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Infra-red spectroscopy and low temperatures
GRYOGENIGS An International Journal of Low Temperature Engineering and Research
VOL.
3 NO.
1
MARCH
1963
INFRA-RED SPECTROSCOPY AND LOW TEMPERATURES D. H. P A R K I N S O N Royal Radar Establishment, Malvern, Worcs, U.K. Received 15 November 1962
I N solid state physics one of the first problems which has to be faced repeatedly is the acquisition of knowledge of the energy level scheme for the ground and excited states of the system in which we are interested. In general, the topics of study are concerned with the behaviour of the outer electrons of the atoms of the solid or with the modes of vibration of the crystal itself, and often involve both. Spectroscopy of the solid state is an obvious tool for investigating energy level schemes where transitions can be induced by radiation. The need to vary the temperature is also obvious; to resolve easily energy differences AE the temperature Tmust be such that k T < A E = hv, where v is the observing frequency. A number of fields of interest spring to mind: (1) The study of the electronic energy gap I at the Fermi surface in superconductors below their transition temperatures; (2) The optical absorption in paramagnetic salts due to excitations of the magnetic ions both with and without the use of magnetic fields; and (3) The optical absorption in semiconductors, a topic which has received great attention and in which a very large effort has been expended by numerous laboratories. This third field will be taken as the main theme for discussion in this paper because of the wide variety of phenomena involved. However, this subject has become large and it will only be possible to discuss a very limited part. An arbitrary division is to divide the subject into those effects which are observed without magnetic fields and those which are observed with magnetic fields; here we concentrate almost entirely on CRYOGENICS
- MARCH
1963
the former, and consider absorption spectroscopy primarily, the area covered being the work of one laboratory (R.R.E.). Throughout, an important factor in a proper understanding of what is happening is an accurate knowledge of the temperature dependence of the effects which are discussed. The spectral range of interest is the Whole region lying between the visible and centimetdc parts of the electromagnetic spectrum. For easy reference the conversion Table I is very useful. Table 1 Wavelength (B) Wavelength (ram) Wave number v (cm -t) Photon energy (eV)
I I0 -s I0' 1-24
I0 10 -I I0' 1"24><, I0 -x
Equivalent temperature
1-44:< 10 ~ 1"44× 10~
I0 z I0 ~ I0 -t I I0 a I0 1-24× I0 -i 1'24× I0 -s
1.44x 101
1"44x 101
Although the emphasis here is on studies in semiconductors, the techniques used are wholly or partly of much wider application. Simple description of semiconductors The main features to be expected in the absorption spectrum of a semiconductor can be described qualitatively in terms of the crudest model, thus permitting a ready appreciation of some of the difficulties to be encountered. Using an energy band scheme for describing the electronic energy levels they fall into two bands, and in a completely pure and perfect crystal at absolute zero the situation can be represented as in |
Figure l(a). The lower band, the valence band, is completely filled and the upper is empty, the 'forbidden energy" gap AEg lying between being the first leading parameter of the system which is required. Obviously if radiation is passed through the material a spectrum
I / / e / .e ~ ' / /
S /i
/ --
,,ond v's / / J - r
J.'/
"_1.
r/A, ./ -,, • . e / / / " '¢/ f / / ,// // /// / r~. , / / / , / / /'/" , /s /// ,/s / I i
J
J
J
I
J
J
.
4-+,.1:,++++
~Impurity
levels .+--t-++++
/G , ' / / /// / •
~. ,",~o I'~nce" b~ond /'~,' J/s" /t
~'o s J s . s
s/,
(.)
(b)
Figure 1. Elementary description of the electronic band structure of a semiconductor: (a) Pure; (b) With impurity levels
with a very sharp step or edge will result, the edge corresponding to frequencies v, such that h v = AEs: at higher frequencies there will be high absorption, at lower, none, in this ideal state. This edge is known as the 'intrinsic edge' and in practice is rounded off (Figure 2) and can exhibit considerable structure. Typical values for AEg are given in Table 2. Table 2
AEs(eV)
Ge
Si
lnSb
GaAs
GaP
0"72
l "06
0-I 8
1-35
2"24
As is well known, when impurities are present, extra energy levels appear in the forbidden gap (Figure l(b)). In germanium and silicon, impurity atoms having more than four electrons outside closed shells usually form 'donor levels' near the conduction band (e.g. t h e G r o u p V elements), while those with less than four atoms (e.g. Group III elements) form acceptor levels near the valence band. As the impurity atoms differ more in valency from those of the host lattice, so the impurity centres become more complex and can produce energy, levels lying near the centre of the forbidden gap. In germanium, typical Group III and V impurity levels can be as little as ~0.01 eV from the nearest band. In silicon, such levels are more separated from the nearest bands. In a semiconductor with, for example, predominating simple donor impurities present (N-type), at reasonably low temperatures some of the donor levels will be occupied and some will be excited, having given electrons to the conduction band. :2
In this situation, as the photon energy decreases and the spectrum is examined beyond the intrinsic edge, a general background absorption (dashed curve in Figure 2) will be observed, due to the presence of thermally excited free carriers in the material. This absorption is obviously temperature dependent, but in the purest materials at very low temperatures may be almost negligible. The presence of impurity levels also produces noticeable absorption effects corresponding to their separation from the energy bands; crudely, some sort of 'edges' can be expected. However, Coulomb forces between the localized impurity centre and the associated electron give rise to series of excited states similar to those of the hydrogen atom. The result is a fairly broad absorption rising as the wavelength increases to a hump on which one or two narrow peaks can be observed in the manner of the curve shown in Figure 2. There are then three distinct types of contribution to the infra-red spectrum arising from electronic energy levels. In addition there are other absorption bands on the longwave side of the intrinsic edge in the same region as the impurity levels which are due to the crystal lattice itself (Figure 2). A lattice vibration can be imagined as temporarily polarizing an atom in the lattice owing to the nucleus and outer electrons not quite moving in phase, electromagnetic radiation then being able to interact with the temporarily polarized lattice. Provided energy and momentum are conserved, processes in which several phonons are involved are possible: these are discussed later. All' four processes combined give rise to a fairly complex spectrum. Experimentally the situation is made more difficult owing to the fact that water vapour bands exist throughout the spectral range from 1.5 la up to the centimetric region. Typical lattice and impurity absorption bands can extend from as near as 10 p to 150 !1 or more. Two of the features of the absorption spectrum, namely the intrinsic edge and
10S
'E
10 4
l/edge 102
/ I " ~
Envelope of / spectrum
1 i --
--
--
Free carriers . . . . .
.... Lattice
Impurity bands
levels
Figure 2. Components o]" the inJ'ra-red absorption spectrum of a semiconductor CRYOGENICS
• MARCH
1963
lattice bands, will be discussed, for they exemplify the way in which considerable information can be extracted from accurately known experimental data. However, it is worth considering the experimental situation first.
Experimental situation In Figure 2, which is entirely schematic but represents a room temperature spectrum, it can be seen that the absorption coefficient varies through roughly five orders of magnitude. With the compound semiconductors such as gallium arsenide at the top of the intrinsic edge the absorption is greater than 10 cm-L At the foot of the intrinsic edge in very pure specimens, e.g. silicon and germanium, at low temperatures the absorption can fall to values near 10-2 or 10-3 cm -j. A very wide variation in thickness in prepared specimens must therefore be expected, from a few microns up to a few centimetres or millimetres at least, and such specimens have to be prepared with plane parallel surfaces. For finding the behaviour of a particular absorption process as a function of temperature, measurement at the easily obtainable fixed points is usually adequate, e.g. 4.2, 2"0, 77, and 290 ° K. However, the cryostat design must be such that the specimen is effectively cooled to the temperature of the cooling bath. In Figure 3 the cell of a typical absorption cryostat is shown (Roberts2). The features to note are that the specimen is housed in a chamber filled with helium exchange gas and sealed front and rear by
infra-red transmitting windows. The use of exchange gas in this way is far more satisfactory than reliance on direct thermal contact at the edges of the specimen as used in some other designs. In later versions of this type of cryostat the windows are sealed at their edges using thin polyethylene gaskets. 3 If the beam irradiating the specimen is properly defined by the nitrogen shield (Figure 3 inset) helium evaporation rates can be cut to less than 100 cm3/hr through the reduction in stray thermal radiation to the helium vessel.
I
J
L Specimen • - -
L.:
~
FIDI
(o)
BSpecimen
I
I
el--
"-
D°,
B Mirror D2
(b)
B Specimen
D2
(c)
I N
S Figure 3. Schematic arrangement o/'the lower part o f an infra-red absorption cell, showing the helium t,essel, nitrogen shield N, infra-red transmitting windows He, and specimen position S in a t,essel containing helium exchange gas. P is a push rod for changing the specimen position. 0 is an improt'ed nitrogen shield aperture ]br better beam definition CRYOGENICS • MARCH 1963
Figure 4. Schematic layout of measuring techniques
For the simplest experiments the procedure is outlined in Figure 4(a). Suppose the radiation emerges from the monochromator with intensity Io and falls on the specimen; the transmitted signal t can be measured in a suitable detector D~, e.g. a Golay cell. By removing the specimen, 1, can be measured. The reflection losses at the specimen can be calculated if the refractive index is known and does not vary over the spectral range in question. The cryostat must be provided with means for accurately positioning the specimen so that the same area is always illuminated, and also for removing it completely from the beam as necessary (Roberts2). The experimental procedure can then be either point by point, monitoring Io between each measurement of t, or it may consist of measuring lo and then t in successive continuous runs through the spectral range in question. The latter method greatly depends on the stability of the light source. 3
Monochromators as such will not be discussed here; however, the usual light sources are Nernst filaments, Globars, or high pressure mercury lamps. Most of the energy is produced in the visible region and only a minute fraction usefully employed (~-10 -lo) in the infra-red measurement. Beyond about 10-15 ~t the resolution which can be obtained in an experiment is limited entirely by the intensity of radiation from the emergent slits and the absorption in the specimen. To get sufficient signal the slit width may have to be as much as several millimetres. Stringent precautions must also be taken to ensure spectral purity. Unwanted short wavelength radiation can produce large spurious signals, particularly when experiments are performed with air occupying the radiation path. Normally water vapour absorption will be automatically corrected for in the experiment, but with stray shortwave radiation present, which is differently absorbed or reflected at the specimen from the longwave radiation, spurious values for t can occur. The water vapour problem can be alleviated by using 'dry' atmospheres in the radiation path, but the only satisfactory solution is to use an all-vacuum system where the residual gas pressure can be reduced to ~ 0 . I mm or less. If the vacuum system is good enough, a vacuum common with that protecting the helium crystal can be used, thus eliminating the outer infra-red transmitting windows in the cryostat with a corresponding increase in signal. K.R.S. 5 windows give reflection losses as high as 50 per cent. The refractive index usually varies through the spectral range of interest, is complex, and is often not accurately known; this simple method of experiment is then quite unsatisfactory and it is necessary to measure the reflected signal from the specimen directly. A useful method 4 for use in the infra-red region is illustrated in Figure 4(b). The specimen, placed in a 'three position' holder, is arranged so that the specimen, a blank space, and a freshly surfacesilvered mirror can be placed in the beam. A 'beam splitter' of an infra-red transmitting material with a reflection coefficient of, say, 30 per cent (e.g.K.R.S. 5) is placed at A so that reflection from the specimen can be directed to a second detector D2. Measurements then proceed as follows: with the specimen in position the transmitted signal t and reflected signal r, as seen in D t and D2, can be measured; with the specimen removed, Io can be measured; with the mirror in position the reflected signal in D2 can be measured as a standard of ~99 per cent against which to compare the reflected signal from the specimen• This system can be satisfactory provided the reflection of the mirror remains evenly high throughout the spectral region. Reflections from windows of the cells and detectors must not interfere with the resulting measurement; ideally, the detectors should appear black at the wavelengths concerned. If the infra-red transmitting windows are everywhere tilted through a small angle 4
(~5 degrees) unwanted signals of this type can be deflected from the system. An alternative system avoiding the beam splitter but using a small angle of incidence is shown in Figure 4(c). This is a more satisfactory experimental arrangement in which all the unwanted reflections can be eliminated. More elaborate systems can be devised in such a way that the true reflection can be measured with the condition that the surface preparation must be such that the two sides of the specimen are equally reflecting. If they differ, then the geometric mean between the reflectivities can be found.
The intrinsic edge When an intrinsic edge is examined under high resolution, structure appears from which a great deal of information concerning the electronic energy bands can be deduced. This was first made clear by work on germanium by Macfarlane, McLean, Roberts, and Quarrington. s-~ Using a band description the electronic energy in a solid is dependent on the propagation vector k of the Bloch waves ~Uk(r) = U(r) exp(ik.r). Following the usual custom of working in terms of k space, in any particular direction x, curves can be drawn representing the variation of energy as a function of kz. The simplest theoretical situation is that shown in Figure 5(a), where the conduction and valance bands are parabolic; in general they will have some more complex shape. The probability of the transition of an electron from a state in the valance band of wave vector k; to another of wave vector k2 can
onductlon
AE~___ i
,,
•
(o)
•
k1=k2
kx "
(b)
km
Figure 5. Energ)' band structure o f semiconductors: (a) Simple structure; (b) More complex, such as germanium CRYOGENICS
• MARCH
1963
be readily deduced by considering the appropriate matrix element, and if q is the wave vector of the incident radiation it is easily shown that k2-k,+q
=
0
which is equivalent to expressing the conservation of momentum; q is very small indeed and we can write k2 = k~. In terms of Figure 5(a) only vertical or direct transitions such as PQ are allowed, by direct interaction with radiation. Allowance must be made for the Coulomb interaction between the hole left in the valance band, which is equivalent to a positive charge, and the excited electron. The result is a series of excited states similar to those of the hydrogen-like atoms, the electron-hole pair in an excited state being termed an 'exciton'. One of the most graphic examples is found in cuprous oxide, where many lines can be seen (NikitineS). These, however, occur on the visible part of the spectrum. In germanium the situation is more complicated than in Figure 5(a). In the [111] direction, the form of the relations connecting E and k is shown in Figure 5(b). In the conduction band the lowest minima are at the zone edge in the [111] direction. The valance band is made up from three distinct branches, two of which show degeneracy at k = 0 and the third is split off by spin-orbit coupling (Elliottg). Vertical or direct transitions have their smallest value at k = 0 and in this region exciton formation is observed. The intrinsic edge has a knee at the top at room temperature which sharpens into the peak shown in Figure 6 as the temperature is lowered. This peak represents the first excited state of the direct exciton. The whole front of the intrinsic edge is in fact the envelope of absorptions due to exciton formation which merges into and overlies the absorption from band to band proper as the photon energy increases. By detailed analysis involving the shapes of the absorption curves the value of the 'direct intrinsic gap' (AEo in Figure 5(b)) as a function o f temperature and the exciton binding energy have been deduced in this instance. The value of AEo at T = 0 ° K was 0.89 eV and the exciton binding energy 0.007 eV. The effects mentioned here in germanium were missed in a number of earlier researches owing to the use of too low resolving powers. Silicon does not show a corresponding peak owing to differences in the band structure. It is unlikely that direct excitons are formed in that material. At the foot of the intrinsic edges in silicon and germanium there are pronounced 'tails' where distinct small knees in the absorption curve can be seen, which sharpen as the temperature falls, and one set of which disappears at the lowest temperature. It has been shown that these features are due to indirect transitions with exciton formation. An indirect transition is one in which the electron makes the transition to the conduction band by interacting with both a photon and CRYOGENICS
• MARCH
1963
a phonon, i.e. a transition such as X Y in Figure 5(b), where the intrinsic gap AEg is shown. The momentum required, supplied by either the emission or absorption of phonons, a n d knees corresponding to both processes can be seen. Those due to phonon absorption disappear at the lowest temperature, as might be expected. Transitions due to phonon absorption will appear as the photon energy reaches A E g - E , - E z , where E~ is the phonon energy involved and Ez the exciton binding energy. Similarly, processes involving phonon emission will begin where hv = A E g + E ~ - E z . By studying the shapes of the absorption curves it is possible to distinguish quite clearly between 'allowed' and 'forbidden' transitions in these processes. The variation in AEg with temperature and the phonon energies involved can also be deduced. There is an extensive and thorough review of the work on silicon and germanium by McLean. *°
4 x ~0~
3 x 10~
o.8~
o,~;o
hv
"
(eV)
Figure 6. Absorption dae to tile direct exciton in germanium (After Macfarlane, McLean, Quarrington, attd Roberts 6.7)
More recently other examples of exciton formation have been found, such as that of Sturge 1~ in gallium arsenide (Figure 7). In this material the band structure is similar to that of germanium except that the minima in the conduction band away from the centre are higher than those at the centre. The intrinsic gap AEg corresponds to the centre of the zone and is found to be 1.521 eV at 21 ° K with an exciton binding energy of 0.0034 eV. By carrying out experiments on very thin specimens direct transitions have also been observed from the split-off valence band to the conduction band in both germanium la and gallium arsenide H (LM in Figure 5(b)). The values for the spin-orbit splitting at the centre of the zone are 0.29 eV in germanium and 0-35 eV in gallium arsenide. 5
f-.
0'9
energy phonon absorbed with the emission of a higher energy phonon. The first of these processes gives bands in the shorter wavelength part of the spectrum, e.g. tip to near lO0 p, while the second type of process gives bands at longer wavelengths. The first are 'summation bands' and the second, 'difference bands'. If K' is the photon wave vector and kp the phonon wave vector
08
K ' + n K = Z+_k I, p
0"7
where K is a reciprocal lattice vector and n an integer. K' is very small compared with K and k v, and can be ignored, so we can write
1"2 x 104 1.1
Q • •
e
$$
1.0
0"6
nK = Z + kp
I
1 "50
1:52
1'~i4
'
~
1"56
(eV)
Figure 7. Absorption due to exciton formation in gallium arsenide (After Sturge t t)
Lattice bands in semiconductors
In the course of the work revealing the presence of indirect transitions in germanium and silicon it became evident that our knowledge of the lattice vibration spectrum was far too scanty. The absorption of infra-red radiation in ionic crystals by interaction with lattice vibrations appeared to be understood, and the question arose of whether the absorption bands in the infra-red spectrum of semiconductors attributed to lattice vibrations Could yield information concerning the details of the vibration spectrum. Lax and Burstein la considered the mechanism involved in homopolar crystals on the basis of charge deformation, followed by Mashkevich and Tolpygo. TM In the diamond type crystals the electric moment M can be expressed as a power series in the atomic displacements o f the form M = E A ~+~, Z.x~~B;~ + ~, Z.xt~C .~ x;~+
...
The selection rules governing the two-phonon processes in silicon and germanium demand that the two phonons should come from distinct branches of the spectrum. In the zinc-blende structure of the Group III-V compounds there is no centre of symmetry and one-phonon processes are possible, while the twophonon selection rules are relaxed. There is a characteristic temperature dependence for each type of process, which is an important factor in identifying the bands seen in the spectrum. In any process the net absorption is the difference between the forward and reverse processes, e.g. in the two-phonon summation absorption bands the difference between the absorption of one photon with the emission of two photons and the induced emission of one photon with the absorption of two phonons. For every phonon absorbed there is a temperature dependent factor F = l/[exp (E/kT)- 1] and for every phonon emitted the factor is 1 +F. As the temperature is lowered, processes involving the absorption of a phonon will become less and less probable because there are fewer and fewer phonons present. One can also reason that three-phonon processes are less probable than two-phonon processes,
Bi
In this expression the coefficients A, B, and C are in general tensors of the first, second, and third rank. The first term is constant and does not give rise to optical processes but the second and third terms are of interest. The second term corresponds to a first-order moment and in general gives rise to one-phonon processes, i.e. one photon is absorbed and one phonon emitted. However, in a crystal with a centre of symmetry this term is zero; thus with the diamond type lattice single-phonon processes should not be observed. The third term corresponds to two-phonon processes and higher terms involving three or more phonons exist. In all processes both energy and wave vector (cf. momentum) must be conserved. In the twophonon processes a photon can be absorbed with the emission of two phonons, or a photon and a low 6
~
Longltudinol ocoustic Tronsverseocoustic
ct Figure 8. Conventional plot o f relation between frequency and wave number f o r lattice vibrations on a reduced zone scheme CRYOGENICS
• MARCH
1963
which themselves must be less probable than onephonon, provided selection rules permit all processes. An energy versus wave vector or frequency versus wave number diagram can be constructed for phonons (Figure 8) just as for electrons. It is customary to call the lower branches 'acoustic' and the upper branches 'optical' and to distinguish between transverse and longitudinal in the symmetry direction in cubic crystals. If there is one atom per unit cell there are only
t
Figure 9. Density o f vibrational states as a fimction o f energy for an individual branch o f Figure 8
then assuming that the highest energy bump corresponded to three transverse optical modes (selection rules are relaxed for three-phonon process) and the next highest to two transverse optical and one longitudinal optical, etc., an extremely consistent breakdown into the different mean energies for each mode was found. In Figure 10 only one set of measurements taken at one temperature is shown to simplify the diagram. This work was shortly afterwards confirmed remarkably by the neutron scattering data of Brockhouse. TM More recently Stierwalt and Potter ~7 have measured the spectral emittance of silicon (as well as other semiconductors) and have produced a curve which is so near that of Figure 10 that it is difficult to distinguish between them. Work of this type has been extended to germanium, indium antimonide, and gallium arsenide (Johnson et al.4,1a,19). With the Group III-V compounds the spectrum is more complex owing to a relaxation of the two-phonon selection rules, and the absorption more intense; in Figure 11 the two- and three-phonon bands for gallium arsenide are shown. In Table 3 there is a summary of the chief results from this work.
Table 3 acoustic branches; with two atoms per unit cell there are a set of acoustic and optical branches as well, a new set appearing for each atom in the cell. Summing the density of states along any particular branch of Figure 8 a curve as in Figure 9 will be obtained in which there is a sharp maximum corresponding to a frequency (or energy) towards the edge of the zone. This maximum as a function of energy will be very sharp, because even if the branches in Figure 8 were straight lines, seven-eighths of the states would lie at energies greater than half the mean energy. In fact dispersion is present and branches have the shape shown: as the acoustic and optical branches flatten towards the zone edge so the maxima in the density of states are sharpened. In considering the likely shape of the lattice absorption bands it is evident that the curves for the density of states for different branches will have to be combined, though weighted according to selection rules. Thus in analysing any spectrum it should be possible to break the energies corresponding to different peaks into combinations of the energies corresponding to the maxima for the density of states of the different phonon branches. This was first carried out successfully by Johnson, is who examined the spectrum of silicon under high resolution (Figure 10). In particular, the three small bumps marked A, B, and C were identified as threephonon processes from their temperature dependence, CRYOGENICS
• MARCH
1963
Mean energies (eV) Mode
Transcerse acoustic
Germanium Silicon Indium antimonide G a lliu m arsenide
0.0084 0-0172 0.0053 0"0097
Longitudinal Longitudinal Transoerse optical acoustic optical 0"0226 0-0416 0"0145 0'023
0-0299 0"0512 0'019 0"029
0"35 0'0527 0"022 0"034
and 0'0316
The Raman frequencies can also be deduced from the two- and three-phonon spectra by noting the
IE
/ 2tO
4~0
60 150 100 Wa~ numbers
120
140
Figure 10. Lattice vibration summation absorption bands silicon (After Johnsont s)
16u
of
7
points at which they cease fairly abruptly at the high energy side of the spectrum. These are shown in Figure 10, where they can be seen quite easily as the points P and Q, formed by short linear extrapolations. On the energy scale P is twice the Raman energy and Q, three times.
"Two-phonon bands
~
E
,~ h
10
Three- )honon bands
o.~o.
~' ~
-
-
photon is negligible. The reststrahlen reflections occur at frequencies corresponding to R. Thus if one-phonon absorptions are to be seen at all they are hidden in the region of intense reflection. Near the region of such reflections the refractive index is not constant and is complex ( n - i k ) , in particular k varies rapidly and can exceed n. Experimentally the reflection from the specimen and the transmission through the specimen have the form shown in Figure 12. Under these circumstances it is essential at least to use a system of measurement as shown in Figure 4(b) or (c), or preferably to devise a system in which the absolute reflection coefficient can be determined. Further difficulties arise in this experimental range because the wavelength of the observing frequency (in gallium arsenide the reststrahlen reflection is at ~35 la) in the material is comparable with the thickness of the specimen.
~
L. 1.0
40
60 Wave number
80 r (mm-1)
R
Figure 11. Lattice vibration summation absorption bands o f gallium arsenide (After Johnson et al. 2°)
I I ! ! !
In all four materials the transverse acoustic mode is of very much lower energy than all others (only about one-third of that of the longitudinal acoustic). In the curve for the total density of states there is thus a sharp low-energy maximum. On warming from absolute zero these low-energy modes are heavily excited before there is an appreciable contribution from any other. This is the reason for the pronounced minimum in the curve of the Debye temperature Oo, derived from specific heat data plotted as a function of temperature for germanium and other diamond type crystals (Johnson and Lock20). Once the data shown in Table 3 are known for a given material it is possible to predict with accuracy where the minimum in the OD/Tcurve will be. The most important difference between the lattice absorption processes of the compound semiconductors and the simple Group IV elements is that in the former there is the possibility of one-phonon absorption processes, and pronounced reststrahlen reflections occur. One-phonon processes, the absorption of one photon and the emission of one phonon, correspond to radiation interacting with states very near to R in Figure 8, because the wave vector K' of the 8
! ,P
0
A
Figure 12. Schematic representation o f the reflection and transmission signals in the reststrahlen region
Cochran 2~published an account ofthe shell model for atoms in a crystal lattice, and in particular proposed that for germanium the atom could be represented as a central massive core bound isotropically to a charged spherical shell of electrons. The relative displacement of shell and core could account for the polarization of the atom. The potential energy of the model is a quadratic function of the displacement and polarization. Using this model as a basis for the calculations of the lattice vibration spectrum, good agreement with the neutron scattering results can be obtained and good agreement with the positions of the maxima in the spectra are referred to by Johnson and Cochran. 22 This model has also been extended to include the Group III-V compounds. CRYOGENICS
• MARCH
1963
Semiconductors and the detection of radiation
From the elementary diagram in Figure 1 it is obvious that when current carriers are excited by radiation a change in electrical conductivity, i.e. photoconductivity, will be observed. This may be due to short wavelength radiation inducing electronic transitions across the forbidden gap AEg, intrinsic photoconductivity, or by longer wavelength radiation exciting carriers from impurity levels. Detectors of radiation based on intrinsic photoconductivity have been known for a long time and because of the fairly large energies involved (short wavelengths) will work at room temperature, although cooling will usually enhance their performance. The sensitivity of any photoconductive detector may be no better than that of bolometric devices such as the Golay cell, but their advantage lies in the possibility of very short time constants (e.g. 10-~ sec). The first advances well into the far infra-red region using detectors based on the excitation of carriers from impurities were made with copper 23 and zinc 24 doped germanium. With zinc in germanium, acceptor levels lie at about 0.033 eV above the valence band and a good photoconductive response can be observed out to about 40 la. Cooling with such detectors now becomes a necessity because AEg is small. A notable advance into the very far infra-red region resulted with the use of antimony doped germanium by Fray and Oliver. 2s Antimony as an impurity yields a very shallow donor level ~0-01 eV below the conduction band. Cooled with liquid helium a photoconducted response can be observed out to ~135 H, near which wavelength the signal falls rapidly as the wavelength increases. A problem arises at these low temperatures which can limit the time constant of the detector, as the resistivity of the specimen increases and the RC time constant of the circuit containing the sensitive
t
S /~Es--m~H*c!i., ,~/"/'
---"~
Figure 13. Energy-momentum relation in z direction in magnetic field applied along z direction, kz is still a good quantum number. The broken line indicates unperturbed curves • MARCH
m
,
(~
•
Helm c Broodened Landau
~ " ~
levels
f
Exposed H
>>
0
~ impurity
level
Figure 14. Simple picture o f bunching o f electronic level in a magnetic
:eld
In a recent development Putley26 has extended the use of photoconductivity to the millimetre wavelength range, and depends on the use of a magnetic field with very pure N-type indium antimonide. Under the action of a magnetic field the electronic band structure cannot be represented simply by a diagram as in Figure 5. If the field is applied in the z direction, k= and k u are no longer good quantum numbers, and regarded semi-classically, the electronic motion in the x,y plane is quantized. E, k curves in the style of Figure 5 can only be drawn for the kz direction and are, in the simplest instances, of the style of Figure 13. The separation AEs between the curves corresponds to the cyclotron resonance frequency and AEs = He/m*c
E
CRYOGENICS
element can then become the limiting factor for the overall time constant. The performance of the antimony doped germanium detector was fairly good, the minimum detectable signal was ~ 3 × 10-1° W/cm 2 per unit cycle at 800 c/s.
1963
(see, for example, the review by Lax27); here m* is the effective electronic mass. Unless the crystal is very pure and the temperature very low, the curves showing the relation between E and kz will tend to be smeared out, i.e. the collision time will be short and, in general, on the diagram of the density of states (Figure 14) the energy levels will be bunched. The purest specimens of indium antimonide are N-type, with a residual impurity believed to be tellurium giving a level very near the bottom of the conduction band. The effect of a magnetic field on the impurity !evel is very much smaller than the effect on the conduction band itself. The conduction band effective mass in indium antimonide is ,-4).013 and consequently the separation AEs is large. At quite low field the impurity level is 'exposed' by the lowest levels 9
in the conduction band moving upwards on the energy scale. In the diagram of Figure 13 an impurity level is associated with each band produced by the magnetic field. The development of an impurity level with a field dependent excitation energy was shown by Putley 2a in a series of measurements of the Hall coefficient and conductivity of yery pure N-type indium antimonide. He applied these results to the development of a very long wave photoconductive detector. In a magnetic field photoconduction should be observable up to wavelengths corresponding to the ionization energy of the 'exposed' impurity level. These ideas have worked out in practice 26 and a schematic representation of a detector based on this effect is shown in Figure 15. The minimum detectable signal reported is 1-2 x 10 -12 W/unit cycle at 1 mm wavelength with a time constant < 10 -6. The resistivity of the sensitive specimens remains quite low at a few kilogauss, about 5 kl). Consequently the RC time constant is not a limiting factor.
Incident rod~..ion
[~tector leods in screen'mg tube
~ll,
,:::
i
t
M,,~
i
i
::
at l"S ° K
le detector solenoid
Figure 1.5. Infra-red detector based on N-type indium antimonide in a magnetic fieM (After Pulley26.27)
10
Rollin 29 at the Clarendon Laboratory has reported that photoconductivity can also be observed at these long wavelengths in zero field, which is presumably due to the excitation of free electrons within the band itself. Conclusions In this review the subject matter has been selected on the basis of the work of one laboratory (R. R.E.) and it will be appreciated that there are extensive investigations which have not been discussed but which would be included in any comprehensive review under this title. The work of the group at the M.I.T. Lincoln Laboratory in studying the magneto-optical absorption spectra of semiconductors must be mentioned (in particular, see the reviews by Lax and Zwerdling3o and Zwerdlin#l). Nevertheless, there is adequate material here to show how the combination of two powerful experimental techniques, infra-red and low temperature, can be used not only to solve problems concerned with the physical structure of materials but also to provide new experimental tools.
REFERENCES I. GINSBERG, D. M., and TINKHAM, M. Phys. Rev. 118, 990 (1960) 2. ROBERTS,V. J. sci. Instrum. 32, 294 (1955) 3. ROBERTS,V. J. sci. lnstrum. 36, 99 (1959) 4. COCHRAN,W., FRAY, S. J., JOHNSON,F. A., QUARRINGTON, J. E., WILLIAMS,N. J. appI. Phys. (Suppl.) 32, 2102 (1961) 5. MACFARLANE,G. G., MCLEAN, T. P., ROBERTS, V., and QUARRINGTON,J. E. Phys. Reu. 108, 1377 (1957) 6. MACFARLANE,G. G., MCLEAN, T. P., ROBERTS, V., and QUARR1NGTON,J. E. Proc. phys. Soc. Lond. 71,863 0958) 7. MACFARLANE,G. G., MCLEAN, T. P., ROBERTS, V., and QUARRINGTON,J. E. d. Phys. Chem. Solids B, 388 (1959) 8. NIKrrINE, S. Progress in Semiconductors 6, pp. 223, 269 (Heywood, London, 1961) 9. ELLIOTT,R. J. Phys. Rev. 95, 226 (1954), 96, 280 (1954) 10. MCLEAN, T. P. Progress in Semiconductors 5, p. 55 (Heywood, London, 1960) 11. STURGE,M. Phys. Rev. 127, 768 (1962) 12. HOBDEN,M. J. Phys. Chem. Solids 23, 821 (1962) 13. LAX, M., and PUP.STEIN,E. Phys. Rev. 97, 39 (1955) 14. MASHKEVICH,V. S., and TOLPYGO,K. B. Soviet Phys. JETP 5, 435 (1957) 15. JOHNSON,F. A. Proc. phys. Soc. Lond. 73, 265 (1959) 16. BROCKHOUSE,B. N. Phys. Rev. Left. 2, 256 (1959) 17. STIERWALT,D. L., and POTrER, R. F. Proc. Conf. Physics of Semiconductors, Exeter (To be published) 18. FRAY, S. J., JOHNSON, F. A., and JONES, R. H. Proc. phys. Soc. Lond. 76, 939 (1960) 19. FRAY, S. J., JOHNSON, F. A., QUARP.INGTON,J. E., and WILLIAMS,N. Proc. phys. Soc. Lond. 77, 215 (1961) 20. JOHNSON,F. A., and LOCK,J. M. Proc. phys. Soc. Lond. 72, 914 (1958) 21. COCHRAN,W. Proc. roy. Soc. A253, 200 (1959) 22. JOHNSON,F. A., and COCHRAN,W. Proc. phys. Soc. Lond. In press 23. KRAU~, P. W., MCGLAUGHLIN,L. D., and McQUISTAN, R. B. Elements of Infra-red Technology, p. 407 (Wiley, New York, 1962) CRYOGENICS • MARCH 1963
24. BURNSTEIN,E., et al. Phys. Rev. 93, 95 (1962); see also
28. PUTLEY,E. H. Electronics Wkly No. 114, 13 (1962)
reference 23 25. FRAY, S. J., and OLIVER,J. F. C. J. sci. lnstrum. 36, 195 (1959) 26. PUTLEY, E. H. Proc. Int. Conf. Photoconductivity, New York, p. 241 (Pergamon, London, 1962) 27. LAX,B. Rev. rood. Phys. 30, 122 (1958)
29. ROLUN,B. V. Proc. phys. Soc. Lond. 77, 1102 (1961); see also GOODWIN,D., and JONES,R. H. J. appl. Phys. 32, 2065 (1961) 30. LAX,B., and ZWERDLING,S. Progress in Semiconductors 5, p. 221 (Heywood, London, 1960) 31, ZWERDLING,S., KLEINER,W. H., and THERIAULT,J. P. J. appl. Phys. (Suppl.) 32, 2118 (1961)
CONFERENCE THE 1963 Cryogenic Engineering Conference will be sponsored jointly by the University of Colorado and the National Bureau of Standards, and will be held in Boulder, Colorado, on 19-21 August. Preregistration information and a copy of the programme will be available about the middle of July. The deadline for both abstract (not over 200 words) and a preliminary manuscript is 1 May 1963. They should be sent directly to: The Conference Secretary, Professor K. D. Timmerhaus, Cryogenic Engineering Conference, Chemical Engineering Department, University of Colorado, Boulder, Colorado.
CRYOGENICS • MARCH 1963
11
VOL.
3 NO.
1
MARCH
1963
INFRA-RED SPECTROSCOPY AND LOW TEMPERATURES D. H. P A R K I N S O N Royal Radar Establishment, Malvern, Worcs, U.K. Received 15 November 1962
I N solid state physics one of the first problems which has to be faced repeatedly is the acquisition of knowledge of the energy level scheme for the ground and excited states of the system in which we are interested. In general, the topics of study are concerned with the behaviour of the outer electrons of the atoms of the solid or with the modes of vibration of the crystal itself, and often involve both. Spectroscopy of the solid state is an obvious tool for investigating energy level schemes where transitions can be induced by radiation. The need to vary the temperature is also obvious; to resolve easily energy differences AE the temperature Tmust be such that k T < A E = hv, where v is the observing frequency. A number of fields of interest spring to mind: (1) The study of the electronic energy gap I at the Fermi surface in superconductors below their transition temperatures; (2) The optical absorption in paramagnetic salts due to excitations of the magnetic ions both with and without the use of magnetic fields; and (3) The optical absorption in semiconductors, a topic which has received great attention and in which a very large effort has been expended by numerous laboratories. This third field will be taken as the main theme for discussion in this paper because of the wide variety of phenomena involved. However, this subject has become large and it will only be possible to discuss a very limited part. An arbitrary division is to divide the subject into those effects which are observed without magnetic fields and those which are observed with magnetic fields; here we concentrate almost entirely on CRYOGENICS
- MARCH
1963
the former, and consider absorption spectroscopy primarily, the area covered being the work of one laboratory (R.R.E.). Throughout, an important factor in a proper understanding of what is happening is an accurate knowledge of the temperature dependence of the effects which are discussed. The spectral range of interest is the Whole region lying between the visible and centimetdc parts of the electromagnetic spectrum. For easy reference the conversion Table I is very useful. Table 1 Wavelength (B) Wavelength (ram) Wave number v (cm -t) Photon energy (eV)
I I0 -s I0' 1-24
I0 10 -I I0' 1"24><, I0 -x
Equivalent temperature
1-44:< 10 ~ 1"44× 10~
I0 z I0 ~ I0 -t I I0 a I0 1-24× I0 -i 1'24× I0 -s
1.44x 101
1"44x 101
Although the emphasis here is on studies in semiconductors, the techniques used are wholly or partly of much wider application. Simple description of semiconductors The main features to be expected in the absorption spectrum of a semiconductor can be described qualitatively in terms of the crudest model, thus permitting a ready appreciation of some of the difficulties to be encountered. Using an energy band scheme for describing the electronic energy levels they fall into two bands, and in a completely pure and perfect crystal at absolute zero the situation can be represented as in |
Figure l(a). The lower band, the valence band, is completely filled and the upper is empty, the 'forbidden energy" gap AEg lying between being the first leading parameter of the system which is required. Obviously if radiation is passed through the material a spectrum
I / / e / .e ~ ' / /
S /i
/ --
,,ond v's / / J - r
J.'/
"_1.
r/A, ./ -,, • . e / / / " '¢/ f / / ,// // /// / r~. , / / / , / / /'/" , /s /// ,/s / I i
J
J
J
I
J
J
.
4-+,.1:,++++
~Impurity
levels .+--t-++++
/G , ' / / /// / •
~. ,",~o I'~nce" b~ond /'~,' J/s" /t
~'o s J s . s
s/,
(.)
(b)
Figure 1. Elementary description of the electronic band structure of a semiconductor: (a) Pure; (b) With impurity levels
with a very sharp step or edge will result, the edge corresponding to frequencies v, such that h v = AEs: at higher frequencies there will be high absorption, at lower, none, in this ideal state. This edge is known as the 'intrinsic edge' and in practice is rounded off (Figure 2) and can exhibit considerable structure. Typical values for AEg are given in Table 2. Table 2
AEs(eV)
Ge
Si
lnSb
GaAs
GaP
0"72
l "06
0-I 8
1-35
2"24
As is well known, when impurities are present, extra energy levels appear in the forbidden gap (Figure l(b)). In germanium and silicon, impurity atoms having more than four electrons outside closed shells usually form 'donor levels' near the conduction band (e.g. t h e G r o u p V elements), while those with less than four atoms (e.g. Group III elements) form acceptor levels near the valence band. As the impurity atoms differ more in valency from those of the host lattice, so the impurity centres become more complex and can produce energy, levels lying near the centre of the forbidden gap. In germanium, typical Group III and V impurity levels can be as little as ~0.01 eV from the nearest band. In silicon, such levels are more separated from the nearest bands. In a semiconductor with, for example, predominating simple donor impurities present (N-type), at reasonably low temperatures some of the donor levels will be occupied and some will be excited, having given electrons to the conduction band. :2
In this situation, as the photon energy decreases and the spectrum is examined beyond the intrinsic edge, a general background absorption (dashed curve in Figure 2) will be observed, due to the presence of thermally excited free carriers in the material. This absorption is obviously temperature dependent, but in the purest materials at very low temperatures may be almost negligible. The presence of impurity levels also produces noticeable absorption effects corresponding to their separation from the energy bands; crudely, some sort of 'edges' can be expected. However, Coulomb forces between the localized impurity centre and the associated electron give rise to series of excited states similar to those of the hydrogen atom. The result is a fairly broad absorption rising as the wavelength increases to a hump on which one or two narrow peaks can be observed in the manner of the curve shown in Figure 2. There are then three distinct types of contribution to the infra-red spectrum arising from electronic energy levels. In addition there are other absorption bands on the longwave side of the intrinsic edge in the same region as the impurity levels which are due to the crystal lattice itself (Figure 2). A lattice vibration can be imagined as temporarily polarizing an atom in the lattice owing to the nucleus and outer electrons not quite moving in phase, electromagnetic radiation then being able to interact with the temporarily polarized lattice. Provided energy and momentum are conserved, processes in which several phonons are involved are possible: these are discussed later. All' four processes combined give rise to a fairly complex spectrum. Experimentally the situation is made more difficult owing to the fact that water vapour bands exist throughout the spectral range from 1.5 la up to the centimetric region. Typical lattice and impurity absorption bands can extend from as near as 10 p to 150 !1 or more. Two of the features of the absorption spectrum, namely the intrinsic edge and
10S
'E
10 4
l/edge 102
/ I " ~
Envelope of / spectrum
1 i --
--
--
Free carriers . . . . .
.... Lattice
Impurity bands
levels
Figure 2. Components o]" the inJ'ra-red absorption spectrum of a semiconductor CRYOGENICS
• MARCH
1963
lattice bands, will be discussed, for they exemplify the way in which considerable information can be extracted from accurately known experimental data. However, it is worth considering the experimental situation first.
Experimental situation In Figure 2, which is entirely schematic but represents a room temperature spectrum, it can be seen that the absorption coefficient varies through roughly five orders of magnitude. With the compound semiconductors such as gallium arsenide at the top of the intrinsic edge the absorption is greater than 10 cm-L At the foot of the intrinsic edge in very pure specimens, e.g. silicon and germanium, at low temperatures the absorption can fall to values near 10-2 or 10-3 cm -j. A very wide variation in thickness in prepared specimens must therefore be expected, from a few microns up to a few centimetres or millimetres at least, and such specimens have to be prepared with plane parallel surfaces. For finding the behaviour of a particular absorption process as a function of temperature, measurement at the easily obtainable fixed points is usually adequate, e.g. 4.2, 2"0, 77, and 290 ° K. However, the cryostat design must be such that the specimen is effectively cooled to the temperature of the cooling bath. In Figure 3 the cell of a typical absorption cryostat is shown (Roberts2). The features to note are that the specimen is housed in a chamber filled with helium exchange gas and sealed front and rear by
infra-red transmitting windows. The use of exchange gas in this way is far more satisfactory than reliance on direct thermal contact at the edges of the specimen as used in some other designs. In later versions of this type of cryostat the windows are sealed at their edges using thin polyethylene gaskets. 3 If the beam irradiating the specimen is properly defined by the nitrogen shield (Figure 3 inset) helium evaporation rates can be cut to less than 100 cm3/hr through the reduction in stray thermal radiation to the helium vessel.
I
J
L Specimen • - -
L.:
~
FIDI
(o)
BSpecimen
I
I
el--
"-
D°,
B Mirror D2
(b)
B Specimen
D2
(c)
I N
S Figure 3. Schematic arrangement o/'the lower part o f an infra-red absorption cell, showing the helium t,essel, nitrogen shield N, infra-red transmitting windows He, and specimen position S in a t,essel containing helium exchange gas. P is a push rod for changing the specimen position. 0 is an improt'ed nitrogen shield aperture ]br better beam definition CRYOGENICS • MARCH 1963
Figure 4. Schematic layout of measuring techniques
For the simplest experiments the procedure is outlined in Figure 4(a). Suppose the radiation emerges from the monochromator with intensity Io and falls on the specimen; the transmitted signal t can be measured in a suitable detector D~, e.g. a Golay cell. By removing the specimen, 1, can be measured. The reflection losses at the specimen can be calculated if the refractive index is known and does not vary over the spectral range in question. The cryostat must be provided with means for accurately positioning the specimen so that the same area is always illuminated, and also for removing it completely from the beam as necessary (Roberts2). The experimental procedure can then be either point by point, monitoring Io between each measurement of t, or it may consist of measuring lo and then t in successive continuous runs through the spectral range in question. The latter method greatly depends on the stability of the light source. 3
Monochromators as such will not be discussed here; however, the usual light sources are Nernst filaments, Globars, or high pressure mercury lamps. Most of the energy is produced in the visible region and only a minute fraction usefully employed (~-10 -lo) in the infra-red measurement. Beyond about 10-15 ~t the resolution which can be obtained in an experiment is limited entirely by the intensity of radiation from the emergent slits and the absorption in the specimen. To get sufficient signal the slit width may have to be as much as several millimetres. Stringent precautions must also be taken to ensure spectral purity. Unwanted short wavelength radiation can produce large spurious signals, particularly when experiments are performed with air occupying the radiation path. Normally water vapour absorption will be automatically corrected for in the experiment, but with stray shortwave radiation present, which is differently absorbed or reflected at the specimen from the longwave radiation, spurious values for t can occur. The water vapour problem can be alleviated by using 'dry' atmospheres in the radiation path, but the only satisfactory solution is to use an all-vacuum system where the residual gas pressure can be reduced to ~ 0 . I mm or less. If the vacuum system is good enough, a vacuum common with that protecting the helium crystal can be used, thus eliminating the outer infra-red transmitting windows in the cryostat with a corresponding increase in signal. K.R.S. 5 windows give reflection losses as high as 50 per cent. The refractive index usually varies through the spectral range of interest, is complex, and is often not accurately known; this simple method of experiment is then quite unsatisfactory and it is necessary to measure the reflected signal from the specimen directly. A useful method 4 for use in the infra-red region is illustrated in Figure 4(b). The specimen, placed in a 'three position' holder, is arranged so that the specimen, a blank space, and a freshly surfacesilvered mirror can be placed in the beam. A 'beam splitter' of an infra-red transmitting material with a reflection coefficient of, say, 30 per cent (e.g.K.R.S. 5) is placed at A so that reflection from the specimen can be directed to a second detector D2. Measurements then proceed as follows: with the specimen in position the transmitted signal t and reflected signal r, as seen in D t and D2, can be measured; with the specimen removed, Io can be measured; with the mirror in position the reflected signal in D2 can be measured as a standard of ~99 per cent against which to compare the reflected signal from the specimen• This system can be satisfactory provided the reflection of the mirror remains evenly high throughout the spectral region. Reflections from windows of the cells and detectors must not interfere with the resulting measurement; ideally, the detectors should appear black at the wavelengths concerned. If the infra-red transmitting windows are everywhere tilted through a small angle 4
(~5 degrees) unwanted signals of this type can be deflected from the system. An alternative system avoiding the beam splitter but using a small angle of incidence is shown in Figure 4(c). This is a more satisfactory experimental arrangement in which all the unwanted reflections can be eliminated. More elaborate systems can be devised in such a way that the true reflection can be measured with the condition that the surface preparation must be such that the two sides of the specimen are equally reflecting. If they differ, then the geometric mean between the reflectivities can be found.
The intrinsic edge When an intrinsic edge is examined under high resolution, structure appears from which a great deal of information concerning the electronic energy bands can be deduced. This was first made clear by work on germanium by Macfarlane, McLean, Roberts, and Quarrington. s-~ Using a band description the electronic energy in a solid is dependent on the propagation vector k of the Bloch waves ~Uk(r) = U(r) exp(ik.r). Following the usual custom of working in terms of k space, in any particular direction x, curves can be drawn representing the variation of energy as a function of kz. The simplest theoretical situation is that shown in Figure 5(a), where the conduction and valance bands are parabolic; in general they will have some more complex shape. The probability of the transition of an electron from a state in the valance band of wave vector k; to another of wave vector k2 can
onductlon
AE~___ i
,,
•
(o)
•
k1=k2
kx "
(b)
km
Figure 5. Energ)' band structure o f semiconductors: (a) Simple structure; (b) More complex, such as germanium CRYOGENICS
• MARCH
1963
be readily deduced by considering the appropriate matrix element, and if q is the wave vector of the incident radiation it is easily shown that k2-k,+q
=
0
which is equivalent to expressing the conservation of momentum; q is very small indeed and we can write k2 = k~. In terms of Figure 5(a) only vertical or direct transitions such as PQ are allowed, by direct interaction with radiation. Allowance must be made for the Coulomb interaction between the hole left in the valance band, which is equivalent to a positive charge, and the excited electron. The result is a series of excited states similar to those of the hydrogen-like atoms, the electron-hole pair in an excited state being termed an 'exciton'. One of the most graphic examples is found in cuprous oxide, where many lines can be seen (NikitineS). These, however, occur on the visible part of the spectrum. In germanium the situation is more complicated than in Figure 5(a). In the [111] direction, the form of the relations connecting E and k is shown in Figure 5(b). In the conduction band the lowest minima are at the zone edge in the [111] direction. The valance band is made up from three distinct branches, two of which show degeneracy at k = 0 and the third is split off by spin-orbit coupling (Elliottg). Vertical or direct transitions have their smallest value at k = 0 and in this region exciton formation is observed. The intrinsic edge has a knee at the top at room temperature which sharpens into the peak shown in Figure 6 as the temperature is lowered. This peak represents the first excited state of the direct exciton. The whole front of the intrinsic edge is in fact the envelope of absorptions due to exciton formation which merges into and overlies the absorption from band to band proper as the photon energy increases. By detailed analysis involving the shapes of the absorption curves the value of the 'direct intrinsic gap' (AEo in Figure 5(b)) as a function o f temperature and the exciton binding energy have been deduced in this instance. The value of AEo at T = 0 ° K was 0.89 eV and the exciton binding energy 0.007 eV. The effects mentioned here in germanium were missed in a number of earlier researches owing to the use of too low resolving powers. Silicon does not show a corresponding peak owing to differences in the band structure. It is unlikely that direct excitons are formed in that material. At the foot of the intrinsic edges in silicon and germanium there are pronounced 'tails' where distinct small knees in the absorption curve can be seen, which sharpen as the temperature falls, and one set of which disappears at the lowest temperature. It has been shown that these features are due to indirect transitions with exciton formation. An indirect transition is one in which the electron makes the transition to the conduction band by interacting with both a photon and CRYOGENICS
• MARCH
1963
a phonon, i.e. a transition such as X Y in Figure 5(b), where the intrinsic gap AEg is shown. The momentum required, supplied by either the emission or absorption of phonons, a n d knees corresponding to both processes can be seen. Those due to phonon absorption disappear at the lowest temperature, as might be expected. Transitions due to phonon absorption will appear as the photon energy reaches A E g - E , - E z , where E~ is the phonon energy involved and Ez the exciton binding energy. Similarly, processes involving phonon emission will begin where hv = A E g + E ~ - E z . By studying the shapes of the absorption curves it is possible to distinguish quite clearly between 'allowed' and 'forbidden' transitions in these processes. The variation in AEg with temperature and the phonon energies involved can also be deduced. There is an extensive and thorough review of the work on silicon and germanium by McLean. *°
4 x ~0~
3 x 10~
o.8~
o,~;o
hv
"
(eV)
Figure 6. Absorption dae to tile direct exciton in germanium (After Macfarlane, McLean, Quarrington, attd Roberts 6.7)
More recently other examples of exciton formation have been found, such as that of Sturge 1~ in gallium arsenide (Figure 7). In this material the band structure is similar to that of germanium except that the minima in the conduction band away from the centre are higher than those at the centre. The intrinsic gap AEg corresponds to the centre of the zone and is found to be 1.521 eV at 21 ° K with an exciton binding energy of 0.0034 eV. By carrying out experiments on very thin specimens direct transitions have also been observed from the split-off valence band to the conduction band in both germanium la and gallium arsenide H (LM in Figure 5(b)). The values for the spin-orbit splitting at the centre of the zone are 0.29 eV in germanium and 0-35 eV in gallium arsenide. 5
f-.
0'9
energy phonon absorbed with the emission of a higher energy phonon. The first of these processes gives bands in the shorter wavelength part of the spectrum, e.g. tip to near lO0 p, while the second type of process gives bands at longer wavelengths. The first are 'summation bands' and the second, 'difference bands'. If K' is the photon wave vector and kp the phonon wave vector
08
K ' + n K = Z+_k I, p
0"7
where K is a reciprocal lattice vector and n an integer. K' is very small compared with K and k v, and can be ignored, so we can write
1"2 x 104 1.1
Q • •
e
$$
1.0
0"6
nK = Z + kp
I
1 "50
1:52
1'~i4
'
~
1"56
(eV)
Figure 7. Absorption due to exciton formation in gallium arsenide (After Sturge t t)
Lattice bands in semiconductors
In the course of the work revealing the presence of indirect transitions in germanium and silicon it became evident that our knowledge of the lattice vibration spectrum was far too scanty. The absorption of infra-red radiation in ionic crystals by interaction with lattice vibrations appeared to be understood, and the question arose of whether the absorption bands in the infra-red spectrum of semiconductors attributed to lattice vibrations Could yield information concerning the details of the vibration spectrum. Lax and Burstein la considered the mechanism involved in homopolar crystals on the basis of charge deformation, followed by Mashkevich and Tolpygo. TM In the diamond type crystals the electric moment M can be expressed as a power series in the atomic displacements o f the form M = E A ~+~, Z.x~~B;~ + ~, Z.xt~C .~ x;~+
...
The selection rules governing the two-phonon processes in silicon and germanium demand that the two phonons should come from distinct branches of the spectrum. In the zinc-blende structure of the Group III-V compounds there is no centre of symmetry and one-phonon processes are possible, while the twophonon selection rules are relaxed. There is a characteristic temperature dependence for each type of process, which is an important factor in identifying the bands seen in the spectrum. In any process the net absorption is the difference between the forward and reverse processes, e.g. in the two-phonon summation absorption bands the difference between the absorption of one photon with the emission of two photons and the induced emission of one photon with the absorption of two phonons. For every phonon absorbed there is a temperature dependent factor F = l/[exp (E/kT)- 1] and for every phonon emitted the factor is 1 +F. As the temperature is lowered, processes involving the absorption of a phonon will become less and less probable because there are fewer and fewer phonons present. One can also reason that three-phonon processes are less probable than two-phonon processes,
Bi
In this expression the coefficients A, B, and C are in general tensors of the first, second, and third rank. The first term is constant and does not give rise to optical processes but the second and third terms are of interest. The second term corresponds to a first-order moment and in general gives rise to one-phonon processes, i.e. one photon is absorbed and one phonon emitted. However, in a crystal with a centre of symmetry this term is zero; thus with the diamond type lattice single-phonon processes should not be observed. The third term corresponds to two-phonon processes and higher terms involving three or more phonons exist. In all processes both energy and wave vector (cf. momentum) must be conserved. In the twophonon processes a photon can be absorbed with the emission of two phonons, or a photon and a low 6
~
Longltudinol ocoustic Tronsverseocoustic
ct Figure 8. Conventional plot o f relation between frequency and wave number f o r lattice vibrations on a reduced zone scheme CRYOGENICS
• MARCH
1963
which themselves must be less probable than onephonon, provided selection rules permit all processes. An energy versus wave vector or frequency versus wave number diagram can be constructed for phonons (Figure 8) just as for electrons. It is customary to call the lower branches 'acoustic' and the upper branches 'optical' and to distinguish between transverse and longitudinal in the symmetry direction in cubic crystals. If there is one atom per unit cell there are only
t
Figure 9. Density o f vibrational states as a fimction o f energy for an individual branch o f Figure 8
then assuming that the highest energy bump corresponded to three transverse optical modes (selection rules are relaxed for three-phonon process) and the next highest to two transverse optical and one longitudinal optical, etc., an extremely consistent breakdown into the different mean energies for each mode was found. In Figure 10 only one set of measurements taken at one temperature is shown to simplify the diagram. This work was shortly afterwards confirmed remarkably by the neutron scattering data of Brockhouse. TM More recently Stierwalt and Potter ~7 have measured the spectral emittance of silicon (as well as other semiconductors) and have produced a curve which is so near that of Figure 10 that it is difficult to distinguish between them. Work of this type has been extended to germanium, indium antimonide, and gallium arsenide (Johnson et al.4,1a,19). With the Group III-V compounds the spectrum is more complex owing to a relaxation of the two-phonon selection rules, and the absorption more intense; in Figure 11 the two- and three-phonon bands for gallium arsenide are shown. In Table 3 there is a summary of the chief results from this work.
Table 3 acoustic branches; with two atoms per unit cell there are a set of acoustic and optical branches as well, a new set appearing for each atom in the cell. Summing the density of states along any particular branch of Figure 8 a curve as in Figure 9 will be obtained in which there is a sharp maximum corresponding to a frequency (or energy) towards the edge of the zone. This maximum as a function of energy will be very sharp, because even if the branches in Figure 8 were straight lines, seven-eighths of the states would lie at energies greater than half the mean energy. In fact dispersion is present and branches have the shape shown: as the acoustic and optical branches flatten towards the zone edge so the maxima in the density of states are sharpened. In considering the likely shape of the lattice absorption bands it is evident that the curves for the density of states for different branches will have to be combined, though weighted according to selection rules. Thus in analysing any spectrum it should be possible to break the energies corresponding to different peaks into combinations of the energies corresponding to the maxima for the density of states of the different phonon branches. This was first carried out successfully by Johnson, is who examined the spectrum of silicon under high resolution (Figure 10). In particular, the three small bumps marked A, B, and C were identified as threephonon processes from their temperature dependence, CRYOGENICS
• MARCH
1963
Mean energies (eV) Mode
Transcerse acoustic
Germanium Silicon Indium antimonide G a lliu m arsenide
0.0084 0-0172 0.0053 0"0097
Longitudinal Longitudinal Transoerse optical acoustic optical 0"0226 0-0416 0"0145 0'023
0-0299 0"0512 0'019 0"029
0"35 0'0527 0"022 0"034
and 0'0316
The Raman frequencies can also be deduced from the two- and three-phonon spectra by noting the
IE
/ 2tO
4~0
60 150 100 Wa~ numbers
120
140
Figure 10. Lattice vibration summation absorption bands silicon (After Johnsont s)
16u
of
7
points at which they cease fairly abruptly at the high energy side of the spectrum. These are shown in Figure 10, where they can be seen quite easily as the points P and Q, formed by short linear extrapolations. On the energy scale P is twice the Raman energy and Q, three times.
"Two-phonon bands
~
E
,~ h
10
Three- )honon bands
o.~o.
~' ~
-
-
photon is negligible. The reststrahlen reflections occur at frequencies corresponding to R. Thus if one-phonon absorptions are to be seen at all they are hidden in the region of intense reflection. Near the region of such reflections the refractive index is not constant and is complex ( n - i k ) , in particular k varies rapidly and can exceed n. Experimentally the reflection from the specimen and the transmission through the specimen have the form shown in Figure 12. Under these circumstances it is essential at least to use a system of measurement as shown in Figure 4(b) or (c), or preferably to devise a system in which the absolute reflection coefficient can be determined. Further difficulties arise in this experimental range because the wavelength of the observing frequency (in gallium arsenide the reststrahlen reflection is at ~35 la) in the material is comparable with the thickness of the specimen.
~
L. 1.0
40
60 Wave number
80 r (mm-1)
R
Figure 11. Lattice vibration summation absorption bands o f gallium arsenide (After Johnson et al. 2°)
I I ! ! !
In all four materials the transverse acoustic mode is of very much lower energy than all others (only about one-third of that of the longitudinal acoustic). In the curve for the total density of states there is thus a sharp low-energy maximum. On warming from absolute zero these low-energy modes are heavily excited before there is an appreciable contribution from any other. This is the reason for the pronounced minimum in the curve of the Debye temperature Oo, derived from specific heat data plotted as a function of temperature for germanium and other diamond type crystals (Johnson and Lock20). Once the data shown in Table 3 are known for a given material it is possible to predict with accuracy where the minimum in the OD/Tcurve will be. The most important difference between the lattice absorption processes of the compound semiconductors and the simple Group IV elements is that in the former there is the possibility of one-phonon absorption processes, and pronounced reststrahlen reflections occur. One-phonon processes, the absorption of one photon and the emission of one phonon, correspond to radiation interacting with states very near to R in Figure 8, because the wave vector K' of the 8
! ,P
0
A
Figure 12. Schematic representation o f the reflection and transmission signals in the reststrahlen region
Cochran 2~published an account ofthe shell model for atoms in a crystal lattice, and in particular proposed that for germanium the atom could be represented as a central massive core bound isotropically to a charged spherical shell of electrons. The relative displacement of shell and core could account for the polarization of the atom. The potential energy of the model is a quadratic function of the displacement and polarization. Using this model as a basis for the calculations of the lattice vibration spectrum, good agreement with the neutron scattering results can be obtained and good agreement with the positions of the maxima in the spectra are referred to by Johnson and Cochran. 22 This model has also been extended to include the Group III-V compounds. CRYOGENICS
• MARCH
1963
Semiconductors and the detection of radiation
From the elementary diagram in Figure 1 it is obvious that when current carriers are excited by radiation a change in electrical conductivity, i.e. photoconductivity, will be observed. This may be due to short wavelength radiation inducing electronic transitions across the forbidden gap AEg, intrinsic photoconductivity, or by longer wavelength radiation exciting carriers from impurity levels. Detectors of radiation based on intrinsic photoconductivity have been known for a long time and because of the fairly large energies involved (short wavelengths) will work at room temperature, although cooling will usually enhance their performance. The sensitivity of any photoconductive detector may be no better than that of bolometric devices such as the Golay cell, but their advantage lies in the possibility of very short time constants (e.g. 10-~ sec). The first advances well into the far infra-red region using detectors based on the excitation of carriers from impurities were made with copper 23 and zinc 24 doped germanium. With zinc in germanium, acceptor levels lie at about 0.033 eV above the valence band and a good photoconductive response can be observed out to about 40 la. Cooling with such detectors now becomes a necessity because AEg is small. A notable advance into the very far infra-red region resulted with the use of antimony doped germanium by Fray and Oliver. 2s Antimony as an impurity yields a very shallow donor level ~0-01 eV below the conduction band. Cooled with liquid helium a photoconducted response can be observed out to ~135 H, near which wavelength the signal falls rapidly as the wavelength increases. A problem arises at these low temperatures which can limit the time constant of the detector, as the resistivity of the specimen increases and the RC time constant of the circuit containing the sensitive
t
S /~Es--m~H*c!i., ,~/"/'
---"~
Figure 13. Energy-momentum relation in z direction in magnetic field applied along z direction, kz is still a good quantum number. The broken line indicates unperturbed curves • MARCH
m
,
(~
•
Helm c Broodened Landau
~ " ~
levels
f
Exposed H
>>
0
~ impurity
level
Figure 14. Simple picture o f bunching o f electronic level in a magnetic
:eld
In a recent development Putley26 has extended the use of photoconductivity to the millimetre wavelength range, and depends on the use of a magnetic field with very pure N-type indium antimonide. Under the action of a magnetic field the electronic band structure cannot be represented simply by a diagram as in Figure 5. If the field is applied in the z direction, k= and k u are no longer good quantum numbers, and regarded semi-classically, the electronic motion in the x,y plane is quantized. E, k curves in the style of Figure 5 can only be drawn for the kz direction and are, in the simplest instances, of the style of Figure 13. The separation AEs between the curves corresponds to the cyclotron resonance frequency and AEs = He/m*c
E
CRYOGENICS
element can then become the limiting factor for the overall time constant. The performance of the antimony doped germanium detector was fairly good, the minimum detectable signal was ~ 3 × 10-1° W/cm 2 per unit cycle at 800 c/s.
1963
(see, for example, the review by Lax27); here m* is the effective electronic mass. Unless the crystal is very pure and the temperature very low, the curves showing the relation between E and kz will tend to be smeared out, i.e. the collision time will be short and, in general, on the diagram of the density of states (Figure 14) the energy levels will be bunched. The purest specimens of indium antimonide are N-type, with a residual impurity believed to be tellurium giving a level very near the bottom of the conduction band. The effect of a magnetic field on the impurity !evel is very much smaller than the effect on the conduction band itself. The conduction band effective mass in indium antimonide is ,-4).013 and consequently the separation AEs is large. At quite low field the impurity level is 'exposed' by the lowest levels 9
in the conduction band moving upwards on the energy scale. In the diagram of Figure 13 an impurity level is associated with each band produced by the magnetic field. The development of an impurity level with a field dependent excitation energy was shown by Putley 2a in a series of measurements of the Hall coefficient and conductivity of yery pure N-type indium antimonide. He applied these results to the development of a very long wave photoconductive detector. In a magnetic field photoconduction should be observable up to wavelengths corresponding to the ionization energy of the 'exposed' impurity level. These ideas have worked out in practice 26 and a schematic representation of a detector based on this effect is shown in Figure 15. The minimum detectable signal reported is 1-2 x 10 -12 W/unit cycle at 1 mm wavelength with a time constant < 10 -6. The resistivity of the sensitive specimens remains quite low at a few kilogauss, about 5 kl). Consequently the RC time constant is not a limiting factor.
Incident rod~..ion
[~tector leods in screen'mg tube
~ll,
,:::
i
t
M,,~
i
i
::
at l"S ° K
le detector solenoid
Figure 1.5. Infra-red detector based on N-type indium antimonide in a magnetic fieM (After Pulley26.27)
10
Rollin 29 at the Clarendon Laboratory has reported that photoconductivity can also be observed at these long wavelengths in zero field, which is presumably due to the excitation of free electrons within the band itself. Conclusions In this review the subject matter has been selected on the basis of the work of one laboratory (R. R.E.) and it will be appreciated that there are extensive investigations which have not been discussed but which would be included in any comprehensive review under this title. The work of the group at the M.I.T. Lincoln Laboratory in studying the magneto-optical absorption spectra of semiconductors must be mentioned (in particular, see the reviews by Lax and Zwerdling3o and Zwerdlin#l). Nevertheless, there is adequate material here to show how the combination of two powerful experimental techniques, infra-red and low temperature, can be used not only to solve problems concerned with the physical structure of materials but also to provide new experimental tools.
REFERENCES I. GINSBERG, D. M., and TINKHAM, M. Phys. Rev. 118, 990 (1960) 2. ROBERTS,V. J. sci. Instrum. 32, 294 (1955) 3. ROBERTS,V. J. sci. lnstrum. 36, 99 (1959) 4. COCHRAN,W., FRAY, S. J., JOHNSON,F. A., QUARRINGTON, J. E., WILLIAMS,N. J. appI. Phys. (Suppl.) 32, 2102 (1961) 5. MACFARLANE,G. G., MCLEAN, T. P., ROBERTS, V., and QUARRINGTON,J. E. Phys. Reu. 108, 1377 (1957) 6. MACFARLANE,G. G., MCLEAN, T. P., ROBERTS, V., and QUARR1NGTON,J. E. Proc. phys. Soc. Lond. 71,863 0958) 7. MACFARLANE,G. G., MCLEAN, T. P., ROBERTS, V., and QUARRINGTON,J. E. d. Phys. Chem. Solids B, 388 (1959) 8. NIKrrINE, S. Progress in Semiconductors 6, pp. 223, 269 (Heywood, London, 1961) 9. ELLIOTT,R. J. Phys. Rev. 95, 226 (1954), 96, 280 (1954) 10. MCLEAN, T. P. Progress in Semiconductors 5, p. 55 (Heywood, London, 1960) 11. STURGE,M. Phys. Rev. 127, 768 (1962) 12. HOBDEN,M. J. Phys. Chem. Solids 23, 821 (1962) 13. LAX, M., and PUP.STEIN,E. Phys. Rev. 97, 39 (1955) 14. MASHKEVICH,V. S., and TOLPYGO,K. B. Soviet Phys. JETP 5, 435 (1957) 15. JOHNSON,F. A. Proc. phys. Soc. Lond. 73, 265 (1959) 16. BROCKHOUSE,B. N. Phys. Rev. Left. 2, 256 (1959) 17. STIERWALT,D. L., and POTrER, R. F. Proc. Conf. Physics of Semiconductors, Exeter (To be published) 18. FRAY, S. J., JOHNSON, F. A., and JONES, R. H. Proc. phys. Soc. Lond. 76, 939 (1960) 19. FRAY, S. J., JOHNSON, F. A., QUARP.INGTON,J. E., and WILLIAMS,N. Proc. phys. Soc. Lond. 77, 215 (1961) 20. JOHNSON,F. A., and LOCK,J. M. Proc. phys. Soc. Lond. 72, 914 (1958) 21. COCHRAN,W. Proc. roy. Soc. A253, 200 (1959) 22. JOHNSON,F. A., and COCHRAN,W. Proc. phys. Soc. Lond. In press 23. KRAU~, P. W., MCGLAUGHLIN,L. D., and McQUISTAN, R. B. Elements of Infra-red Technology, p. 407 (Wiley, New York, 1962) CRYOGENICS • MARCH 1963
24. BURNSTEIN,E., et al. Phys. Rev. 93, 95 (1962); see also
28. PUTLEY,E. H. Electronics Wkly No. 114, 13 (1962)
reference 23 25. FRAY, S. J., and OLIVER,J. F. C. J. sci. lnstrum. 36, 195 (1959) 26. PUTLEY, E. H. Proc. Int. Conf. Photoconductivity, New York, p. 241 (Pergamon, London, 1962) 27. LAX,B. Rev. rood. Phys. 30, 122 (1958)
29. ROLUN,B. V. Proc. phys. Soc. Lond. 77, 1102 (1961); see also GOODWIN,D., and JONES,R. H. J. appl. Phys. 32, 2065 (1961) 30. LAX,B., and ZWERDLING,S. Progress in Semiconductors 5, p. 221 (Heywood, London, 1960) 31, ZWERDLING,S., KLEINER,W. H., and THERIAULT,J. P. J. appl. Phys. (Suppl.) 32, 2118 (1961)
CONFERENCE THE 1963 Cryogenic Engineering Conference will be sponsored jointly by the University of Colorado and the National Bureau of Standards, and will be held in Boulder, Colorado, on 19-21 August. Preregistration information and a copy of the programme will be available about the middle of July. The deadline for both abstract (not over 200 words) and a preliminary manuscript is 1 May 1963. They should be sent directly to: The Conference Secretary, Professor K. D. Timmerhaus, Cryogenic Engineering Conference, Chemical Engineering Department, University of Colorado, Boulder, Colorado.
CRYOGENICS • MARCH 1963
11