quantities. The regions themselves are defined by wavelengths, from the nanometers of the ultraviolet-visible to the micrometers of the infrared. Laser lines are also named by wavelength. In photoluminescence work, spectra are often plotted against emitted photon energy in units of electron volts ( e V ) for easy comparison with semiconductor gap and impurity energies, although wavelength and wavenumber are sometimes used. In infrared and Raman work, the unit of frequency is usually wave-number/, the reciprocal of the wavelength in units of c m - 1 , which gives quantities of convenient magnitude. Some recent far infrared work is given in units of terahertz ( 1 0 1 5 H z ) . Table 2.1 gives conversions among these quantities, in the hope of minimiz­ing confusion among the different usages.

T a b l e 2.1 Conversions among wavelength, frequency, radian frequency, photon energy, and wavenumber of light.*

Quantity λ (μηι) (10 1 2 Hz)

ω (10 1 3 rads _ 1 )

ήω (eV) / Λ

(cm" 1 ) 1 μηι 300 188 1.24 104

10 μηι 30 18.8 0.124 1000 100 μηι 3 1.88 0.012 100

101 2 Hz 300 0.628 0.00413 33.3 10 1 4 Hz 3 62.8 0.413 3330

l m e V 1240 0.242 0.152 8.07 l e V 1.24 242 152 8070

10 cm- 1 1000 0.3 0.188 0.0012 100 cm- 1 100 3.0 1.88 0.0124

1000 cm- 1 10 30.0 18.8 0.124 lO^rn" 1 1 300 188 1.24

* The equivalent for each quantity on the left appears under each column heading. Useful conversions: 1 meV = 8.071 cm - 1 ; photon energy = 1.239 μπιβΥ/λ, where Λ is wavelength.