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Quantum wells in semiconductors — An industrial view
Physiea 127B (1984) 219-224 North-Holland. Amsterdam
Q U A N T U M ~EII,1LS IN S E M I C O N D U C ' ] [ ' O I ~ - A N I N D U S T R I . , ~ V I E W B.L.H. WILSON Plessey Research (Ca,wietl)Lid, Allen Clark l~esearch Centre, Cow,well. Towce~ter,Norlhan~s, UK Properties of quantttm wells for industrial utilisation include a novel density of states carve, variaole energy ~.apand modulation doping. C~l.rr.entembodimentsof these ideas in devices are reviewed and some problem ~;reasMd]ealed.
1. Fealmres of qu~ta~l~]~ well~ Features avzilable ~'o~'exploitation include: i) .An energy gap, Ea,i controlled by well thickness, L, intermediate between the barrier and the confined ~uaterial. ,mqenerally both materials should have the same l~Lttice constant, as strained-layer superlatt ces generate dislocations in operation [1]. For infinite barrlet~ llhe wavefunctions are
tit = ( 2/ V) UZu~(r, z) exp Ikr sin k,z ; k= ==n~rtL, I
(1)
where k, v are vectors in the plane. The energy levels above the gap a~Ie
h2k ~. E-~2-~+nzE~;
h~ra E0~l:~;
E,=Ea+En.
(2) ii) Counting states i~j k shows a sharp onset in the density of states at[ E,, = E, + E,, a~d further steps for each value of In. The density oil states of given spin is p~(E)= m*/(2~'h "~) per unit area or p(E)=m*/2;,rh2L pell unit volume. It approaches the 2D density for the original compound when E = t~=E~[. Comparinl~~, a 3D compound of the same ~nergy gap and effective mass, whose density of states rises as ~/2~£,the 2D material has, a higher p(E) if E < E o . Thus degeneracy is delayed in aan'ow wells. The sharp steps in pfE) are as,,;oclated with steps iin recombination and scattering].
iii) Carriers fall into the wells giving rise to concentrated sources of radiation and phonons. As emission of phonons from a levet E is proport~onal to the final state density p(E-h~o), it is relatively favoured over absorption by the constant density of states. iv) Welt shapes are distorted by depletion of free carriers. The latter sit in nan'ow roughly triangular wells. vi) In modulation doping the barrier region only is doped. Carriers in the well undergo less impurity scattering. vi) Multiple quantum wells, MQW, may have a barrier of imermediate height from the outer barrier to aid collection and transport from well to well. Single quantum wells may likewise be surrounded by an outer well or graded region, fig. 1. Small isolated wells are found not to capture carriers efficiently. vii) Appreciable penetn~tion of the barrier lifts the degeneracy of k.- and leads, as in superlattines, to banding and r~sonant tunnelling. Superlattice phenomena are not covered in this paper. viii) Strong fields ir: the plane of the well can Iead to carrier excitation over the barrier and subsequent collection by another electrode ('real-space transfer'). ix) Strong fields across MQWs Mad to production of hot carriers at each high--low barrier and controlled impact ionisation, e.g. solid-sltate photomultiplier. x) Wells act as etalons for phonons. xi) Excitoas have higher binding energy in wells. Impurity levels are also deepened if I[he well size is less than the Bohr radius bat depend on the position in the well.
0378-4363/84/$0300 ~ , Elsevier Science Publishers B.V. (North-Ho]land Physic{s Pubiishing Division)
220
Jg.L.H. Wilson I Quantum wells in semiconductors-an industrial vie)v
Ca)
~~.
_ ~
J
..,i-.-- -. L
FI~. 2. Levels in a finite GaAIAs/GaAs well. Fig, t, Ouamum w¢ll,~, a) Modified ~ingle; b) multiple: c) modified multlple; dl grildcd index,
2. New 2D compounds Real barriers are of finite height, V(~, so that k~ = ~ can become comparable with the inverse decay leng',h of the evanescent w~ve in the barrier, ~ Vo - E ) . Thi:~ causes a ~;ll.,ift i.n the eigenvalues lrom the series n~E0, as e x e m p lified in fig. 2 for GaAIAs. The number of levels is finite, though there is always one b o u n d state in a rectangular well. Band-edge values for m* in the barrier material or some suitable continuation may be u:~ed, and q, and d~l,[dx matched at the boundary. If the states in the barrier are widely different from those in the well r,;fleetion may be strong and the infinite well a fair approximation, c f Osbourn and Smith [2]. Within the well allowance can be m a d e for non-parabolicity, e.g. by taking
t,2(k~ + k ~-)/2m * = E(1 ~-~E),
(3)
levels for electrons in an InSb/CdTe well as calculated by Welzenis and Ridley [3]. It appears quite possible to double the energy gap of I:nSb for accessible welt depths. Similar examples could be generated in the c a d m i u m mercury telluride and lead tin telluride systems. Photodetectors based on these materials would be less susceptible to Moss-Burs~ein shift, but the higher density of states would give poorer performance at high temperatures owing to thormall3/generated carriers. Some alloys are dl;ficult to grow because of an actual miscibiSty gap or because optimal temperatures are different for the end members. T h u s
0,3
eV 0.'/
where m , is the band-edge mass. N o n parabolicity causes the density of states to decrease slightly with energy between the steps. Mosl striking effects are found with small m 5 Fig. 3 shows the height of the lowest energy
'FOD Fig. 3. Conduction znni.'.,).
20r~
300
400
500
band levels for" ]nSb well la,fter W
B,L.H. Wilaon 1 Quantum wells in ,~.¢miconductors-an industrial view wh;.le l n G a A l A s carl be grown by MI~E, the optimal temper~tture tJ0r InGaAs ~s below that of InAiAs. Growth ,hi ~altemate tex-nary layers at temperatures of 530 ° and 550 ° respectively, gives a gap up to 0.474 eV (in a 15 ~ well) above the bulk band edge [411. Q W materials ma)r help to shift useful emission i,zto the visibte iln such systems ,%; Ga,tdAs, and possibly InGaP where there is evklenee for a miscibility gap. The ez:tra wavelength range will not be large, as lhe high mass indirect minima will move in the welt less than the indirect minima, but lasers also benefit from the effect:, described below.
3. Low fln'esho~ld
I:Ls~rs
The aharp onset in the density of st,,tes affects the threshold anti sll:ability of lasers. Consider initially a direct gap semiconductor with completely degenerate palrabotie conduetio'rt and valence bands with density of Stales p¢(E) and p~(E). For heavy doping tile assumption of a constant matrix elem;ent is usually made so that the recombination r(E) is proportional to S p,.(13')o~(E'-E) dE". For 3D semiconductors the fintegral rises as Jrlz from the threshold while for 2D with ccJnstaat density of stmes as E. Quantum wells are often lightly doped, when k selection obtains, anld assuming now a constant matrix element for the periodic part of the wave function, the rate is proportional to the reduced density of states p~(ll0, q.~is will be a step function in 2D and rise as E from the threshold in 3D. In each case the threshold is sharper in 2D and for /g-selection xhe recombination and thus the gain coefficient, g, rises to its maximum value for n =-1 immediate'!y. In practice the densities of states must be weighted by the fra~:tion occupied by carriers. FoUowing Lasher and Stern [5] we write for k-s~lection r(~) -
p,od(E)fo(1
g (E) - o,~AE)(L-
-
f~), [~l,
(4) (5)
for spontaneous and (net) stimulated emission,
f,p,fp
22!
2D
f'PJP~ E
Fig. 4. Density o~ sta;es function p(E) weighted by the Fermi-Dirac dhtribution f in 2D and 3D. where f¢ and [~ are the Fermi-Dirae ft~netions for the conduction and valence band in the~aal equilibrium. P,~d(E) again increases as .JE above threshold for 3D or is a step function for 2D. Fig. 4 shows the weighted p(E) for the 2 cases, concentrating on one band. For 2D the peak emission is at tJ~e threshold, while for 3D it is displaced. For 3D spontaneous emission from hlgh energy states is more important which increases the threshold current. Specific expressions have been given by Dutta [6]. E~periment~,dLly the gain has been found to rise 4 times more rapidly with recombination rate in OW lasers than in DHS. At threshold the net gain, G, in the active region of the laser just offsets the losses due to free cartier absorption and reflectivity, that is 1 G ~ Fg-~-~ln
1 ~" = 0,
when g
=
gth.
(6)
For a single well the confinement factor, fl, the fraction of the laser power which is coupled to the active region, is low and gth and n , will be high typically 10 "a and 2.5 × 10 ~s. Modified multipie wells with opt~mised total thickness have
222
13.L.H. Wilson t Quantum wells in semicon&teWrs- ,,n industrial view
given threshold current densities as low as 250 A cm -a [7]. Reduction i,n the barrier in the M M Q W has several effects: it changes mean refractive index and hence U, 0rid helps to equalis¢ the Fermi level in each well. Below threshold the (negative) net gain in a Iaser was found to increase by only 2 ,~ as g varied by 30 dB compared with 27 A. in conventional DHS lasers [8], Calculations of O(T)--exp T / T . show T~)~-330 K: whiIe experimental values lie in range 1 6 0 - 4 0 0 K compared with ] 60-180 for conventional DH l a s e r . Other favourable features of MQW lasers inelude lower shot noise, improved spectral purity and reduced spectral shift under heavy modulation [9] and the ,~uppression of relaxation oscillations. Single wells need to move the Fermi levels further into the bang than multiple wells at threshold. They thus should, show more spontaneous recombination and higher threshold current. There is however no problem of matching wells.
4. Reduced Auger processes in QW lasers .~ The low value of T,, in InGaAsP and InGaAIAs lasers intended for the 1.3-1.6 tam band is a disadvantage that is usually attributed to Auger processes; ti~e hot carrier may be created in the conduction band, the light bole or the split off band. Higher temperatures and the consequent wider range of k-value:~ favour the Auger process as it eases the demand that momentum and energy should be conserved in the overall process. The situation remains obscure, All agree thai as the Auger processes vary a:, tt a, their relative importance diminishes when quantum wells reduce the carrier den.~ity al threshold. Chiu and Yariv []0] calculate the lifetime for the hot etcclron process, using pmabolic bands and both confined slates and cont;inuous states for a finite well. They find the Auger lifetime to be 10f} thnes higher in the 2D case as energy and momentum conservation is harder to achieve
/
Barrlor height
./ ..,,.,.x"n= I kz
Fig. 5. Conservalion of energy and momentum in an Auger transition in a finite well in the E-k: plane.
when only discrete values of kz are possible for the bound particles, fig. 5. Dutta [1 t] calculated all three processes for an infinite well. (Confinement for the high energy states m a y be less important for large quantum numbers but the low energy states will be quite different in a finite well.) For a typical 1.3 v,m M Q W he finds To is reduced from 330 for radiation alone to 130 K by Auger processes around 325 K for paraholic be~ds, but that nonparabolicity greatly reduces the effects. A similar caleulatior~ by Sugimu:ra [12] for infinite wells and parabotie bands attempted to optimize number and thickness of well and gave low values of threshold current. 0.34 kA cm-" at 300 K for 1.3 ~m, but again strong temperature dependence. I'honon-assistcd Auger processes are not expected to be much affected by quantisation in wells as arbitrary values of q= can be obtained from phonons. Experimentally, Temkin el al. [13~] have shown that GalnAs/GaAIAs .l.5-1.6p, m MQW well lasers have To ~ 6 6 - 7 4 ° C when optically pumped and 70-83 ° when electrically pumped, and that the best threshold current, 2.4 kA/cm -~, comoares well with quaternary DH lasers ira InGaAsP or A[GalrtAs.
B. L,H. Wilson / Quantum ~'clls in semiconductors- an industrial view
5. ]~le~vy excitation in QW lasers kinder heavy (optical) pumping Q W lasers in sho~t high loss cavi!ties exhibit ~lasing near the bottom of bands w~th n as high as 5 [14]. It appears energy loss by optical phonon emission is easier within a ba'ad than from band to band. Sometimes there are phonon shifted lasing peaks one or two phonons ibelo~ peaks ~ss0ciated with the no-phonon e-hh and e-lh transitions. The appearance of phonon replicas is affected by the den:;ity of optical pumping and by whether all or only a part of the la~er cavity is pumped. Some L O phonon line~ appear at quite low injection (optically pumped but with ~'~,,,~2.2 kA cm -:) where a M Q W laser appears to operate: below the energy gap anti 2 L O phonons below the n = 1 heavy hole tr~!.asition. We appear to have coherent phonon as:l;isted iasing: :~ometimes the threshold of the phelnon replica is abrupt. Other explanations have beteft advanced for the phon~n shift. Band gap nan,owing does not account for the small shifts exhibited with injected current. The combination of I.~mission :and absorption in a passive layer can givte rise to llwo appare~tt emission peaks. Such considerati(~ns do not seem to explain for the della phenomena observed by Holonyak [14] and his coworkers, most of which haw~ not been ~ulty explaine¢~ or confirmed.
6. ~attedng Ridley [15] ha.s exlamined the consequences of the approximate conservation of the Z eorr~ponent of crystal momentum in wells folr diff~.~rent types of se;'~ttering, lle found that scatteri~ng increases inversely with well width, and thai: a higher scattlering rate '~ssociated with momentum relaxation occurred for sta*,es of energy at least htotl~ above the lowest bound state owing to the filnite density of states at the bottom of the well. Thi~.'. reduced the mobility f r o ~ the bulk by a factor Ul:, ~o 5 or so, fig. 6, and reduced the eflfect of raodulation doping. The sharp threshold for phonon eraission led to negative resistance ;itt fields around 2 kV"em in
223
0i "~0~ =. 10 -=
•10 SO
r
100
L-_
150
I
t
200
250
300
TK
Fig. 6. Temperature dependence of electron mobilit'~, lirnitcd. in ]lnSb well by unscreened pol~tr o p t i c a l phonon scattering. The parameter is Eo/h~t,o: huh_o = 24.2 mcV. of. fig. 3 (after We;zenis).
GaAs at 77 K, comparable with those which give int,cr-vatley transfer. In triangular wells the effects of modulatiot~J doping have been improved by imposing an uadoped high gap region of thickness d near the heterojunctions; impurity scattering is reduced as d -sl2. At low temperatures interface states dominate the mobility in GaAIAs heterostructures. Application to the HEMT has excited great interest. A t room temperature switching at just over 10 pS for ring oscillators, is however little better in power and delay than for well-designed cc~nventiona! Ga~A,s FETs. As the transmit t.irne is conventionally mostly dependent on peak velocity, which is less changed, this is not surprising. Those benefits which do obtain seem to stern mostly from the high Schottky barrier height of GaAIAs relative both to the free sttrfaee and to GaAs. At low temperatures and short gate lengths the effects of velocity overshoot may be more favourabie in the HEMT. F'ven in long samples Inoue et al. [16] have measured ~x~ electron velocity of 3.6x107 at 2.3kV/cm a: 4 . 2 K in a modulation-doped GaAIAslGaAs heterostructure.
224
7. ~ b l e m
B,L.H, Wilson / Quantum wells in semiconduct~s-an indl,~t:ia'l view areas
Capture of carriers by the wells is little understood. Carriers are not fttlly collected by sirtgJe wells with L < 1(~0/~., an observmion consistent with a single transit with a veloe:ity ot! 107 c m ¢ x a n d a 0.1 pS electron-phonon relaxation time. Holonyak }'14] has described it in terms of p h o non emission without considering; the n a t u : e of the electronic states. Even above the gap the wells act as scatterers and affect the incident wave function. W h e n first captured high lying X and L states are likely to be occupied, with relatively slow transfer to high-lying F bound ,';tates. Optical phonon emission allows transitions between the states, tending to conserve n where possible, helped o u t by e l e c t r o n - e l e c t r o n co[lislons. Apart from tunnelling through thin barriers, carrier transoort between adjace~at wells has been considered ['17] in an elementary model involving leakage above the intervening barrier and diffnsive flow, although diffusion is scarcely adequate for the short distances imtolved. For example, wells 100 .~, wide are well ~,ouplecl by a barrier 0.3 eV high and 2 0 0 ~ wide. In laser operation, stimulated emission followed by ab-
sorption in a m~ighbouring welt couples t h e m closely even tho~Lgh spontaneous emission in a n arbitrax]./direction produces negligible coupling.
&cknowtedgemev'~s T h e author t h a n k s hi~ coll¢agu¢~ for u ~ f t t l discussions and apologi,,;es for the paucity of references possible in a review of thi[s niiture. Refe,renees []J [2] [3] [4] [5]
M,D, Camras et al., LA.P. 54 (1983) 5~3, G.C. Osbourn et al. Phy,~. Rev, BZ9 (1079} 2124,. R,G. W¢]zenls el M,, SoI~d.SL Elee. 27 (1984) 113, D.F, Welch el al.. Appl. Pllys, Lctt. 43 (1983) 762. G. Lasher eta]., Phys. Rev. 133 (1956) A3.53.
[6] N.K. Durra, El. Lett. 18 (1'982) 452, [7] W.T. Tsang, AppI. Phys, Lt,tt. 39 (198I) 736. [8] N.K, Durra et eL. ]EEE J QE-19 (1983) 1243, [9] H. [wamura maI., El. Le'ct. lq (19,83} 180. [10"] L.C, Chiu et aI., IE.E~' J QF~-I.8 (1982) 1406, [11] N,K. Durra, J.A.P. 154(198:0 1236. [12] A, Sugmura. A. Ph. 42 (]9:~3) 17. [I3] I-I. Ternkin et el., Appl. Phi, Lett, 42 (1983) 845. I~1~-]N. HolooyaI~ et aL, ',IEEEJ ~E-16 (t980) 170. (1~] B. Ridley, J. Phys. C.I5 (t98 "~)5899. [16] M, [ao~e et M,, Jap~ J, ApT,l, Phys, 22 (1983) L,213, [17] N.K. Dutta. IEEE J QE;-19 {1983)794.
Q U A N T U M ~EII,1LS IN S E M I C O N D U C ' ] [ ' O I ~ - A N I N D U S T R I . , ~ V I E W B.L.H. WILSON Plessey Research (Ca,wietl)Lid, Allen Clark l~esearch Centre, Cow,well. Towce~ter,Norlhan~s, UK Properties of quantttm wells for industrial utilisation include a novel density of states carve, variaole energy ~.apand modulation doping. C~l.rr.entembodimentsof these ideas in devices are reviewed and some problem ~;reasMd]ealed.
1. Fealmres of qu~ta~l~]~ well~ Features avzilable ~'o~'exploitation include: i) .An energy gap, Ea,i controlled by well thickness, L, intermediate between the barrier and the confined ~uaterial. ,mqenerally both materials should have the same l~Lttice constant, as strained-layer superlatt ces generate dislocations in operation [1]. For infinite barrlet~ llhe wavefunctions are
tit = ( 2/ V) UZu~(r, z) exp Ikr sin k,z ; k= ==n~rtL, I
(1)
where k, v are vectors in the plane. The energy levels above the gap a~Ie
h2k ~. E-~2-~+nzE~;
h~ra E0~l:~;
E,=Ea+En.
(2) ii) Counting states i~j k shows a sharp onset in the density of states at[ E,, = E, + E,, a~d further steps for each value of In. The density oil states of given spin is p~(E)= m*/(2~'h "~) per unit area or p(E)=m*/2;,rh2L pell unit volume. It approaches the 2D density for the original compound when E = t~=E~[. Comparinl~~, a 3D compound of the same ~nergy gap and effective mass, whose density of states rises as ~/2~£,the 2D material has, a higher p(E) if E < E o . Thus degeneracy is delayed in aan'ow wells. The sharp steps in pfE) are as,,;oclated with steps iin recombination and scattering].
iii) Carriers fall into the wells giving rise to concentrated sources of radiation and phonons. As emission of phonons from a levet E is proport~onal to the final state density p(E-h~o), it is relatively favoured over absorption by the constant density of states. iv) Welt shapes are distorted by depletion of free carriers. The latter sit in nan'ow roughly triangular wells. vi) In modulation doping the barrier region only is doped. Carriers in the well undergo less impurity scattering. vi) Multiple quantum wells, MQW, may have a barrier of imermediate height from the outer barrier to aid collection and transport from well to well. Single quantum wells may likewise be surrounded by an outer well or graded region, fig. 1. Small isolated wells are found not to capture carriers efficiently. vii) Appreciable penetn~tion of the barrier lifts the degeneracy of k.- and leads, as in superlattines, to banding and r~sonant tunnelling. Superlattice phenomena are not covered in this paper. viii) Strong fields ir: the plane of the well can Iead to carrier excitation over the barrier and subsequent collection by another electrode ('real-space transfer'). ix) Strong fields across MQWs Mad to production of hot carriers at each high--low barrier and controlled impact ionisation, e.g. solid-sltate photomultiplier. x) Wells act as etalons for phonons. xi) Excitoas have higher binding energy in wells. Impurity levels are also deepened if I[he well size is less than the Bohr radius bat depend on the position in the well.
0378-4363/84/$0300 ~ , Elsevier Science Publishers B.V. (North-Ho]land Physic{s Pubiishing Division)
220
Jg.L.H. Wilson I Quantum wells in semiconductors-an industrial vie)v
Ca)
~~.
_ ~
J
..,i-.-- -. L
FI~. 2. Levels in a finite GaAIAs/GaAs well. Fig, t, Ouamum w¢ll,~, a) Modified ~ingle; b) multiple: c) modified multlple; dl grildcd index,
2. New 2D compounds Real barriers are of finite height, V(~, so that k~ = ~ can become comparable with the inverse decay leng',h of the evanescent w~ve in the barrier, ~ Vo - E ) . Thi:~ causes a ~;ll.,ift i.n the eigenvalues lrom the series n~E0, as e x e m p lified in fig. 2 for GaAIAs. The number of levels is finite, though there is always one b o u n d state in a rectangular well. Band-edge values for m* in the barrier material or some suitable continuation may be u:~ed, and q, and d~l,[dx matched at the boundary. If the states in the barrier are widely different from those in the well r,;fleetion may be strong and the infinite well a fair approximation, c f Osbourn and Smith [2]. Within the well allowance can be m a d e for non-parabolicity, e.g. by taking
t,2(k~ + k ~-)/2m * = E(1 ~-~E),
(3)
levels for electrons in an InSb/CdTe well as calculated by Welzenis and Ridley [3]. It appears quite possible to double the energy gap of I:nSb for accessible welt depths. Similar examples could be generated in the c a d m i u m mercury telluride and lead tin telluride systems. Photodetectors based on these materials would be less susceptible to Moss-Burs~ein shift, but the higher density of states would give poorer performance at high temperatures owing to thormall3/generated carriers. Some alloys are dl;ficult to grow because of an actual miscibiSty gap or because optimal temperatures are different for the end members. T h u s
0,3
eV 0.'/
where m , is the band-edge mass. N o n parabolicity causes the density of states to decrease slightly with energy between the steps. Mosl striking effects are found with small m 5 Fig. 3 shows the height of the lowest energy
'FOD Fig. 3. Conduction znni.'.,).
20r~
300
400
500
band levels for" ]nSb well la,fter W
B,L.H. Wilaon 1 Quantum wells in ,~.¢miconductors-an industrial view wh;.le l n G a A l A s carl be grown by MI~E, the optimal temper~tture tJ0r InGaAs ~s below that of InAiAs. Growth ,hi ~altemate tex-nary layers at temperatures of 530 ° and 550 ° respectively, gives a gap up to 0.474 eV (in a 15 ~ well) above the bulk band edge [411. Q W materials ma)r help to shift useful emission i,zto the visibte iln such systems ,%; Ga,tdAs, and possibly InGaP where there is evklenee for a miscibility gap. The ez:tra wavelength range will not be large, as lhe high mass indirect minima will move in the welt less than the indirect minima, but lasers also benefit from the effect:, described below.
3. Low fln'esho~ld
I:Ls~rs
The aharp onset in the density of st,,tes affects the threshold anti sll:ability of lasers. Consider initially a direct gap semiconductor with completely degenerate palrabotie conduetio'rt and valence bands with density of Stales p¢(E) and p~(E). For heavy doping tile assumption of a constant matrix elem;ent is usually made so that the recombination r(E) is proportional to S p,.(13')o~(E'-E) dE". For 3D semiconductors the fintegral rises as Jrlz from the threshold while for 2D with ccJnstaat density of stmes as E. Quantum wells are often lightly doped, when k selection obtains, anld assuming now a constant matrix element for the periodic part of the wave function, the rate is proportional to the reduced density of states p~(ll0, q.~is will be a step function in 2D and rise as E from the threshold in 3D. In each case the threshold is sharper in 2D and for /g-selection xhe recombination and thus the gain coefficient, g, rises to its maximum value for n =-1 immediate'!y. In practice the densities of states must be weighted by the fra~:tion occupied by carriers. FoUowing Lasher and Stern [5] we write for k-s~lection r(~) -
p,od(E)fo(1
g (E) - o,~AE)(L-
-
f~), [~l,
(4) (5)
for spontaneous and (net) stimulated emission,
f,p,fp
22!
2D
f'PJP~ E
Fig. 4. Density o~ sta;es function p(E) weighted by the Fermi-Dirac dhtribution f in 2D and 3D. where f¢ and [~ are the Fermi-Dirae ft~netions for the conduction and valence band in the~aal equilibrium. P,~d(E) again increases as .JE above threshold for 3D or is a step function for 2D. Fig. 4 shows the weighted p(E) for the 2 cases, concentrating on one band. For 2D the peak emission is at tJ~e threshold, while for 3D it is displaced. For 3D spontaneous emission from hlgh energy states is more important which increases the threshold current. Specific expressions have been given by Dutta [6]. E~periment~,dLly the gain has been found to rise 4 times more rapidly with recombination rate in OW lasers than in DHS. At threshold the net gain, G, in the active region of the laser just offsets the losses due to free cartier absorption and reflectivity, that is 1 G ~ Fg-~-~ln
1 ~" = 0,
when g
=
gth.
(6)
For a single well the confinement factor, fl, the fraction of the laser power which is coupled to the active region, is low and gth and n , will be high typically 10 "a and 2.5 × 10 ~s. Modified multipie wells with opt~mised total thickness have
222
13.L.H. Wilson t Quantum wells in semicon&teWrs- ,,n industrial view
given threshold current densities as low as 250 A cm -a [7]. Reduction i,n the barrier in the M M Q W has several effects: it changes mean refractive index and hence U, 0rid helps to equalis¢ the Fermi level in each well. Below threshold the (negative) net gain in a Iaser was found to increase by only 2 ,~ as g varied by 30 dB compared with 27 A. in conventional DHS lasers [8], Calculations of O(T)--exp T / T . show T~)~-330 K: whiIe experimental values lie in range 1 6 0 - 4 0 0 K compared with ] 60-180 for conventional DH l a s e r . Other favourable features of MQW lasers inelude lower shot noise, improved spectral purity and reduced spectral shift under heavy modulation [9] and the ,~uppression of relaxation oscillations. Single wells need to move the Fermi levels further into the bang than multiple wells at threshold. They thus should, show more spontaneous recombination and higher threshold current. There is however no problem of matching wells.
4. Reduced Auger processes in QW lasers .~ The low value of T,, in InGaAsP and InGaAIAs lasers intended for the 1.3-1.6 tam band is a disadvantage that is usually attributed to Auger processes; ti~e hot carrier may be created in the conduction band, the light bole or the split off band. Higher temperatures and the consequent wider range of k-value:~ favour the Auger process as it eases the demand that momentum and energy should be conserved in the overall process. The situation remains obscure, All agree thai as the Auger processes vary a:, tt a, their relative importance diminishes when quantum wells reduce the carrier den.~ity al threshold. Chiu and Yariv []0] calculate the lifetime for the hot etcclron process, using pmabolic bands and both confined slates and cont;inuous states for a finite well. They find the Auger lifetime to be 10f} thnes higher in the 2D case as energy and momentum conservation is harder to achieve
/
Barrlor height
./ ..,,.,.x"n= I kz
Fig. 5. Conservalion of energy and momentum in an Auger transition in a finite well in the E-k: plane.
when only discrete values of kz are possible for the bound particles, fig. 5. Dutta [1 t] calculated all three processes for an infinite well. (Confinement for the high energy states m a y be less important for large quantum numbers but the low energy states will be quite different in a finite well.) For a typical 1.3 v,m M Q W he finds To is reduced from 330 for radiation alone to 130 K by Auger processes around 325 K for paraholic be~ds, but that nonparabolicity greatly reduces the effects. A similar caleulatior~ by Sugimu:ra [12] for infinite wells and parabotie bands attempted to optimize number and thickness of well and gave low values of threshold current. 0.34 kA cm-" at 300 K for 1.3 ~m, but again strong temperature dependence. I'honon-assistcd Auger processes are not expected to be much affected by quantisation in wells as arbitrary values of q= can be obtained from phonons. Experimentally, Temkin el al. [13~] have shown that GalnAs/GaAIAs .l.5-1.6p, m MQW well lasers have To ~ 6 6 - 7 4 ° C when optically pumped and 70-83 ° when electrically pumped, and that the best threshold current, 2.4 kA/cm -~, comoares well with quaternary DH lasers ira InGaAsP or A[GalrtAs.
B. L,H. Wilson / Quantum ~'clls in semiconductors- an industrial view
5. ]~le~vy excitation in QW lasers kinder heavy (optical) pumping Q W lasers in sho~t high loss cavi!ties exhibit ~lasing near the bottom of bands w~th n as high as 5 [14]. It appears energy loss by optical phonon emission is easier within a ba'ad than from band to band. Sometimes there are phonon shifted lasing peaks one or two phonons ibelo~ peaks ~ss0ciated with the no-phonon e-hh and e-lh transitions. The appearance of phonon replicas is affected by the den:;ity of optical pumping and by whether all or only a part of the la~er cavity is pumped. Some L O phonon line~ appear at quite low injection (optically pumped but with ~'~,,,~2.2 kA cm -:) where a M Q W laser appears to operate: below the energy gap anti 2 L O phonons below the n = 1 heavy hole tr~!.asition. We appear to have coherent phonon as:l;isted iasing: :~ometimes the threshold of the phelnon replica is abrupt. Other explanations have beteft advanced for the phon~n shift. Band gap nan,owing does not account for the small shifts exhibited with injected current. The combination of I.~mission :and absorption in a passive layer can givte rise to llwo appare~tt emission peaks. Such considerati(~ns do not seem to explain for the della phenomena observed by Holonyak [14] and his coworkers, most of which haw~ not been ~ulty explaine¢~ or confirmed.
6. ~attedng Ridley [15] ha.s exlamined the consequences of the approximate conservation of the Z eorr~ponent of crystal momentum in wells folr diff~.~rent types of se;'~ttering, lle found that scatteri~ng increases inversely with well width, and thai: a higher scattlering rate '~ssociated with momentum relaxation occurred for sta*,es of energy at least htotl~ above the lowest bound state owing to the filnite density of states at the bottom of the well. Thi~.'. reduced the mobility f r o ~ the bulk by a factor Ul:, ~o 5 or so, fig. 6, and reduced the eflfect of raodulation doping. The sharp threshold for phonon eraission led to negative resistance ;itt fields around 2 kV"em in
223
0i "~0~ =. 10 -=
•10 SO
r
100
L-_
150
I
t
200
250
300
TK
Fig. 6. Temperature dependence of electron mobilit'~, lirnitcd. in ]lnSb well by unscreened pol~tr o p t i c a l phonon scattering. The parameter is Eo/h~t,o: huh_o = 24.2 mcV. of. fig. 3 (after We;zenis).
GaAs at 77 K, comparable with those which give int,cr-vatley transfer. In triangular wells the effects of modulatiot~J doping have been improved by imposing an uadoped high gap region of thickness d near the heterojunctions; impurity scattering is reduced as d -sl2. At low temperatures interface states dominate the mobility in GaAIAs heterostructures. Application to the HEMT has excited great interest. A t room temperature switching at just over 10 pS for ring oscillators, is however little better in power and delay than for well-designed cc~nventiona! Ga~A,s FETs. As the transmit t.irne is conventionally mostly dependent on peak velocity, which is less changed, this is not surprising. Those benefits which do obtain seem to stern mostly from the high Schottky barrier height of GaAIAs relative both to the free sttrfaee and to GaAs. At low temperatures and short gate lengths the effects of velocity overshoot may be more favourabie in the HEMT. F'ven in long samples Inoue et al. [16] have measured ~x~ electron velocity of 3.6x107 at 2.3kV/cm a: 4 . 2 K in a modulation-doped GaAIAslGaAs heterostructure.
224
7. ~ b l e m
B,L.H, Wilson / Quantum wells in semiconduct~s-an indl,~t:ia'l view areas
Capture of carriers by the wells is little understood. Carriers are not fttlly collected by sirtgJe wells with L < 1(~0/~., an observmion consistent with a single transit with a veloe:ity ot! 107 c m ¢ x a n d a 0.1 pS electron-phonon relaxation time. Holonyak }'14] has described it in terms of p h o non emission without considering; the n a t u : e of the electronic states. Even above the gap the wells act as scatterers and affect the incident wave function. W h e n first captured high lying X and L states are likely to be occupied, with relatively slow transfer to high-lying F bound ,';tates. Optical phonon emission allows transitions between the states, tending to conserve n where possible, helped o u t by e l e c t r o n - e l e c t r o n co[lislons. Apart from tunnelling through thin barriers, carrier transoort between adjace~at wells has been considered ['17] in an elementary model involving leakage above the intervening barrier and diffnsive flow, although diffusion is scarcely adequate for the short distances imtolved. For example, wells 100 .~, wide are well ~,ouplecl by a barrier 0.3 eV high and 2 0 0 ~ wide. In laser operation, stimulated emission followed by ab-
sorption in a m~ighbouring welt couples t h e m closely even tho~Lgh spontaneous emission in a n arbitrax]./direction produces negligible coupling.
&cknowtedgemev'~s T h e author t h a n k s hi~ coll¢agu¢~ for u ~ f t t l discussions and apologi,,;es for the paucity of references possible in a review of thi[s niiture. Refe,renees []J [2] [3] [4] [5]
M,D, Camras et al., LA.P. 54 (1983) 5~3, G.C. Osbourn et al. Phy,~. Rev, BZ9 (1079} 2124,. R,G. W¢]zenls el M,, SoI~d.SL Elee. 27 (1984) 113, D.F, Welch el al.. Appl. Pllys, Lctt. 43 (1983) 762. G. Lasher eta]., Phys. Rev. 133 (1956) A3.53.
[6] N.K. Durra, El. Lett. 18 (1'982) 452, [7] W.T. Tsang, AppI. Phys, Lt,tt. 39 (198I) 736. [8] N.K, Durra et eL. ]EEE J QE-19 (1983) 1243, [9] H. [wamura maI., El. Le'ct. lq (19,83} 180. [10"] L.C, Chiu et aI., IE.E~' J QF~-I.8 (1982) 1406, [11] N,K. Durra, J.A.P. 154(198:0 1236. [12] A, Sugmura. A. Ph. 42 (]9:~3) 17. [I3] I-I. Ternkin et el., Appl. Phi, Lett, 42 (1983) 845. I~1~-]N. HolooyaI~ et aL, ',IEEEJ ~E-16 (t980) 170. (1~] B. Ridley, J. Phys. C.I5 (t98 "~)5899. [16] M, [ao~e et M,, Jap~ J, ApT,l, Phys, 22 (1983) L,213, [17] N.K. Dutta. IEEE J QE;-19 {1983)794.