FREE CARRIER MODULATION
AND AMPLIFICATION OF LIGHT
IN SEMICONDUCTORS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE DELFT OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H. J. DE WIJS, HOOGLERAAR IN DE AFDELING DER MIJNBOUW-KUNDE VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN ~P DINSDAG 13 OKTOBER 1964
DES NAMIDDAGS TE 4 UUR
door
RALPH MA YNARD GRANT
geboren te Detroit, Michigan, United States of America
DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. H. G. VAN BUEREN
A portion of the work described in this thesis was performed in the University of Michigan's Institute of Science and Technology, Willow Run, Micliigan, U.S.A., the major part in the Reactor Institute at Delft, The Netherlands. For the tech-nical and financial assistance received from these laboratories, the author ex-presses his thanks.
Chapter I Introduction
Chapter II
Theory of Optical Absorption By Photoelectrically Induced Free Charge Carriers
1. Introduction
13
27
27
2. Derivation of Absorption Coefficient 303. Derivation of the Conducti vity 34
4. Free Carrier Absorption (Quantum Mechanics) 38 5. Application To Photoelectrically Generated Absorption 45
a. Derivation of Transmission in aSolid Containing induced free Carriers
b. Determination of n'(O): steady State Density of Photoelectrically Generated Free Carriers c. Establishment of a Free Carrier Modulation
Parameter M*
Chapter III
45
52
63
Experimental Equipment and General Technique 67
1. Introduction 67
2. Optical Setup 67
3. Multiple Internal Reflection 70
4. Electrical Arrangement 74
5. Apparatus Comments 76
a. Introduction 76
b. The Need For High Intensity Infrared Sources 77
c. The Nernst Filament 78
d. The Carbon Arc 78
e. Monochromator Calibration 81
f. Constant Temperature Crystal Holders 83 g. General Technique of Obtaining T he Free Carrier
Modulation Parameter M 85
Chapter IV
P hotoelectricall y Jnduced Free Carrier Modulation and
Amplification of Light In Cadmium Sulfide 88
1. Introduction 88
2. Optical Properties of CdS 88
a. Absorpbed and Transmitted Wavelength
b. Free Carrier Lifetime and Surface Properties 89
c. Effective Mass Values 91
d. Mobility Values 92
e. Index of Refraction Values 92
f. Conclusions 93
3. Theoretica1 Computer Calculations 93
4. Free Carrier Modulation In CdS - Measured With the
Carbon Arc As A Source of Transmitted Light 95
a. Introduction 95
b. Methods and Results 95
5. Free Carrier Modulation In CdS - Measured With the Nernst Filament As A Source Of Transmitted Light 102
a. Introduction 102
b. Method and Results 102
6. Additiona1 Re1ated Results 108
a. Excitation Light Calibration for. Nernst Filament
Data 108
b. Chopping Of Excitation Light 108
7. Discussion of Results And Conclusions 111 8. Light Amplification By Stimu1ated Absorption(LASA) 123 Chapter V
Photoe1ectrically Induced Free Carrier Modulation of
Light In Silver Bromide 128
1. Introduction 128
2. Optica1 Properties of AgBr 128
3. Theoretica1 Computation of M for AgBr 131
4. Experimenta1 Setup 132
5. Discussion of Results and Conclusions 133
S~mmy lTI
Samenvatting 138
A
d'
e g H hs
p s T TT
r v < v>w
LIST OF SYMBOLS
: vector potential : velocity of light: diffusion constant for holes and electrons res-pectively
: average penetration depth of free carriers charge of the electron
energy between trap level and conduction band : electric field intensity
energy
generation rate of carriers (holes or electrons) per unit volume due to photons
generation rate from energy level E. to E. where
1 )
i and j may be replaced by c which refers to the conduction band energy level, v which refers to a valence band energy level, and t which refers to a trap energy level
magnetic field intensity
Plank's constant
capture cross section for a free electron when unoccupied
: capture cross section for a free hole when occupied by an electron : recombination velocity : absolute temperature : transmission : relative transmission : velocity : average velod ty : r.m.s. noise voltage : energy : absorption probability
10 S : a.c. signal vOltage
Lr.
SO : detector signal voltage due to transmitted light
I S S n
*
m m M I N nwith crystal corttaining no generated free carriers : detector signal voltage due to transmitted light with crystal containing generated free carriers : signal voltage at drum position n
: V-l
: conduction current : diffusion current
combined conduction and diffusion current for the type of carrier referred to by index i
: incident light intensity of wavelength À.
1
: transmitted light intensity where medium does not contain generated carriers
: transmitted light intensity where medium does con-tain generated carriers
initial light intensity of wavelength À.z
transmitted light intensity of wavelength À.
z Boltsmans constant
: imaginary part of index of refraction
: absorption coefficient for light of wavelength À.z
: diffusion length
effective electronic mass electron mass
modulation with multiple internal reflection geo-metry
modulation per single perpendicular pass of trans-mitted light beam
total steady state number of generated free carrier pairs per square centimeter of illuminations by excitation
number of incident quanta per square centimeter per sec.
equilibrium value of the electron density
a. 1 a o a j3 E E o E r
e
7): equilibrium density traps occupied by electrons : excited electron density
: 6. n
= n -
no= excess electron carrier density
: density .of imperfection (trap) centers: density of states in conduction band : density of states in valence band : real part of index of refraction : equilibrium value of the hole density : equilibrium value of excited hole density
: 6. p
=
p - p o= excess hole carrier density
. : final momentum: initial momentum : space charge density
: number of reflections at surface containing gener-ated free carriers
: a.c. voltage reading
: net rate of decay by recombination
: recombination or decay rate where subscript i and j
are spec ified as in g .. above
IJ
: absorption coefficient due to impurity scattering : absorption coefficient due to lattice scattering
: free carrier absorption coefficient due to generated carriers
: free carrier absorption coefficient due to naturally present free carrier
: total free carrier absorption coefficier.t due to both natural and generatep free carriers
: constant
: permittivity of the medium
: permittivity of free space
: relative permittivity (E = E / E )
r 0
: internal reflection dngle : complex index of refraction
H
v
v o-(cu ) 0-o 'T . n 12: absorption coefficient for light of wavelength À2
: transmitted wavelength
: excitation wavelength, capable of generating free
carrier from the valence band : fundamental absorption wavelength
: mobility of holes and electrons respectively : drift mobility
: micron (10-4 cm)
: permeability of medium : permeability of free space : relative permeability : frequency
: quantuin yield
: frequency dependent conductivity : d.c. conductivity
: angle at which the end of the crystal has been cut : bulk lifetime of holes
: bulk lifetime of electrons : effective lifetime of holes : eHective lifetime of electrons
eHective lifetime of domina te carrier (usually electrons)
'T mean time between charge carrier collisions
c
'I' 1 : transmitted light irradiance in watts/cm2
'1'2
: excitation light irradiance in
watts/cm2 cu = 27TV : circular frequencyChapter I
INTRODUCTION
The control of the intensity óf infrared radiation transmitted through the semiconductors germanium and silicon, by variations in the concentration of electrically induced free charge car-riers, which absorb the radiation being transmitted, has been studied bya number of research workers since the early 1950's. The fact that free charge carriers can absorb electromagnetic radiation was known long before free carrier absorbtion was used as a means of controlling the transmission of radiation through semicond uctors.
In 1880 Lorentz(l,2) showed that a qualitative theory of the optical properties of ins ulators could be obtained by ad opt-ing a simple atomic model. This model was based on the as-sumption that insulating materials contain electrons which are bound to equilibrium positions by harmonic forces. This theory was extended by Drude, Zener and Kronig (3) to the case of metals containing free electrons. N ow, the qualitati ve absorp-tion coefficient of a semiconductor for wavelengths longer than
. the fundamental absorption edge, (wavelength longer than that which is required to induce a valence to conduction band tran-sition) can also be obtained from this theory. Hence, it is to be expected that at such wavelengths absorption, in intrinsic undamaged material, should be predominately due to free charge carriers; and recent inves tigations have prod uced an essential verification of the qualitative aspects of this theory. (4,5)
On the basis of this principle Lehovec (6) was the first to
propose, in the literature, the possibility of electrically induced free carrier mod ulation of infrared radiation. It is interes ting to note, however, that Wallace (7) pointed out the same technique
to a group of colleagues much earlier (in 1949), but did not publish his technique. In a paper preseilted to the Institute of Radio Engineers in 1952, Lehovec described a device based on what he called the "principle of a photo amplifier" which consisted of a photomodulator and a phototransistoT. He sugges-ted carrier injection into p-n junctions as a means of creating large charge carrier densities , and he described his principle as follows: 'A light beam of wavelength longer than that of the absorption edge of the semiconductor is passed through the region in which carriers are injected Oeft part of Figure 1.1).
14 L~HT BEAM>~~~~~~~~~~~ (unmodulated À- 1,5 microns) LIGHT BEAM (modulated)
FIG. l.I. Principle of a photoamplifier, consiSting of a photo-modulator and a phototransistor (after Lehovec).
The intensity of the light beam emerging from the semiconduc-tor is mod ulated by the modulated absorption of the semicon-ductor due to the injected carriers. The modulated light beam can be transformed back into an electric signal using, for ex-ample, a phototransis tor (rig ht part of F igure 1.1)'.
In the same year G ibson (8) published a paper containing
èxperimental evidence of carrier injection mod ulation where the technique was essentially the same as that of Lehovec's (see Figure 1.2).
! - + --1I-_ IIECHANICAlIITEIHIUPTIN8 DIK
FIG. 1.2. Experimental arrangement of Gibson
The absorption of infrared radiation by injected free carriers in germanium p-n junctions was olso reported during this same year by both Newman (g) and Briggs and Fletcher (I 0), but for the purpose of studying the free carrier absorption characteristics, not for the purpose of modulating light. Since Lehovec's
origi-nal proposal of the technique of modulating light by carrier injection and the demons tration by Gibson, many additional refinements have been reported by Gibson (11,12) and Kruse
(13,14).
Methods of electrically modulating infrared light beams other than by carrier injection have also been developed in recent years, such as modulation by: varying mobilities using high electric fields, electro-optical Kerr effects, magneto-optical Faraday effects and high electric field absorption edge shifting effects. These methods, including carrier injection modulation, have been recently sUl!lmarized by Moss (15).
Harrick has employed the experimental arrangements of
Lehovec and Gibson; not for the purpose of modulating infrared radiation, but rather to analyze the absorption of th is radiation as a means of measuring carrier density characteristics imme
-diately adjacent to current carrying p-n junctions. The current versus carrier density characteristics provided him with a sensitive test of p-n junction theory (16,17,18). He has also
employed this technique to determine various semiconductor
surface properties , s uch as recombination velocity (19,20).
FIG. 1.3. Schematic diagram of the infrared set-up used to probe out carrier distributions in semi-conductors ( After Harrick).
Harrick in 1956 (21), followed by Huldt and Staflin in 1958 (22,
23) and Kessler in 1959 (24) have demonstrated that infra-red absorption by free carriers, produced in germanium and
silicon by photo effects, can also be quite useful in the
deter-mination of a number of semiconductor bulk properties; such as
16
z
l.
Inf,~.d
V'S,;,~I<::f-t..-.1
-
.
---,.-9
~
FIG. 1.4. Schematic diagram of lifetime
measuring technique (after Harriek).
"
recombination and scattering mechanisms, valence band struc-ture and absorption cross sections. F igure I.4, which is the experimental arrangement used by Harrick, and Figure 1.5, which shows theresults obtained by Huldt and Staflin (25,26),illustrate this statement. It is interesting to note in F igure 1.5 that the relative amount of free carrier absorption although clearly dis-tinguishable is quite small in magnitude.
-.
% uu.:
lt
1.00 o u vi m 0.05 ~ La. o 0: ~ 0.02 WAVE NUMBER 3 -1 "10 CMFIG. 1.5 Increase, due to illumination, of the absorption coefficient of germanium as a
func-tion 'of wave number. Curve b: antimony-doped,
7T-type Ge (resistivity 20 ohm xm). (After Huldt and Staflin).
The new aspects of the$e investigations, compared to the usual techniques, was that now the free carriers were photo-electrically injected by a source of visible light rather than
electrically injected by a p-n junction. Kessler (27), in on attempt to verify con tradictory resul ts presen ted in 1953 by Newman (28), Briggs and Fletcher (29) and Gibson (30) for absórption measurements on e1ectrically injected carriers, hos meas ured absorption cross sections for free carriers in silicon and germanium by the photo generátion technique.
1t is th us a weIl established fact that free carriers con be photoe1ectrically produced in the elemental semiconductors when they are illuminated with light hoving a wavelength shor-ter than, or equal to, the fundamental absorption edge wave-length. T he above photo gener(]tion experiments have demon-strated that free charge carriers con be generated in sufficient quantity to bring about a noticeable absorption of infrared radiation, and despite the fact that the absorptioll was not very great in magnitude it appeared to be a promising manner in which to modulate infrared light beams.
Photon induced free carrier modulation of infrared light in germanium was originally observed and reported by the pre-sent author (31-33) in 1962 and was described by him in the following terms. Light hoving a wavelength shorter th on the fundamental absorption edge wavelength is readily absorbed, whereas light with wavelength À. 1 longer than the fundamental wavelength is not readily absorbed by other than free carriers. T he free carrier absorption coefficient a, as predicted by the Drude-Zener theory is,
1.1
where K is a constant and n is the concentration of free carriers. The free carrier absorption coefficient is proportiona1 to n, and hence varying the concentration n by the absorption of ionizing radiation can thus result in a change in the absorption of radia-tion with wavelength 1\.1. A device based on this principle, in which the optical density for radiation at one wavelength can be changed by the effect of radiation at another wavelength, is called a modulator. When Fresnel losses are disregarded, the transmission T of such a device will be
18
-Knr-2 d
T = J/J =.e 1
o 1.2
w hen d is the radiation path 1ength in the device, J is the transmitted light intensity and Jo is the incident intensity. If
the free carrier density generated by incident short wave1ength radiation is appreciab1y greater than the equilibrium carrier density, n in the above equation can be replaced by ns' the density of free carriers produced by radiation.
The relative free-carrier modulation depth is defined as the free carrier induced absorption , or
-kn
r-
2d'M=l-T ~l-e s I
s 1.3
where Fresnel ~osses have been neglected. The modulation depth is crititall~ dependent upon ns ' hence to produce a sig-l1:if~cant change in carrier density an intense light source is used so as to produce a maximum photon flux density. The increase in the number of free carriers/cm2 cis a result of the absorption of short-wavelength li.ght is equal to G7 and
1.4
w here G is the s teady state rate of generation of free carriers
per unit area due to the absorption of short wavelength photons, d' is their average penetration depth and 7 is the mean effec-tive lifetirne of the carriers near the surface. Now, by using experimental data reported for n-type germanium the value of the constant K in the above equations can be determined. Thus, at a given wavelength, M can be readily estimated as a function of incident shortwave power.
As an example, we compute the theoretica I modulation depth for a typical practical case. To deterrnine the free carrier gener-ation rate G per square centimeter, let us assume (for the pur-pose of later comparison with an experimental measurement) that 0.7 watts/cm 2 of excitation power with a wavelength less
than 0.7 J-L are absorbed by the germanium crystal. Wavelengths longer than 0.7 J-L are cut oH by a filter to make possible un-ammgious measurements of the free carrier absorption. The power absorbed is:
hc
GE=G-=- =0.7watts/cm2 1.5
i\
where E is the average energy of the yhotons, h is Plank's constant, c is the velocity of light and Ä. is the average wave-length of the free carrier exciting photons. Upon assuming that ~ ~ 0.55 J-L the generation rate G becomes equal to 2.0 x 1018 free carriers/cm2/sec. Further, taking T = 6.5 X 10-4 seconds (this value also being chosen since later comparison w ill be made to a sample having this lifetime), the value of GT" and hence nsd' becomes,
n d'
=
G T=
2.0 x 10 15,s 1.6
free carriers/cm2• The value of K(O.5 x 10-10) can be estimated by using the data for a and n presented by Spitzer (34) for n
s
type germanium (0. = 85 cm-1 for ns = 1.7 x 1018 carriers/cm3)
at a wavelength of ten microns. Hence the free carrier modula-tion to be expected in n-type germanium at this wavelength can be estimated according to the above criteria. For an average infrared wavelength of ten microns, the value of M becomes from equations 1.3, 1.4 and 1.6,
-Ki\ ~G T
M
=
1 - e=
0.06 1.7Hence in this case and if the Drude-Zener theory is assumed to be valid, a modulation depth of six percent should be expec-ted under the conditions mentioned.
General experimen tal techniques to meas ure the effect are the following. Infrared radiation from a Nernst filiment, chopped at 90 cycles per second; was focused by a spherical mirror onto
20
the surface of the germanium modulator, and passed through the long axis by multiple internal reflections and onto a thermistor bolometer (see F igure 1.6). A filter limited the detectors
effec-tive sensitivity to the 8 to 14 J.1. spectral range.
Mirror Water Cell
Vc?
IncandescentLamp Bolometer Coated Ge lnfrared Detector Bandpu. FUter
FIG. 1.6. Photon Induced Free Carrier Modulation Experiment 111 Germarü;}m.
T he 90-cps component of the voltage across the detector was a measure of the intensity of the transmitted infrared ra-diation falling on the detector. T he signal from the detector was amplified by appropriate electronics, and the 90-cps com-ponent was measured by a wave analyzer with a 6-cps band-width. The amplitude of the signal was then finally recorded on a strip chart.
The excess free carriers were generated by directing un-chopped radiation from two Sylvania Sun Guns onto the sides
of
the modulator. T he radiation was filtered by a water cellto remove photons whose wavelength was greater than 1.4 J.1.
(0,7 J.1. in the case where copper chloride is added to the water).
That is, except for ambien t laboratory ill umination, only photons
with energy sufficient to cause valence to conduction band transitions were incident on the sides of the modulator. Water was continuously circulated through the water cell to avoid heating effects •
The relative modulation M, given by
S1. . r. - S'1. . r.
1.8 S. 1. r.
where S1.r. . is the 90- cps signal voltage input to the wave ana-lyzer due to transmitted infrared radiation with the shorter-wavelength light oH, S' Lr. is the 90-cps signal voltage input to the wave analyzer due to transmitted infrared light in the presence of short wavelength light generated carriers. The average indication of the wave analyzer, RL r. is the sum of the signal plus noise according to:
~
1.r.. =v'S~
1.r. +V2 n 1.9where Vn is the noise voltage measured by the wave analyzer.
Hence, when equations 1.8 and 1.9 are combined, the relative modulation becomes:
YR~
-
V~
- YR'
.
2
_ v
2
1. r. 1. r. nM=
1.10YR~
v
2
1. r. n _ _ y.where (V n ") is the rms noise voltage, R.1. . r •• is the wave ana-lyzer meter reading due to noise and transmitted infrared light with the shorter wavelength light off and
R;.
r. is the wave ana-lyzer meter reading due to noise and transmitted light with the shorter ,,!,avel~ngth light on.Single crystal n type germanium with a resistivity of 47 to 51 ohm-cm and a bulk lifetime of 650 J.L sec. was cut into slabs 2 cm square and 1.25 cm thick. Forty-five degree angle fa ces were placed on the ends of the slab to fa·cilitate multiple in-temal reflections and to increase effective aperature area per given crystal thickness. The multiple internal reflection tech-nique need not be used; however, it was much easier to polish shorter crystals and use multiple internal reflections than it was to polish a long ultra thin slab for direct transmission. The germanium slabs were electrochemically polished (35) and chemically treated (36) to increase free carrier lifetimes which had been significantly reduced by diamond saw cutting and
22
The results of the experiment, obtained wi th 0.7 watts/cm 2 of filtered shorter wavelength radiation incident on the sides of the modulator, are shown in F igure I. 7. In this F igure an abrupt reduction in infrared transmission due to the generation of free carriers is noted when radiation from the modulating source is directed onto the modulator. The shorter wavelength light produces excess free carriers both by inducing direct op-tica! transitions and by heating the modulator. These two effects
result in modulation near unity, as shown in Figure 1.7. It was possible to distinguish cm almost mstantaneous photon-generated free carrier modulation from modulation due to thermal
excita-tion of carriers. For purpose of discussion, recorder tracing
30
Note: ModulatIng RadIatIon Filter-None No Coollng
60 90
Time . Seconcls ----..
FIG •. 1.7.- Transmission of uncooled modulator from 8 to 14 J.L, with unfiltered modulating radiation.
120
I. 7 can be separated into ti ve regions. Region I represen ts the infrared transmission through the modulator when the shorter wavelength light was not incident upon the sample. The value
of R. l.r. (equation 1.9) was 2.7 volts in that particular measure-ment. Region II, represents the rapid change in transmission due to free carrier modulation when the short wavelength light is turned on. There is a 54% decrease in transmission in that region. On the basis of the theoretical calculations of the mo-dulation depth which were previously given we would expect
to observe 100 instead of 54 percent. (The theoretical results calculated were 6 percent, however in the experiment the
transmitted light experienced 14 multiple intemal surface re-flections, R
=
14, and as shown in the section on multiple in-temal reflections in chapter III the theoretica I value for a sing Ie transmiss ion - which was the case calculated above -must be multiplied by2V2
R, to obtain the theoretica I M value for the case where intemal reflections are employed. Hence yielding a value of 100 percent as obtained with formula 3.5 of chapter lIl.) There is obviously poor agreement between the theoretical prediction and the experimental results however this might be accounted for, by considering the effective free carrier lifetime T. In the theoretical calculations we used forT the bulk lifetime of the particular germanium sample used.
However, the true value of T at the surface is undoubtedly lower,
and hence could account for the dis.crepancy. Better agreement
between experimental and theoretical .results were obtained
with another crystal, however before proceeding to that com- •
parison further comments wil! be made in re gard to region III
of Figure 1.7. T-his region répresents a slower decrease in trans-mission due to the thermal excitation of free carriers. In region
I
III the value of R L.r. goes down to the rfDs noise level, which is of the order of 0.1 volt. The value of ,R 1.r. as taken from the recorded meter reading is 0.1 volt, (V2n )Yo is 0.1 volt, and R'l .r. is 2.7 volts, hence trom equation 1.10 the modulation depth in region III approaches unity. T he thermal effe cts of regions III and IV can be red uced by either of two techniques: the sample can be placed in a gas flow bath at constant temperature, or saturated solutions of copper chloride can be placed in the water cell filter to cut out all wavelengths longer than 0.7 f-L.
When the latter technique was used, curves such as that shown in Figure 1.8 were obtained.
FIG. 1.8. Transmission of un-cooled modulator from 8 to 14 J.1-, with filter. H
20 solution of
CuC1 2 (t-cutoff
= 0.7 J.1-) used
to filter modulatiI)g radiation.,. 00
..
24
In ugure 1.9, distilled water was again used as a filter, but the sample was cooled by directing a stream of air over it. T he effect of the air stream was to reduce the change in tem-perature due to the modulating source. That the slow drift in transmission following activation of the modulator is a thermal effect can be seen by com?aring Figure 1.9 to Figure 1.7. The pronounced drift in transmission evidenced in Figure 1.7 is markedly reduced in F igure 1.9. Except for a change of sign, the electrical conductivity varies much as does the infrared transmission (see Figure 1.10). Figure 1.11 shows that cooling
Note: 1) Modulator Cooled by Alrstream 2) Modulatlng Radl-atlon fUter: dl.-tUled H20 (Acutof! = Lilt) / Modulatlng Souree On / Modulatlng Souree """"""' ... -...1 Of! FIG. 1.9. Transmission of cooled modulator from 8 to 14 Jl.. Modulator cooled by stream of.. air; distilled H 20 (Àcutoff
=
1.1 Jl.) used to filter modulating radiation.oL---~---~30~----~----~6·0 ' " Modulatlng Souree On . / Modulatlng /" Sou ree Of! OL---~----~3~O---~----~6~O Time - Seconds ~
FIG. 1.10 Electrical conductance of uncooled modulator.
the modulator results in minimizing the slow drift in conducti-vity due to thermal effects, just as with infrared transmission. There can be little doubt that the changes in infrared transmis-sion and the changes in electrical cond uctivity have a common origin. /COOI~ Alntram
i
UCM'"""'
/ $ource....
~ On ~"
Time .~on<b_"
- - - - 1 FIG. 1.11. Electrical conductance of modulator, with and without cooling.
A second sample of germanium was measured with approxi-mately 0.6 watts/cm2 incident upon 8 surface reflections where
the transmitted wavelength was 3.0 IJ-, and a 30 percent
modu-lation was obtained. T he theoretica 1 prediction was 11 percent where again it was assumed as in the previous calculation (equation 1.7) that T ~ 6.5 x 10-4 seconds, K = 0.5 x 10-10 and ns calculated from equation 1.6 was equal to 1.7 x 1018• A second experimental mod ulation value was obtained at 2.0
IJ-and resulted in a value of 24 percent. The corresponding,
calculated theoretica 1 value was 5 percent. The agreement between these experimental and theoretical va lues of modu-lation, 30 as compared to 11 percent at 3.0 IJ- and 24 as com-pared to 5 percent at 2.0 IJ-, were certainly as good as could be expected from the type of prediction made. One should not expect experimental and theoretical values of modulation to be closer to each other than one order of magnitude due to un-certainties in T.
The results of the above experiment clearly demonstrate that photon induced free carrier modulation of the transmission of infrared radiation is indeed possible. Modulations of 100% were observed, thereby bringing about a complete control of transmitted light by free carrier absorption which has been photo-electrically rather than electrically induced. These mo-dulators have notabie advantage over the ~dectrical ones de-monstrated by Gibson (38,39) and Kruse (40,41). For example,
26
no electrical contacts are attached to the crystal which can alter its basic properties. Also space charge effects, trouble-some to electrical modulators, are no longer present. In addi-tion the device con tains two dimensioncil iJTlage transfer charac-teristics which would be difficult to accomplïsh with a junction type device (42). A modulator in which photon induced free
carrier light modulation occurs is essenfiaIly, for purposes of analogy, a "Light Triode" where the modulating light, or "photon pump", has the same control over the transmitted light beam that the grid voltage in a vacuum tube triode pro-duces over the plate current. The change in light intensity of the incident primary light beam w hich passes through the semi-conductor slab can be controlled by varying the intensity of the secondary light beam (modulating signal) in a manner simi-lar to the change in plate current being controlled by the varia-tion in the grid voltage in the vacuum tube.
In the following chapters measurements, based in part upon the author's previously described work with modulation in germanium, shall be described and discussed for the modulation of visible,as weIl as infrared light, in other materials especially cadmium sulfide.
Attention will also he given to the
À-1 dependence such that a test of the Drude-Zener theory may be obtained. The primary objective, therefore, of this study is the optical control of light throughout the spectrum between the visible and the far infrared by a controlled variation in photon induced free carrier modulation, rather than by the previously demonstrated electrically induced free carrier modulation of near infrared in germanium and s.ilicon. T he periodic group IV semiconductors are in general not as practical as the periodic group I-VII and Ir-VI compounds which have fundamental absorption edges in regions of the spectrum which are of much greater practical si gnificance tor modulation purposes.
Chapter II
THEORY OF OPTICAL ABSORPTION DY
PHOTOELECTRIC.ALLY INDUCED
FREE CHARGE CARRIERS
1. Introduction
According to Spitzer (43) absorption in semiconductors can be separated into four categories, depending upon the mecha-nism of absorption. These four types of absorption are: (1)
Intrinsic or fundamental absorption which involves electron transitions between the valence and conduction bands and hence across the forbidden energy gap (2) Carrier absorption
in which electrons in the conduction (or valence) band undergo transitions between energy states, but remain in the conduction (or valence) band. In cases where there is an overlapping band structure, that is, when there is more than one energy band in a given electron énergy interval, the process of absorption by free carriers may involve transitions between these bands.
(3) Absorption associated w ith electron transitions between the
valence or conduction band and localized energy states in the forbidden gap. The existence of the localized energy levels may be attributed to the presence of impurities and lattice
imperfections. (4) Lattice absorption resulting from the exci-tation of lattice vibrations'. Bube (44) points out that a fifth weIl known type of absorption must also be included in this list, namely, (5) the formation of excitations. The possibility (6) of absorption via surface states should be mentioned, how-ever since little is known about its importance, we will not
consider it further (see however section 4).
Since light having a wavelength À. 2 ,shorter than the funda-mental absorption edge (the location of the fundamental ab-sorption edge can be somewhat vague in some materials hence it shall be placed at the wavelength at which the maximum change in the absorption coefficient takes place) (45) wave-length À.f ,can induce a valenee to conduction band transition in an intrin sic semicon ductor material, such ligh t is readily absorbed (see Figure Il.l). Light having a wavelength À.! longer than the fundamental wavelength À.f is not readily absorbed by this mechanism since i t cannot induce valence to conduction band transitions. Before considering the À.! > À.f
-
, E -=-o 1000 100 401.1 28 A Wavelength Less Than Àf 1.4 Wavelength At the Fundamental Absorption Edge A Wavelength Greater Than ~Ï À (micron)FIG. Il.l. Intrinsic absorption edge of pure
N-type germanium of 30
0.
cm resitivity.type absorption in detail, we make the assumption that the mechanisms (3) and (4) are not important in our case because a) localized energy states caused by the presence of impurities and lattice imperfections are not present in sufficient quantity to cause significant absorption due to electron transitions be-tween the valence or conduction band and localized energy states in the forbidden gap, ond (b) À 1 is not far greater than Àf such that infrared absorption due to lattice vibrations does
not occur.
Hence on the basis of these assumptions, the most signi-ficant mechanism th at causes absorption of light with a wave-length À 1
>
Àf should be that due to the interaction of the photons w ith free charge carriers. T he absorption of wave-lengths shorter than the absorption edge is relatively Ie ss affected by the presence of free carriers. Under consideration, then, is the optical absorption in intrinsic semiconductors of radiation incapable of exciting a transition across the forbid-den band gap.This type of absorption is qualitatively predicted by the Drude-Ze"ner theory. Lasser, Cholet and Emmons (46) have emphasized the fact that more recent quantum mechanical studies have produced no essential modification of the classi-cal theory, and that "none of the theories is as yet tn very good quantitative agreement with experimental absorption data". As a result of this difficulty it has been the tendency in recent engineering practice to adopt the classical theory of free carrier absorption which in many cases can, with some modifi-cations, yield reasonable good results.
A derivation will be given for the absorption coefficient,
a, of a semiconductor for light of angular frequency w utilizing
the Drude-Zener theory. In addition, some more recent modifi-cations of that theory will be discussed, which res ult in an expressIon which can be extended to predict the case of ab-sorption by optically generated free carriers in semiconductors.
The expressions for the frequency dependent absorption coefficient and conductivity, which are so commonly referred to in recent engineering literature as the basis of free carrier
absorption applications, are:
2.1
and
2.2
where,
a(w) = frequency dependent absorption coefficient,
o-(w) = frequency dependent electronic conductivity
0-0 = nef.L d; the d.c. conductivity (n represents the number of free carriers, e the electronic charge, and f.L
d
the drift mobility of the free carriers, where the drift mobility is defined as the mean velocity (in the direction of the field) of the carriers < v > per
unit field strength E or f.L d
=
<
v>
IE).30
N = real part of the index of refraction, c
=
velocity of light, ClndT
=
mean collision time between charge carriers andc
lattice.
These two equations, 2.1 and 2.2, are given,for examp1e, by Lasser et al. (46) as the theoretical basis for the develop-ment of a scanning system for infrared imaging which depends for its operation, upon free carrier absorption. Before applying these form u1as to the application of optica1 modulation we shall briefly summarize the classical derivations-{47-S1) which lead to equations 2.1 and 2.2.
2. Derivation of Absorption Coefficient
The theory of propagation of electromagnetic waves in conducting media without space charge is based on Maxwell's equations (for a definition of these symbols see the list of symbols at the beginning of this paper):
oH
curl E = - f-Lrf-L 0ot
oE
curl H = aE + E E rOot div H = 0 div E = 0 2.3 2.42.5
2.6
An important relalion required for later use can be obtained by fin ding curl CUrl E from these equations by differentiation from 2.3 and 2.4
oE
o2E
c'Url curl E = - f-L f-L {a - + E E '""t 2
since
curl curl
E
=
grad divE -
'V 2E=
-
'V 2Eone has
2.8
or upon considering propagation in the x direction only
2.9
A general running plane wave solution has the form
E = E
x 0
jw(t - x/v )
e x 2.10
This becomes a solution of our differential equation for a speci-tic form of v x' the velocity of propagation of the wave through the medium in the x direction. Substituting this general solu-tion into equasolu-tion 2.9 results in:
v x jw(t - x / v ) jw(t - x/v x) e x + /-L /-L r E E w2 E e 0 r 0 0 jw(t - x / v ) . cuE e x = 0 - J/-Lr/-L 0 0 2.11
w here, upon di viding out the exponen tial term and solving for
v x ,it becomes clear that the general solution, equation 2.10,
32
r 0 r 0
[
fJ- fJ- W(WE E + jU]Yo'
2.12
By definition TJ = c/v ,where TJ is the complex index of
refrac-x
tion, c is the velocity of light and-V
x is the velocity of pro-pagation of the electromagnetic wave in the x direction. Hence
the index of refraction is:
!
1 [fJ- fJ- W(WE E + jU)]Yo'Yl C r 0 r 0
'/ =
- - -
E E w2 + u2fJ-rfJ-0 r 0
By noting that in a vacuum, where TJ = E r c=l/~ o 0 2.13 becomes, where u ~ 0, 1
Y
1 [fJ- fJ- W(WE E + jU)] TJ- _ _ - - - r 0 r 0 - fJ- fJ- E fJ- (E E w)2 + u 2 r 0 o o r 0which is of the form
where and NK = Uil r-r /2w E 0 2.13 fJ- = 1 andu=O, r 2.14 2.15 2.16 2.17 2.18
Having now found these expressions, we proceed to deter-mine the energy flow per unit area for an electromagnetic wave, given by the Poynting vector of that wave which is thus pro-portional to the vector product of the electric and magnetic vedors. From relations 2.10 and 2.16 we have
c.JxK E=E e 0 c c.JxK H=H e 0 c where N - jK
l/v
= -x c xN j c . J ( t - - ) e c 2.19 xN jc.J(t -e c 2.20 2.21and we find for the amount of energy, W, transported per second through a unit area perpendicular to the direction of wave pro-pagation:
2c.J xK
Woee c 2.22
Consequently, the rate of flow of energy is
dW w,K
oe -2
W.
2.23
dx c
Thus, there is energy independent absorption, characterized by an absorption coefficien t,
2wK
a = - -
2.24
34
and from equation 2.18, considering that the magnetic permea-bility, f..L, r differs only very slightly from unity, we mo:y, with little loss in accuracy, obtain;
2NK=u/WEO 2.25
or
2.26
and hence
a = u /NCE 0 • 2.27
T his is equation 2.1, since N represen ts the real part of the
complex index of refraction.
3. Derivation of the Conducti vity
In order to find equation 2.2, that is to compute the
fre-quency dependent conductivity, we consider the effect of cm alternating electric field on the motion of the electrons in the
material.
The one dimensional equation of motion for a "free" elec-tron is
d2x dx . w t
m - - + y m - = - eEeJ .
dt2 dt 2.28
Since the vibrations of the electron, which are forced by the electromagnetic wave Eeiwt, undergo damping, the equation of
left siçle of equation (2.28) represen ts this force which depends upon the mass m, and the velocity through a damping constant '1, which Lorentz (52) supposed is inversely proportional to the mean length of time (or consequently proportional to a frequency) during which the vibrations of the particle can go on undistur-bed. We shall return to the discussion of '1 later.
The equation of motion, 2.28, can be solved by assuming a general solution of the form x = a e Jw!.. Hence the solution for the frequency dependent position of the electron is
j eE Jwt
x = - - - - - e
-'1+ jw mw
and the curren t density is
- dx ne2jm
J = ne - = - - - Eeiwt dt '1+ jw
This we put equal to 2: E eiw~, where
2.29
2.30
2.31
is the complex conductivity. The real part is what we have called u(w) earl ier. Thus
ne2 y u ( w ) = -w 2 + '1 2 m ne2 u ( w ) = -1 + (w 2 j'12). 2.32 m'1
This is equation 2.2 because it can be written as
cr
cr(w) = 0
1 + w2 72 c
2.33
where cr o
=
ne2/my and T c=
l/y (this is a formal definition ofr
which shall not be used further).c
Combining the two equations (2.1) and (2.2) one has for the frequency dep enden tabsorption coefficien t:
ne2 a(w) = : -CE Nmy o cr o = -C<)2 1 + -y2 1 a(w) cE o
N
w2 1 + -y2 •The quantity y follows from the quasistatie case. Then
my .
2.34
2.35
2.36
But from semiconductor theory we also know that cr 0 can be
written as: cr 0 = nejJ-d and thus e y + -rnjJ-d • 2.37 2.38
We can therefore write nefL d a ( w ) = -cE
N
o CT a(w) = _ _ 0_ CEN
o 2.39 1 2.40In 1951 Becker (53) used equation 2.40 for the frequency dependent free carrier absorption coefficient and found that it
yie1ded values which were approximately two orders of magni-tude lower than the observe.d absorption values in semiconduc-tors. More recently, considerably more extensive measurements have been made by Collins (54) and he also concluded, for the case of germanium, that "the observed absorption values are approxima tely two orders of magnitude higher than predicted by any theoretical treatment". The disagreement between the predicted and measured values pOinted out by Becker and Col-lins encouraged Fan and Frolich (55) , who had been directing their work, to pursue a more extensive theoretical investigation of free carrier absorption. The results of their study indicated that reasonable agreement between the experimental and pre-vious theoretical treatments could be obtained by replacing in 2.40 the free electron mass m, by the average effective mass m* and retaining the d.e. conductivity. For instance, in n type germanium they found good agreement between theory and ex-perimen t by taking m* = 0.10 m (56). Therefore we shall hence-forth also replace m by m* in equation 2.40, resulting in:
CT a(w)=~ cE
N
o 1 2.4138
4. Free Carrier Absorption (Quantum Mechanies)
The result of the Drude theory for the tree carrier absorp-tion coefficient, a, (equation 2.41), has been dèrived on the basis of the classical free eiectron theory apart from the use of the scalar effective mass m*, which according to Kronig (57) should be used instead of the free electron mass when one studies absorption in crystals where the electron moves in a periodic potential. The classical derivation is based on the assumption of a formal damping term or frictional force which is inversely proportional to the mean time between carrier collisions or proportional to the collision frequency of the electrons. In practice electrons in the conduction band of a crystal are slowed down in their motion owing to interaction collisions: (l) with the lattice vibrations (phonons) and (2) with
charged impurities or lattice defects.
The requirement of simultaneous conservation of energy and momentum during a transition trom one electron state in the conduction band to another under absorption of a photon tells us immediately that no absorption can take place without simul-taneous emission or absorption of a phonon, or without inter-action w ith a localized lattice pot~ntial. Let us· first I consider
the interaction with phonons· to be prevalent. According to Schmidt (58) the absorption of a photon by an electron can be described as a twofold process. During the first stage the elec-tron interacts with the photon, and during the next stage it in-tel'acts with the lattice. The sequence of these two stages can also be reversed. Let the initial momentum and energy of the electron be pand E , respectively, and the Hnal v'éilues of o
0
these quantities pand E. Since the energy carried by the pho-nons is small, we know that E '" E + hv where v is the frequency
o
of the absorbed photon. The probability of an absorption process is proportional to the squares of the matrix elements for the first and second transition, respectively, and to the density of states available near the energy of the end state E. The matrix element for the interaction of an electron with a photon is pro-portional to the momentum of the electron because the classical interaction potential between an electromagnetic wave and an
e
-electron is - A • p, where A is the vector potential of the mc
lightwave. The number of states in the conduction band in an energy interval dE around the energy E is, in the
approxima-tion of quasi-free electrons, given by
47T(2m*)3/2
n(E) dE
=
3VE
dE •h
Thl,ls the absorption probability can be written in the form
w
= C p2VEo
+ hv1 0 0
when first the photon is absorbed, or as
w
2 = C p2 VE + hvo 0
2.42
2.43
2.44
w hen firs t the interaction w ith a phonon has taken place. T he constant of proportionality
C,
contains the square of theo
matrix elements for this latter interaction, which can be des-cribed by a relaxation time or collision frequency independent of the electron momenturn. The total absorption probability is thus g iven by the sum of W 1 and W 2' hence upön adding equa-tions 2.43 and 2.44: 2.45 and by putting p2 = 2m*(E o + hv) 2.46 p2 = 2m*E o 0 2.47
40
we have
W = C' (2E + hv) VE + hv •
cr . 0 0 2.48
This ho1ds for e1ectrons with initia1 energy Eo' When we assume
that the e1ectrons in the conduction band have a
Maxwell-Bo1tzmann distribution of energies with temperature T
(non-degenerate semiconductor), we must average W cr over this
dis-tribution and obtain
1
00 E /kT(2E + hv) VE + hvVË e 0 dE
o 0 0 o'
o
2.49
Apart from the absorption process we shou1d a1so consider the
reverse, induced emission process where a photon is emitted
under the influence of the incident photon. This introduces a
factor {1 - è-hv /kT} before the int~ra1 sign._Since in the
case we are interested in hv > > kT
=
E , where E is the meanva1ue of E , (for near infrared radiation °hv
>
10-1<:3 erg and atroom temp:rature kT < 10- 13 erg) we can neg1ect the induced
emission factor and approximate the integra1 by
00
V7T
1
(hv) 3/ 2VE
e -Eo/kT dE = - (hvkT).3/2 2.50 o 0 0 2 ThusW
=C'(hv)3/2, cr 2.51is the absorption probability of a single photon with energy
h v
> >
kT which can be immediately compared to the classicalabsorption coefficient a. Similarly for this absorption
w
C' 8hv (kT) 1/2V1T
2.52a
In this latter case, however, one approaches the classical limit as hv ... O and correspondingly this formula should be equivalent to the Drude formula:
'I
2.53 a =
-7Tm*E o
for the classical absorption coefficient, a, as was derived
*
.
earl ier • It follows by comparison of equation 2.52 and 2.53
that
,V7T
C
= 8 hv(kT)I/2,
ne2 'I 7T E m*N cu 2 + '12 o 2.54Inserting C, into equation 2.51 one finds for the absorption-coefficient according to quantum mechanics, when h v > > kT:
Vi
=V7T
lfiW
a.a 8
r
kT
2.55This formula was first derived by Frolich et al. (59) and shows that, taking only lattice vibrations into account, the classical absorption coefficient should be multipled by the factor
V7T~
.
8
kTIn the frequency and temperature region we are interested in:
3 x 1013 :::;:vs:: 3 x 1014 sec-I, T =3000K, this factor varies * Since reference to and comparison of the resuIts of other authors who have given their equations in the rationalized MKS system of units will be consistantly given in this section, equation 2.34 which has been derived in the cgs system of units has been converted to MKS units (equation 2.53). The correctness of these equations is con-firmed by comparing the results for a given by Planker et al. (61) in cgs units and the-results given by Schmidt(58) in MKS units.
4.2
between 0.5 and 2 and is thus of the order of unity. However,
instead of an inverse square frequency dependence, one should.
rather expect a 1)-3/2, that is a >--.3/2 variation for the free
carrier absorption coefficien t.
Apart from the interaction w ith phonons, however, the scattering of electrons by charged physical or chemical lattiee
defects (impurities, color centers) might also be influential i!:.
determining the free carrier absorption. T he latter infl uenc,:·
has been investigated in some detail by Meyer (60). The
absorp-tion of photons in the neighborhood of charged impurity center< was treated as an inverse Bremsstrahlung process, in w hieh the eiectron is scattered by the charge of the center and theJ. makes a transition under absorption of the photon. It would lead us too far to present this theory here in detail since it is only an approximative theory and the formulas arrived at are
rather complicated. However Planker and Kauser (61) hav~
recently given a very useful summary of the consequences cf
impurity scattering and have compared these results with the formulas derived for lattice scattering as given above. FollowinJ their work, we express the absorption coefficient in term,::; of the quantity
kT kT
A = - >--..
hiJ hc
We then have for lattice scattering only:
n
- - - 'F1(A)
T2 Ji IN
and for impurity scattering only:
nf3
o2.S6
2.5:
2.5:3
Her'e A is a numerical constant of the order of 10-16 in ratio
na-lized MKS units, (mi
J
is the ratio between the free ·electronmass and the effective mass of the carriers, n is the free
carrier density, T is the temperature, /-Ll is the carrier mobility
when only lattice scattering takes place, /-L i this mobility whem
impurity scattering dominates,
IJ
o is a numerical constant ofthe order ·of unity and N is the optical refractive index. The
functions F 1 (A) and F i (A) depend on the approximations used.
For phonon interactions only, Kauer finds:
A
> >
1 Drude-Schmidt, F 1 '" A 2 1 A«
1 : Frohlich, Fl '" - A 3/2 •.
4
/
/
/
/
/
I ;//
.
/b///
'"
0.1FIG. !I.2. Comparison of absorption by free carriers.
- - - Classical theory
- - - - FreUch lattice scattering theory for 11.
«
1- - -Meyer impurity scattering theory
2.59
44
F or impurity scattering only, one finds from Meyer's theory, that for,
A ~ 1 2.61
and for,
A-+oo F i "" 6 A 2 In A. 2.62
These dependences have been plotted in Figure II.2. From this figure it follows that there are in principle appreciable differences to be expected according to the underlying scatter-ing mechanisms, espp.cially for values of A ~ 0.01 and A ~ 1. Since we are primarily concerned with values of A ranging between about 0.2 and 0.02, the differences are not so great in practice. In the absence of precise knowledge as to the predominate form of scattering we are unable to give preference either to the Frolich or the Meyer approximation. We shall therefore use the classical Drude formula, first because it is the simplest one and second because it seems to give more or less the average between two extremes where neither may be
appli-cabie to our problem.
However, if one would like to use our quantitative absorp-tion meas uremehts to be discussed in chapter IV, to derive for instance the effective mass of the charge carriers, one should first make s ure which scattering mechanism preva-ils, and further insert into the proper formula the proper value of the mobility. It was pointed out by Fan (59) that the difference between J.L 1 and J.L i should not be forgotten. T he choice between the mechanisms is facilitated when the precise wavelength dependence of the absorption coefficient has been measured. In the experimental cases to be discussed later in chapter IV we find for CdS on the average a À 1.5 (th us A 1.5) depen-dence, w hich points to phonon interaction as the important contribution to the mechanjsm of absorption +. In some cases we observe approximately a À 2 dependence, indicating a
contri-+ A À1.5 dependence hos a1so been observed in silicon by Kess1er(l44), indicating phonon interactions in silicon.
bution from impurity scattering. T his stands in agreement to former measurements on germanium where a dependence. resem-bling a 'I\. 2 relation was a1so found (43).
5. Application To Photoelectrically Generated Absorption a. Der i vat ion 0 f Tra n s mis s ion i n aso 1 i d con t a i
n-ing induced free carriers
The general equation (2.41) for the frequency dependent absorption coefficient shall now be applied to fit a special case, name1y that of photoe1ectrically induced free carrier absorption.
1t may be safe1y assumed that for wavelengths between the fundamental absorption edge and say 10 to 20 J.-L (the onset of lattice dispersion):
1nserting this into equation 2.41 results in:
or
CT '1\.2
o 1 2.63
2.64
w here the equation has been expressed as a function of the wave1ength '1\.1 of the radiation to be absorbed, rather than of frequency,and the constant
f3
is defined as46
where the weIl known semiconductor relation (Jo = n e J.1 cl has been inserted.
lt is imp'ortant to bear in mind that in general there are two types of free carriers, holes and electrons, hence
we should write or 'À 2e 3 a(À 1) = -47T2C3E o
N
instead of 2.63. We can write this as
2.66
2.67
2.68
however, for the sake of simplicity and because one of the two terms between brackets usually dominates over the other we shall usually refer only to equation 2.63 in the future, but of course bear relationship 2.67 in mind. The absorption coefficient an (À 1) for the electrons usu-ally dominates over a (À 1) for the holes to such an extent that ap(À 1) may be leglected. There are cases, however where ap(À
1)
»
an(À1), for example in silicon where ap(À1) is approximately 500 times as great as an(À1)13,
and then ap (À 1) may not be neglected. It is of further in-terest to note that the drift mobility f-Lcl is strongly tem-perature dependent and hence when measuring a(À 1), it
is important to maintain a constant temperature.
In view of the above derived equations for the absorption coefficient of light of wavelength À 1 we shall now consider a semiconductor slab of thickness 1 in which only naturally occuring free carriers are present. Neglecting Fresnel losses, the light intensity transmitted perpendicularly through such a
slab will be = J 1 (À. 1 ) -(30- ~ 2 1 1 e 0 2.69
w here J 1 (À. 1) (see F igure Il. 3.a) is the incident light intensity of wavelength À. 1 where À.1 > À.
f and J~(À.1) is the transmitted
intensity. This light has a low absorption coefficient, a
o' due
to naturally present free carriers + when the d.c. conductivity CT 0 is relatively small in magnitude.
Semiconductor slab of
thickness 1 with a na-tural absorbtion-coeffi-cient a. FIG. II.3.a. ~O) I J 1 (~ 1) Semiconduct<>r slab of thickness 1 with an in-duced - plus natural ab-sorption-coefficient a(x) ..
FIG. II.3.b. Absorption coefficient versus thickness diagram for a
semiconductor free carrier modulator
J
L
l-IJ. . 'EEJ
o 4X L
FIG. II.3.c. Incremental photon absorption in a semiconductor containing induced free carriers
+ Some of these free carriers may be generated by J 1 (À.
1) due to traps or imperfection levels or heat.
48
N ext we shall illuminate the surface of the semiconductor with light of intensity J 2 and wavelength À
2
<
Àf (see FigureII.3.b). The consequent absorption generates electron-hole pairs or free carriers in the semiconductor material in addition to the naturally present free carriers, bringing about a new absorption coefficient a(x) due to the higher total conductivity which now exists in the slab of thickness 1. Since the absorp-tion is proporabsorp-tional to the density of carriers, the two contri-butions are additive. It is immediately evident that the induced absorption coefficien ta ' (x) due to only the generated carriers is a function of the depth of penetration, x, of the carrier genera-ting light source J 2 (À
2). Hence the total coefficient a(x) may
be written:
,
a(x) = a (x) + ao 2.70 where 2.71,
and n (x) is the concentration of the induced carriers. When we assume that J
2(À2) is only absorbed by creating
new carriers, when there is no excessive recombination of these carriers at the surface, and the carriers do not diffuse appreciabie from the spot where they have been formed, this concentration n' (x) varies with the depth (x) as
H x
n'(x) = n'(o) e- 2 2.72
w here H
2 is the mean absorption coefficient of the e"xciting
light. n' (0) is the tota1 stead y state density of photogenerated carriers at the surface.
Upon the considering an infinitesima1 thickness 6. x within the region O·
