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Surface-plasmon-polariton dispersion of semiconductors with depletion layers
Solid State Communications, Vol. 18, pp. 1123—1125, 1976.
Pergamon Press.
Printed in Great Britain
SURFACE-PLASMON—POLARJTON DISPERSION OF SEMICONDUCTORS WITH DEPLETION LAYERS E.M. Conwell and C.C. Kao Xerox Webster Research Center, Webster, NY 14580, U.S.A. (Received 23 October 1975 by J. Tauc) For a sample with plasma frequency varying exponentially from w~at the surface to a larger value wPb in the bulk, and moderate damping, we find, contrary to other calculations, that the long wavelength surface plasmon dispersion has a single branch that is reentrant at w = w~. THE EFFECT on the long wavelength plasmon dispersion of an inhomogeneous free electron density below
k~.It is possible to write it in a simple form, however, for the case that the quantity v8, the ratio of the surface
the surface of a14 solid has been for studied recently by several Particularly the case of space investigators. charge regions N in with semiconductors, the enough variation carrier concentration depth z is slow soof that it appears reasonable to use a local approximation for the dielectric function e(z). In two of the investigations cited,’3 has been approximated by
value toasminus theachievable bulk valueinofpractice, e(z), is close to zero. When, is easily 1v I ~ 1,bep0d 2d2/c2 I ~ pod, the dispersion relation 3may ebw written, for real,
=
~b + & exp (z/d),
Z <0,
=
Cb
x (1 + pod(l + v 8)
~b being the bulk value. With d a constant of the order
solution of the wave equation with given by (1). For waves propagating in the x-direction with wave vector k~,we assume the form of the solution to be e~~0Z F’~z,~ e i(k~x-wt) z <~ (2) 2)1~. F(z) can only be obwhere = (k~ ~/c solutions, with arbitary contamed Po in the form of~series stants determined so as to satisfy the appropriate boundary counditions. We take the medium above the interface to have dielectric constant 2, independent of frequency and position. H~for z >0 is then given by an expression similar to (2) with F(z) = 1 and Po replaced by ~~P2 = (k~ 2 w2/c2)112. The resulting dispersion relation, obtained by equating tangential components of E and H at the interface, must be solved numerically for w vs — —
—
—
—
—
[Pod+ 6bw2d211 IvsI). C
of the space charge region width in a semiconductor, the form (1) should be a reasonably good description for either an accumulation or depletion layer. Using this 3 concluded that the dispersion has form, Rice et al.” two branches: I, at frequencies w such that e <0 everywhere and II, in the range of w for which goes through O inside the sample. Although their conclusion about branch II was based on an invalid solution of the wave equation,2’5 it did suggest a hunt for additional structure in the frequency range occupied by II. We have studied this frequency range and find that the actual situation is different from their results. To obtain the dispersion, we look for a TM mode
H
1,
—~
2
(1)
‘~
(3) Since we are mainly interested in conduction electron (or hole)soeffects, relatively highis reasonable, frequencies we anduse d large enough that a local theory a free electron model for with the constant scattering time r ~ 11w. For complex ewe have then e(w) = e~,(1 w~of2)+ ie~w2w’2(wT)’ (4) —
where ,,~is the high-frequency dielectric constant and w,, is the local plasma frequency, having the value Wpb at z = w~at the surface. For the case of a depletion layer, to which we limit further discussion, w~, 8
~,
charge-region width depends also on all of theserange factors and on Wpb. Typical values for semiconductors from 250 A, for InSb with Nb I 0’8/cm3, to I jim. In Fig. 1 we show a plot of the dispersion obtained with real, (wpbd/c) = 0.1 and two different values of (JJP8/wPb. The value (wPbd/c) = 0.1 represents d = 300 A for w~ = l015/sec, or 3000 A for WPb = 1014/sec, etc. Since l0’4/sec corresponds toNb a few times 1018/cm3 in InSb, (wPbd/c) = 0.1 represents a larger value of d than would normally be expected in such samples. However, there is experimental evidence that disturbed samples, i.e. samples polished but not etched, or samples
1123
-~
1124
SURFACE-PLASMON—POLARITON DISPERSION OF SEMICONDUCTORS I•C
—
.9
—
‘w~ck
I —iii6i~
i’
5’
/
nr(.8)
Vol. 18, No. 8
logarithmic term dominates. We also find a third branch, labeled 111(0.8), which joins 11(0.8) near (ckX/wPb) = 0.89. (See inset.) For smaller w~we also find a pair of branches just above and below Wps/Wpb and a higher third branch. In the limit w,,9 0, branch III corresponds to the dispersion curve found by Cunningham et al. although theirs differs in detail since they approximate by a piecewise linear function. We have compared the variation of H~with z for the same k~value on branches I, II and III. In all three cases H~decreases as ePoz as z -÷ oo. For branch I H~ decreases monotonically with distance below the surface, while for branches II and III H~first increases, reaching a maximum at = 0, and then decreases monotonically as z decreases further. The difference between II and III is that the peak in H~lies closer to the surface in the former case, corresponding to smaller v8. The maximum inH~corresponds toE~andEs becoming infinite. -~
(8) I~8)
—
,~
w Wpb
.81 7
, /mt8)
—
-
(1U8)
.80 .6
—
-
~
1(8)
/ .88
-
0
I 2
—
-
.92
.~
I
I
I
I
~
k ~
8
0
2
x/Wpb
Fig. I. w vs k~in normalized units for (wPbd/c) = 0.1, = 16, ~2 = 1, (wP5/wpb) = 0.8 and 1.0. The solid + dotted lines in the inset represent an expanded view of the neighborhood of w = w~,8for (w~I2wPb)= 0.8. In the of dotted lines are theand dot— the 4/sec, uresence damping, withreplaced r = 10 bysec w b= 10i dashed one.
on which gratings have been constructed, have larger d’s 6 than normal. The dashed line, WPS/ Wpb = 1.0, represents the homogeneous sample. This plot was obtained from (p 2/e2) = (po/b). As can be seen from equation (3), for small enough d the dispersion will be indistinguishable from that for the homogeneous sample. For w~5/ = 0.8, as can be partly seen with the help of the inset line in Fig. 1, there is w one branch light closely up to co~, which follows the 5and then approaches very closely the horizontal line (w/wpb) = 0.8 although always staying below it. For this branch <0 everywhere, although very close to 0 at the surface. Thus it corresponds to branch I, discussed5earlier, for the in the limit ofcase large of I~/CbI~ what 1.~As earlier, k~(beyond is shown in Fig. 1) branch I goes asymptotically to the frequency w~[e,,,/(e,~,,+ 2)] i/2 at which e~= ~2. The branch labelled 11(0.8) lies everywhere above (w/w~)= 0.8, thus corresponds to e being + at the surface, having gone through 0 just below it. It is readily seen from (3) that such a closely spaced pair of branches, above and below ( WI,s! w~,),is to be expected when w is very close to w~, 5,1v51 ~ 1, and the —
These infinities are, however, the result of neglecting damping. When we allow damping, assuming that the cornplex varies with z according to (1) we obtain formally the same solution for the wave equation, with all quantities now complex. The only change that must be made is to replace ln Iv~Iby in v 5. For moderate damping the only significant effect occurs where the imaginary part of for at the i.e. w surface w~ is greater than the real part of there, 5.This prevents ln V5 from approaching 00 at these frequencies and the dispersion relation may no longer be satisfied. The main effect then is to wipe out portions of the dispersion curve in such regions. Overall, this should result in the three branches becoming a single reentrant curve. In the inset of Fig. 1 we show part of this curve, calculated approximately 14/sec, the dotted linesfor r = 1 Ø_i2sec and w1,1, = I0 representing the regions that have been wiped out. The re-entrant feature is more pronounced for somewhat higher w~,5,and should be more pronounced below room temperature, where r is larger. To find this feature it would beofdesirable to at have in which the position the bands themeasurements surface is varied, most conveniently by an applied electric field. In addition to the branch discussed, at higher frequencies, but still within the range where goes through 0 inside the sample, there may be further branches particularly for larger values of d. These describe guided modes7 rather than surface plasmons, however, since they correspond to solutions that are oscillatory where e> 0. They wifi be discussed in another publication.
Vol. 18, No. 8
SURFACE-PLASMON—POLARITON DISPERSION OF SEMICONDUCTORS
1125
REFERENCES 1.
GUIDOTTI D., RICE S.A. & LEMBERG H.L., Solid State Commun. 15, 113 (1974).
2. 3. 4.
CONWELLE.M.,Phys. Rev. B!!, 1508 (1975). RICE S.A., GUIDOTTI D., LEMBERG H.L., MUR1~HYW.C. & BLOCH A.N., in Advances in Chemical Physics (Edited by PRIGOGINE I. & RICE S.A.), Vol. XXVII. Wiley, NY (1974). CUNNINGHAM S.L., MARADUDIN A.A. &WALLIS R.F.,Phys. Rev. BlO, 3342 (1974).
5.
CONWELL E.M. (submitted for publication).
6.
ANDERSON W.E., ALEXANDER R.W. & BELL R.J.,Phys. Rev. Lett. 27, 1057 (1971).
7.
CONWELL E.M., Solid State Commun. 14,915 (1974).
Pergamon Press.
Printed in Great Britain
SURFACE-PLASMON—POLARJTON DISPERSION OF SEMICONDUCTORS WITH DEPLETION LAYERS E.M. Conwell and C.C. Kao Xerox Webster Research Center, Webster, NY 14580, U.S.A. (Received 23 October 1975 by J. Tauc) For a sample with plasma frequency varying exponentially from w~at the surface to a larger value wPb in the bulk, and moderate damping, we find, contrary to other calculations, that the long wavelength surface plasmon dispersion has a single branch that is reentrant at w = w~. THE EFFECT on the long wavelength plasmon dispersion of an inhomogeneous free electron density below
k~.It is possible to write it in a simple form, however, for the case that the quantity v8, the ratio of the surface
the surface of a14 solid has been for studied recently by several Particularly the case of space investigators. charge regions N in with semiconductors, the enough variation carrier concentration depth z is slow soof that it appears reasonable to use a local approximation for the dielectric function e(z). In two of the investigations cited,’3 has been approximated by
value toasminus theachievable bulk valueinofpractice, e(z), is close to zero. When, is easily 1v I ~ 1,bep0d 2d2/c2 I ~ pod, the dispersion relation 3may ebw written, for real,
=
~b + & exp (z/d),
Z <0,
=
Cb
x (1 + pod(l + v 8)
~b being the bulk value. With d a constant of the order
solution of the wave equation with given by (1). For waves propagating in the x-direction with wave vector k~,we assume the form of the solution to be e~~0Z F’~z,~ e i(k~x-wt) z <~ (2) 2)1~. F(z) can only be obwhere = (k~ ~/c solutions, with arbitary contamed Po in the form of~series stants determined so as to satisfy the appropriate boundary counditions. We take the medium above the interface to have dielectric constant 2, independent of frequency and position. H~for z >0 is then given by an expression similar to (2) with F(z) = 1 and Po replaced by ~~P2 = (k~ 2 w2/c2)112. The resulting dispersion relation, obtained by equating tangential components of E and H at the interface, must be solved numerically for w vs — —
—
—
—
—
[Pod+ 6bw2d211 IvsI). C
of the space charge region width in a semiconductor, the form (1) should be a reasonably good description for either an accumulation or depletion layer. Using this 3 concluded that the dispersion has form, Rice et al.” two branches: I, at frequencies w such that e <0 everywhere and II, in the range of w for which goes through O inside the sample. Although their conclusion about branch II was based on an invalid solution of the wave equation,2’5 it did suggest a hunt for additional structure in the frequency range occupied by II. We have studied this frequency range and find that the actual situation is different from their results. To obtain the dispersion, we look for a TM mode
H
1,
—~
2
(1)
‘~
(3) Since we are mainly interested in conduction electron (or hole)soeffects, relatively highis reasonable, frequencies we anduse d large enough that a local theory a free electron model for with the constant scattering time r ~ 11w. For complex ewe have then e(w) = e~,(1 w~of2)+ ie~w2w’2(wT)’ (4) —
where ,,~is the high-frequency dielectric constant and w,, is the local plasma frequency, having the value Wpb at z = w~at the surface. For the case of a depletion layer, to which we limit further discussion, w~, 8
~,
charge-region width depends also on all of theserange factors and on Wpb. Typical values for semiconductors from 250 A, for InSb with Nb I 0’8/cm3, to I jim. In Fig. 1 we show a plot of the dispersion obtained with real, (wpbd/c) = 0.1 and two different values of (JJP8/wPb. The value (wPbd/c) = 0.1 represents d = 300 A for w~ = l015/sec, or 3000 A for WPb = 1014/sec, etc. Since l0’4/sec corresponds toNb a few times 1018/cm3 in InSb, (wPbd/c) = 0.1 represents a larger value of d than would normally be expected in such samples. However, there is experimental evidence that disturbed samples, i.e. samples polished but not etched, or samples
1123
-~
1124
SURFACE-PLASMON—POLARITON DISPERSION OF SEMICONDUCTORS I•C
—
.9
—
‘w~ck
I —iii6i~
i’
5’
/
nr(.8)
Vol. 18, No. 8
logarithmic term dominates. We also find a third branch, labeled 111(0.8), which joins 11(0.8) near (ckX/wPb) = 0.89. (See inset.) For smaller w~we also find a pair of branches just above and below Wps/Wpb and a higher third branch. In the limit w,,9 0, branch III corresponds to the dispersion curve found by Cunningham et al. although theirs differs in detail since they approximate by a piecewise linear function. We have compared the variation of H~with z for the same k~value on branches I, II and III. In all three cases H~decreases as ePoz as z -÷ oo. For branch I H~ decreases monotonically with distance below the surface, while for branches II and III H~first increases, reaching a maximum at = 0, and then decreases monotonically as z decreases further. The difference between II and III is that the peak in H~lies closer to the surface in the former case, corresponding to smaller v8. The maximum inH~corresponds toE~andEs becoming infinite. -~
(8) I~8)
—
,~
w Wpb
.81 7
, /mt8)
—
-
(1U8)
.80 .6
—
-
~
1(8)
/ .88
-
0
I 2
—
-
.92
.~
I
I
I
I
~
k ~
8
0
2
x/Wpb
Fig. I. w vs k~in normalized units for (wPbd/c) = 0.1, = 16, ~2 = 1, (wP5/wpb) = 0.8 and 1.0. The solid + dotted lines in the inset represent an expanded view of the neighborhood of w = w~,8for (w~I2wPb)= 0.8. In the of dotted lines are theand dot— the 4/sec, uresence damping, withreplaced r = 10 bysec w b= 10i dashed one.
on which gratings have been constructed, have larger d’s 6 than normal. The dashed line, WPS/ Wpb = 1.0, represents the homogeneous sample. This plot was obtained from (p 2/e2) = (po/b). As can be seen from equation (3), for small enough d the dispersion will be indistinguishable from that for the homogeneous sample. For w~5/ = 0.8, as can be partly seen with the help of the inset line in Fig. 1, there is w one branch light closely up to co~, which follows the 5and then approaches very closely the horizontal line (w/wpb) = 0.8 although always staying below it. For this branch <0 everywhere, although very close to 0 at the surface. Thus it corresponds to branch I, discussed5earlier, for the in the limit ofcase large of I~/CbI~ what 1.~As earlier, k~(beyond is shown in Fig. 1) branch I goes asymptotically to the frequency w~[e,,,/(e,~,,+ 2)] i/2 at which e~= ~2. The branch labelled 11(0.8) lies everywhere above (w/w~)= 0.8, thus corresponds to e being + at the surface, having gone through 0 just below it. It is readily seen from (3) that such a closely spaced pair of branches, above and below ( WI,s! w~,),is to be expected when w is very close to w~, 5,1v51 ~ 1, and the —
These infinities are, however, the result of neglecting damping. When we allow damping, assuming that the cornplex varies with z according to (1) we obtain formally the same solution for the wave equation, with all quantities now complex. The only change that must be made is to replace ln Iv~Iby in v 5. For moderate damping the only significant effect occurs where the imaginary part of for at the i.e. w surface w~ is greater than the real part of there, 5.This prevents ln V5 from approaching 00 at these frequencies and the dispersion relation may no longer be satisfied. The main effect then is to wipe out portions of the dispersion curve in such regions. Overall, this should result in the three branches becoming a single reentrant curve. In the inset of Fig. 1 we show part of this curve, calculated approximately 14/sec, the dotted linesfor r = 1 Ø_i2sec and w1,1, = I0 representing the regions that have been wiped out. The re-entrant feature is more pronounced for somewhat higher w~,5,and should be more pronounced below room temperature, where r is larger. To find this feature it would beofdesirable to at have in which the position the bands themeasurements surface is varied, most conveniently by an applied electric field. In addition to the branch discussed, at higher frequencies, but still within the range where goes through 0 inside the sample, there may be further branches particularly for larger values of d. These describe guided modes7 rather than surface plasmons, however, since they correspond to solutions that are oscillatory where e> 0. They wifi be discussed in another publication.
Vol. 18, No. 8
SURFACE-PLASMON—POLARITON DISPERSION OF SEMICONDUCTORS
1125
REFERENCES 1.
GUIDOTTI D., RICE S.A. & LEMBERG H.L., Solid State Commun. 15, 113 (1974).
2. 3. 4.
CONWELLE.M.,Phys. Rev. B!!, 1508 (1975). RICE S.A., GUIDOTTI D., LEMBERG H.L., MUR1~HYW.C. & BLOCH A.N., in Advances in Chemical Physics (Edited by PRIGOGINE I. & RICE S.A.), Vol. XXVII. Wiley, NY (1974). CUNNINGHAM S.L., MARADUDIN A.A. &WALLIS R.F.,Phys. Rev. BlO, 3342 (1974).
5.
CONWELL E.M. (submitted for publication).
6.
ANDERSON W.E., ALEXANDER R.W. & BELL R.J.,Phys. Rev. Lett. 27, 1057 (1971).
7.
CONWELL E.M., Solid State Commun. 14,915 (1974).