Intersubband optical nonlinearity and bistable behavior of semiconductor superlattices

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Superlattices and Microstructures, Vol. 21, No. 2, 1997

Intersubband optical nonlinearity and bistable behavior of semiconductor superlattices G- Z, S- P* Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China (Received 9 November 1995) The optical nonlinearity and bistability (OB) induced by intersubband transitions (ISBTs) in semiconductor superlattices are studied by means of the Kronig–Penney model and the density-matrix method. The OB state equation in a Fabry–Perot configuration is derived. It is shown that the ISBT-induced OB can be realized over a wide frequency range, and its bistable domain and hysteresis loop are much larger than those of a two-level system. Numerical illustration of the OB behavior in a GaAs/Al Ga As superlattice is also given. x 1~x ( 1997 Academic Press Limited

1. Introduction With the advancement of material growth techniques such as molecular beam epitaxy, semiconductor superlattices have attracted much attention due to their novel electronic, optical, and other physical properties. In particular, it has been found that the quantum confinement and tunneling effects on carriers result in the formation of energy subbands in a superlattice and that intersubband transitions (ISBTs) in a superlattice can induce very large optical nonlinearities [1]. Furthermore, it is easy to realize the desired infrared region and optimal susceptibility by changing the superlattice parameters (well and barrier widths, barrier height, etc.) [2]. It has also been proved that the carrier relaxation time in superlattice structures can be very short (e.g. \1 ps) [3]. The ISBTs of a superlattice may thus offer a useful physical mechanism for the manufacture of practical devices. For instance, ISBTs have been extensively studied for the important applications in infrared detection [4], an infrared superlattice laser using ISBTs has also been proposed and discussed [5]. However, as far as we are aware, to date little work has been done on the optical bistability (OB) of ISBTs in superlattices. In this paper we first derive the Maxwell–Bloch (MB) equations based on a two-subband model. We then give stationary state solutions of the MB equations and analyze the characteristics of the ISBT induced optical nonlinearity and bistable behavior.

2. Maxwell‒Bloch equations of a two-subband model We consider a superlattice consisting of N single QWs with well width a and barrier width b. The superlattice wavenumber is given by k\2lp/Nd, where d\a]b is the period of the superlattice and l is any integer between [N/2]1 and N/2. We denote an envelope wave function as D jkT and *Also at CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China.

0749–6036/97/020177]10 $25.00/0

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its corresponding electronic energy as E , where j is a subband index. An analytic coordinate reprejk sentation / (z) of D jkT has been given by the Kronig–Penney model [6], where the coordinate z is jk along the superlattice growth-axis. The energy spacing between two subbands is in the infrared region, and it varies with the change of a, b, and the barrier height V. For a superlattice material with the usual doping concentration (e.g. N \1]1018 cm~3), at room temperature most of the d electrons are in the first (ground) subband. The intersubband transition from the first to second subband can be easily induced by infrared radiation of a suitable wavelength. A useful formulation for studying nonlinear optical effects of ISBTs is the two-subband model, which consists of the first subband MD1kTN and the second subband MD2kTN, each subband comprising N envelope electronic states with different k. In comparison with the usual two-level model, an advantage of the twosubband model is that it can be applied to treat the intersubband problem of a superlattice with any subband widths, i.e. any degree of tunnel broadening [7]. An electronic state in a superlattice D jkT can be approximately separated into a product of the form: D jkT\Dk TD jkT, where Dk T is a bulk Bloch state with k \Mk ,k N being a wavevector in the x-y t t t x y ~ 2k2/2m*, where k{Mk,k N\Mk,k ,k N and m* plane. The eigenenergy of D jkT is given by E k\E ]h j jk t t x y is the effective mass of the electron. When an ISBT occurs, the dipole matrix element can be approximated as [8] S j@k@DkD jkT\k (k)d dk k , where l\ez is the dipole operator, l is its matrix element j{j k{k {t t j{j between envelope states D jkT and D j@kT, and l \l* . j{j jj{ In the presence of a radiation field with electric vector E (x,t), the Hamiltonian of the twoz subband model can be written as 2 H\ ; ; E kc`k c k[; [k (k)E (x,t)c`kc k]c.c.], (1) j j j 21 2 1 z k j/1 k where c`k and c k ( j\1,2) are, respectively, the creation and annihilation operators. The optical beam j j is assumed to be monochromatic and to propagate along the x direction, i.e. 1 E (x,t)\ E(x,t)e~iut]c.c. z 2 The intersubband polarization P induced by the electric field E can be expressed as: z z 1 2 P (x,t)\ Tr(ok)\ ; [k (k)pk]k (k)p*k ]\M s E (x,t), 21 z 0 zz z V V k 12

(2)

(3)

where V is the superlattice volume, v is the susceptibility due to the ISBTs, and pk{q k k\Sc kc`kT zz 2 ,1 2 1 is a density matrix element. The total electronic polarization induced by E can be written as: z P(x,t)\M (v ]v )E (x,t), (4) 0 0 zz z where v denotes the susceptibility without ISBTs. 0 Let us introduce modified Bloch variables as: uk\k (k)pkeiut]k (k)p*ke~iut, 12 21 (5) vk\i[k (k)pkeiut[k (k)p*ke~iut], 12 21 dk\n k[n k , 2 1 where n k{Sc`k c kT ( j\1,2) denotes the population in the electronic state D jkT, and we have included j j j dipole moments l (k) and l (k) in the definition of uk and vk in order to facilitate the following 12 21 calculation. The equation of motion for the expectation value of operator A in the density matrix formalism is

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179 (6)

where C is the decay operator. Substituting separately uk,vk and dk for A in eqn (6), and using the rotating-wave approximation, we derive the following equations: 1 u˙k\[c uk[(u [u)vk] Dk (k)D2Im(E)dk , t k ~ h 21 1 v˙k\[c vk](u [u)uk] Dk (k)D2Re(E)dk , t k ~ h 21

(7)

1 d˙k\[c (dk[d0k )[ [ukIm(E)]vkRe(E) ], l ~ h which are the optical Bloch equations of the two-subband model suitable for studying the intersubband electronic polarization and optical nonlinear effects in semiconductor superlattices. Here ~ is the transition frequency, c and c are, respectively, the dephasing and depopulation x {(E [E )/h k 2k 1k t l rates, and d0k \n0k[n0k with n0k ( j\1 or 2) being the value of n k when the applied electronmagnetic field 2 1 j j is absent, i.e. n0k is given by the Fermi population distribution: n0k\f k\1/M1]exp[(E k[E )/k T]N. j j j j F B The electric field E (x,t) of the optical beam obeys the Maxwell wave equation: z L2E 1 L2E L2P z[ z\l , (8) 0 Lt2 Lx2 c2 Lt2 where P\P(x,t) is the total polarization given by eqn (4). A commonly used configuration of OB devices is a Fabry–Perot resonator (Fig. 1) consisting of two lossless mirrors of reflectivity R and transmissivity T\1[R. A nonlinear medium (e.g. in our case, a superlattice) fills the space between the mirrors. The Fabry–Perot resonator is driven by a stationary coherent field injected into the cavity. In this case the electric-field amplitude E(x,t) in eqn (2) can be expressed as: 1 E(x,t)\ [E (x,t)ei(qx~/f)]E (x,t)e~i(qx~/b)]]c.c., b 2 f

(9)

where E (E ) and / (/ ) are, respectively, the slowly-varying amplitude and original phase of the f b f b forward (backward) travelling field, and q\n x/c is the wavenumber. The absorption coefficient and r refractive-index change due to ISBTs are defined as a(x,E)\(x/2n cM )Im(M v ), r 0 0 zz

(10) Dn(x,E)\(1/2n M )Re(M v ), r 0 0 zz where n \q1]v is the refractive index of the medium without ISBTs. From these two definitions r 0 and eqns (2), (4), (8) and (9), we derive the Maxwell equation in the slowly-varying amplitude approximation as follows: LE n LE E L/ f@b^ r f@b]a(u,E) f@b f@b^a(u,E)E \0 f@b Lx c Lt u Lt (11) L/ n L/ 1 LE u f@b^ r f@b[[a(u,E)]a ] f@b] *n(u,E)\0, c uE c Lt Lx Lt c f@b where the plus (minus) sign is for E and / (E and / ). f f b b

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εf

εi

εt

εr

z

εb y

x

M

M

Fig. 1. Electric fields in a Fabry–Perot cavity. The mirrors M have reflectivity R and transmissivity T\1[R. E , E , and E are the incident, reflected, and transmitted fields, respectively. E and E are the forward and i r t f b backward travelling fields, respectively.

Setting u˙k\v˙k\d˙k\0 and LE /Lt\L/ /Lt\0 in eqns (7) and (11), we find the following f@b f@b stationary solutions of the Maxwell–Bloch equations. ¯ [u ¯ )Re(E)[Im(E) ] Dk (0)D2 k¯2d0k [ (u k uk\[ 21 ] k , ~ 1](u ¯ [u ¯ )2]k¯ 2 (E/E )2 hc t k k s ¯ [u ¯ )Im(E) ] Dk (0)D2 k¯ 2d0k [Re(E)](u k vk\ 21 ] k , ~ 1](u ¯ [u ¯ )2]k¯2 (E/E )2 hc t k k s 1](u ¯ [u ¯ )2 k dk\ d0 , 1](u ¯ [u ¯ )2]k¯2 (E/E )2 k k k s and x E (x)\E (0)expM«[a(x,E)[i Dn(x,E)]xN. f@b f@b c

(13)

Here l (0){l (k)D , and we have defined the normalized dipole moment l¯ \l (k)/l (0), the 21 21 k/0 k 21 21 ~ Ic c /l (0). normalized frequencies x ¯ \x/c and x ¯ \x /c , and a saturation electric field E {h t l 21 t k k t s

3. Intersubband absorption and refractive-index change From eqns (3), (5), (10) and (12), we derive the ISBT-induced absorption coefficient and refractive-index change as follows:

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0.5

0.3 ∆n

α I=0

0.2

I/Is = 0.4

0.4

I/Is = 0.8

0.3

0.0 ∆n

α (104 cm–1)

0.1

–0.1

∆n

0.2

–0.2 0.1 –0.3

0

50

100

150 200 hω (meV)

250

–0.4 300

Fig. 2. Plots of absorption coefficient a(x, E) and refractive-index change Dn(x,E) for a superlattice comprising ˚ and barrier width 25 A ˚ . Here solid, dashed, and 100 periods of GaAs/Al Ga As with well width 100 A 0.8for the optical intensity I/I (\DE/E D2)\0, 0.4, and 0.8, respectively. The satudotted lines represent the0.20 results s s ration intensity I (\2n cM DE D2)\1.53 MW cm~2. s r 0 s

k¯ 2D k k a(u ¯ ,E)\a ; , 0 1](u ¯ [u ¯ )2]k¯2 (E/E )2 k k s k

(14) a c (u ¯ [u ¯ )k¯ 2D k k k *n(u ¯ ,E)\ 0 ; . ¯ [u ¯ )2]k¯ 2 (E/E )2 u 1](u k k s k Here a \N xDl (0)D2/2n cM ~hc represents a typical value of the linear absorption coefficient, and 0 d 21 r 0 t D {2Rk ( f k[f k)/N V is a normalized population difference. k 1 2 d t We can see from eqn (14) that the behavior of a versus x and Dn versus x in the two-subband model is more complicated than that in the two-level model. Equation (14) also shows that a and DDnD decrease as the electric field E increases. This is due to the optical saturation in the coherences (u ,v ) and population differences (d ), as shown in eqn (12). Moreover, since the transition frequency k k k x ¯ , dipole moment l¯ , and population difference D in eqn (14) are functions of superlattice park k k ameters (well and barrier widths, barrier height, etc.) expression (14) indicates that a(x ¯ ,E) and Dn(x ¯ ,E) depend sensitively on the composition and structure of the superlattice. In other words, the ISBT-induced optical properties can be optimized by a suitable design of the superlattice. ˚ We have performed numerical calculations for a superlattice comprising 100 periods of 100 A ˚ Al GaAs wells and 25 A Ga As barriers. The barrier height is determined by [9] 0.20 0.8

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and the electron effective-mass by [10] m*\(0.0665]0.0835x)m 0 with m being the free-electron mass. The other parameters used are the doping concentration 0 N \1.0]1018 cm~3, temperature T\300 K, refractive index n \3.2, dephasing rate ~hc \4.608 meV d r t and depopulation rate ~hc \1.317 meV [7]. Based on the Kronig–Penney model [6] and eqn (14), we l obtain numerical results for the absorption coefficient a(x ¯ ,E) and refractive-index change Dn(x ¯ ,E) as shown in Fig. 2. Here solid, dashed and dotted lines represent a(x ¯ ,E) and Dn(x ¯ ,E) at the optical intensities I/I (\DE/E D2)\0, 0.4, and 0.8, respectively, where I \2n cM DE D2\1.53 MW cm~2 is the s s s r 0 s saturation intensity. As seen from the figure, the linear absorption coefficient a(x ¯ ,0) peaks at ~ hx B117 meV and has a full width at half maximum (FWHM) Dx B34 meV. We know that the a HW linear coefficient of a two level system is a Lorentzian with the FWHM of 2c , so it is clear that the t FWHM of the superlattice, Dx , is much larger than its two-level counterpart. The above menHW tioned wide frequency range of ISBT induced absorption is intimately related to the tunnel broadening effect of superlattices [6]. Based on the Kronig–Penney model, we obtain the subband widths DE 1 and DE and hence the tunnel bandwidth [7] dE(\DE ]DE )\70.5 meV. 2 1 2

4. Behavior of absorptive optical bistability It is usual to distinguish two extreme OB cases called absorptive OB and dispersive OB which correspond to DnB0 and aB0, respectively. Since a superlattice can be easily designed to have a large tunnel bandwidth, the absorption peak (x ) can be far away from the dispersive peak (x ). Thus a d we can discuss separately the absorptive and dispersive OBs of superlattices. In this paper we analyse in detail the absorptive OB only. That is, we assume in the following that Dn\0. From Fig. 1 and eqn (13) and by using the mean field approximation [11], we can write the following relations: E (0)\JTE ]JRE (0) f i b E (0)\JRE (L)e~(ib`aL), (15) b f E (L)\E /JT\E (0)e~(ib`aL), f t f where E and E are, respectively, the incident and transmitted fields, and a\1/L:La(x,E)dx is the i t 0 average total absorption coefficient, with L being the superlattice length in the x direction. In the mean field limit we have aL@1, thus from eqn (15) we can derive the OB state equation (to the thirdorder nonlinearity of the electric field) as follows:

C

D

2 2Cf¯ 2 (u) 1 I \I 1]¯ , (16) ¯ i t f (u)]f (u)I /I 1 2 c s where I \2cM DE D2 is the optical intensity, and I {2n cM DE ]E D2 is the intensity in the cavity, i@t 0 i@t c r 0 f b which is given by the following relation under resonant conditions [n xL/c\lp(l\1, 2, . . .)]: r I \n (1]JR )2I /T. (17) c r t The cooperative parameter C in eqn (16) is defined as 1]R a LF, C{ 2T 0

(18)

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44

40 36 32

C crit

28

24 20 16 12 8 4 –5

0

5 10 (ωa – ω )/ γ t

15

20

Fig. 3. Plot of the critical cooperative parameter C versus frequency detuning (x [x)/c , with the solid lines a two-level t representing the same superlattice system as in Fig. crit 2. The dashed lines represent the system.

where k¯ 2D k k , (19) F\; 1](u ¯ [u ¯ )2 k a k and x ¯ \x /c , with x being the frequency corresponding to the peak of the linear absorption a a t a coefficient. k¯ 2D ¯f (u)\ 1 f(u,0)\ 1 ; k k , 1 ¯ [u ¯ )2 F 1](u F k k

(20)

and k¯4D ¯f (u)\ 1 ; k k . (21) 2 ¯ [u ¯ )2]2 F [1](u k k From eqns (20) and (14) we can see that ¯f (x) is the normalized linear absorption coefficient, and hence 1 we have ¯f (x )\1 and ¯f (x)\1 for xDx . Note that when tunnel broadening can be neglected in 1 a 1 a superlattices with very wide barriers, the present model reduces to the two-level model, and hence the OB state eqn (16) reduces to the well-known result of the latter model, see, e.g. eqn (2.4-1) in Ref. [11].

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2.5

∆Ι / Ι s

2.0

1.5

1.0

0.5

0.0 –5

0

5 10 (ωa – ω )/ γ t

15

20

Fig. 4. Plot of the bistable domain DI versus frequency detuning (x [x)/c . The solid lines represent the same a t superlattice system as in Fig. 2. The dashed lines represent the two-level system.

The critical point (at the onset of the optical bistability) is given by dI /dI \d2I /dI2\0. From i t i t these two equations and eqn (16), we obtain the bistable threshold condition: C ¯f (x)\4. (22) crit 1 Moreover, since the switch-up incident intensity I and switch-down incident intensity I correspond M m to the two extreme points in the plot of I versus I , we have dI /dI D \0. Substituting (16) and i t i t Ii/IM,m (17) into this relation, we obtain T ¯f (u)I 1 s \ ][ (Cf¯ (u)[1)«JCf¯ (u) (Cf¯ (u)[4) ] M@m n (1]JR )2f¯ (u) 1 1 1 r 2 2 2Cf¯ (u) 1 . (23) ] 1] Cf¯ (u)«JCf¯ (u) (Cf¯ (u)[4) 1 1 1 Equations (22) and (23) clearly show that I \I when C\C , corresponding to the onset of M m crit bistability; bistability is impossible for C\4 (because ¯f (x)¹1); for a fixed value of C[4, bistability 1 exists (i.e. I ºI ) in a finite range of the frequency x. Equation (20) shows that when the detuning M m Dx [xD increases, ¯f (x) decreases more slowly than its counterpart in a two-level system (i.e. c2/ a 1 t I

C

D

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185 0.02

1.0 0.01

0.8

It / I s

0

1

2

3

4

0.6

0.4

0.2

0

1

2 Ii / I s

3

4

Fig. 5. Plot of the output intensity I versus input intensity I . The solid lines represent the same superlattice t i system as in Fig. 2. The dashed lines represent the two-level system.

[c2](x [x)2]), hence for a given C ([4), relation I ºI can be satisfied in a larger domain of t a M m Dx [xD. In other words, ISBT-induced bistability in a superlattice can be realized in a wider frea ~ Ac , the OB quency range. In particular, when the tunnel bandwidth [7] dE is so large that dE/h t frequency-range of a superlattice can be much wider than its two-level counterpart. In order to see more clearly the behavior of ISBT-induced OB in a superlattice, we have made numerical calculations for the same superlattice as in Fig. 2. The sample parameters have been given in Section 3, and the reflectivity R\0.98. Based on the Kronig–Penney model and the OB threshold eqn (22), we obtain the critical cooperative parameter C as a function of (x [x), as shown by the crit a solid line in Fig. 3, where the dashed line represents the result of the corresponding two-level system. The figure clearly shows that the bistability can be realized over a wider frequency range in the superlattice system than in the two-level system. Moreover, the calculated value of the cooperative parameter is C\40.1, corresponding to a frequency-detuning range of (x [x)/c \[4.1 to 16.7. a t Figure 4 presents the bistable domain DI\I [I (calculated from eqn (23)) versus (x [x)/c . The M m a t figure indicates that DI of the superlattice (solid line) is larger than that of the two-level system (dashed line). Moreover, as seen from Fig. 3A, the bistable domain DI has its maximum when the incident light is resonant with the absorption peak (i.e. x [x\0), and DI decreases with an increase a of the detuning Dx [xD. The system is at a critical point (DI\0) for certain values of the detuning, a i.e. (x [x)/c \[4.1 or 16.7. This result is consistent with the above analysis of Fig. 3. The figure a t also clearly shows that this frequency-detuning range ([4.1 to 16.7) is much larger than that of the corresponding two-level system (i.e. [2.9 to 3.1, as seen from the figure). Finally, the calculated results of the output intensity versus input intensity are shown in Fig. 5. For clarity we also give, in

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the inset of the figure, an enlarged plot near the up-switch point. From the figure we can see that the superlattice has a larger OB hysteresis loop than its two-level counterpart. In conclusion, based on the two-subband model of envelope states, we have derived the Maxwell–Bloch equations suitable for describing the electronic polarization and nonlinear optical effects induced by intersubband transitions in a semiconductor superlattice. The stationary solutions of the Maxwell–Bloch equations and the optical bistability equation have been derived. Numerical calculations for a GaAs/Al Ga As superlattice have also been given. The calculated results show x 1~x that the ISBT-induced OB is superior to its two-level counterpart in frequency tunability and has a larger hysteresis loop. Moreover, the absorption peak ~hx of the superlattice is very close to the a photon energy of a CO laser (117 meV), so this superlattice structure is suitable for OB operation 2 with a CO laser as light source. We expect that the present study will stimulate further investigations 2 into ISBT-induced OB and its potential in infrared nonlinear optics and device applications.

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