Quantum cascade lasers based on quantum dot superlattice

Original Paper

phys. stat. sol. (a) 202, No. 6, 987 – 991 (2005) / DOI 10.1002/pssa.200460714

Quantum cascade lasers based on quantum dot superlattice Ivan A. Dmitriev*, 1, 2 and Robert A. Suris**, 1 1 2

Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia Institut für Nanotechnologie, Forschungszentrum Karlsruhe, 76021 Karlsruhe, Germany

Received 19 July 2004, revised 14 February 2004, accepted 14 February 2004 Published online 5 April 2005 PACS 42.55.Ah, 42.55.Px, 78.67.–n, 78.67.Hc, 78.67.Pt, 73.63.–b A theory of threshold characteristics of novel quantum cascade lasers with an active medium made of a regular array of quantum dots is presented. Very low threshold current density for the room-temperature operation is predicted. © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1

Introduction

The possibility of an optical amplification by a quantum well (QW) superlattice subject to a strong DC electric field was predicted more than 30 years ago [1]. This idea stimulated a 15-year research effort at Bell Labs which culminated in 1994 with the invention of a quantum cascade (QC) laser [2]. Now, the QC lasers greatly overperform mid-IR diode lasers and are used in a growing number of industrial and scientific applications [3]. The performance of a QW QC laser is, however, fundamentally limited mainly due to a wide QW subband spectrum, corresponding to a free electronic in-plane motion (motion in the perpendicular direction is quantized by QW potential and DC field). The wide spectrum gives rise to a fast non-radiative decay due to phonon emission whose rate is three orders of magnitude larger than the radiative one. Thus, the population inversion needed for a laser operation can be achieved at a high threshold current of a few kA/cm2 only [2]. The continuous in-plane spectrum of QWs also results in a strong free-carrier absorption and corresponding losses due to current excited by the in-plane electric field component of a laser mode. Here we exploit two alternative schemes of a cascade laser based on a quantum dot (QD) superlattice (first proposed in Ref. [4]). The purely discrete spectrum of QDs allows for a significant reduction of the non-radiative decay rate, threshold current, and optical losses.

2

Three-level scheme of QD cascade laser

In Fig. 1 we show a cavity of a laser with an active medium which forms a two-dimensional array of QD chains. We consider periodic chains of identical coupled QDs [5]. By contrast, there is no coupling between the chains. The current through the chains is provided by the tunnel couplings to the leads. The lead inside the active medium can be made by an appropriate quantum well nanostructure [2, 3] in order to reduce the optical loss in the cavity. In this section a phenomenological model of electron transport through a single chain is presented for a scheme utilizing three energy levels in each QD (Fig. 2). This scheme is appropriate for weak coupling * **

Corresponding author: e-mail: [email protected] e-mail: [email protected] © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

988

I. A. Dmitriev and R. A. Suris: Quantum cascade lasers based on quantum dot superlattice cladding layer

j(V)

3

QD chains

OCL

x, j, Ex V

2

Ex(x)

j0

z, k, Ez

cladding layer

3

1

j(V)

Ez(x)

j(V)

2

j(V)

1

y, B –1

0

1

Fig. 1 Laser structure and electric field distribution E x ( x ), Ez ( x ) for the lowest TH mode of the cavity. QDs are placed near the wall of the optical confinement layer where E x ( x ) has a maximum.

Vres

V

Fig. 2 Scheme of the principle of a threelevel QD cascade laser implying weak coupling between QDs. Inset: tunnelling probability near the resonance.

between QDs. Under a strong DC voltage V applied to the chain (with period a and total number of QDs N), the first (ground) level of each QD falls into resonance with the third level of the adjacent QD. The tunnelling current near the resonance is proportional to the population difference of levels 1 and 3 (see inset in Fig. 2): J = ej (V ) [ n1 - n3 ] ,

j (V ) = j0 (1 + dV 2 /V
2 ) -1 .

(1)

Here j0 = 2|D|2 t res is the tunnelling rate strictly at the resonance [Vres
N (e 3 - e1 )/e ], dV = V - Vres the detuning of voltage from the resonance, ∆ the tunnelling matrix element, and 2V
= 2 /eNt res the reso-1 . Levels 1 and 2 are close nance width determined by the decoherence rate of the tunnelling transition t res to the resonance with longitudinal optical phonons, e 2 - e1
w LO , which provides a high transition rate t 21-1 between these levels. By contrast, the energy difference between levels 2 and 3 differs essentially from the phonon energy, so that the corresponding relaxation rate t -1 is substantially lower [6]. Corre-1 ∼ t 21-1 , spondingly, the resonance width is also determined by fast transitions between levels 1 and 2, t res and the condition for weak coupling between QDs validating Eq. (1) is |D|t 21
 . Population inversion of the levels 3 and 2, d ∫ n3 - n2 > 0 , is possible under the condition t 21-1
t -1 . In that case levels 1 and 2 are in thermal equilibrium, n2 (1 - n1 ) = exp ( -
w LO /T ) n1 (1 - n2 ) ,

(2)

where T is the temperature of the phonon bath. Occupation of the third level in dynamical equilibrium is governed by the equation

∂ n3 /∂ t = j (V ) (n1 - n3 ) - t -1 n3 (1 - n2 ) = 0 .

(3)

Equations (1) – (3) allow us to determine the current per chain J and the inversion d at a given temperature, voltage, total occupancy of a dot n = n1 + n2 + n3, and a single microscopic parameter j0 t (Fig. 3):

d (n, T , j (V ) t ) = n3 - n2 ,

J (n, T , j (V ) t ) = et -1 n3 (1 - n2 ) .

(4)

One can see from Eqs. (1) – (3) that the total inversion d Æ 1 can be achieved at small temperatures T
w LO and high tunnelling rates j (V ) t
1. It corresponds to the situation when the first and third QD levels are filled, n1 = n3 = 1 , whereas the second is empty, n2 = 0 : the maximum inversion occurs at n = 2 . Even at room temperature strong inversion d ∼ 1/3 is achievable for j (V ) t > 1 (Fig. 3). The dot occupancy n , corresponding to the maximum inversion, is now shifted to n ∼ 1.5 . In the limit j (V ) t
1, strong tunnelling equalizes occupation of the first and third levels. It follows then from Eqs. (1) –(3) that the current through a chain and the inversion are proportional to each other, J
et -1 (1 - e -
w LO /T )-1 d . © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

(5)

Original Paper

phys. stat. sol. (a) 202, No. 6 (2005) / www.pss-a.com

989

0.5

0.5

d

Jt/e

0

-0.5

-1 0

1

n

2

3

0

a)

0

1

n

2

3

b)

Fig. 3 Inversion d (n, T , j (V ) t ) (a) and current J (n, T , j (V ) t ) (b) through a QD chain [Eq. (4)] vs. total QD occupancy n at T = 300 K and for j(V) t = {100, 3, 1, 0.4, 0.15, 0.01} from top to bottom. Here we used
w LO = 36 meV .

Note that this simple relation holds reasonably well down to j (V ) t ∼ 1. Both the current and the inversion saturate very quickly with increasing j (V ) t and, practically, there is no need to make this parameter substantially larger than unity (see Fig. 3).

3

Two-level scheme of QD cascade laser

The three-level scheme can be realized using a QD array with non-equidistant energy levels in an individual QD. Different spacings between levels 1 and 2 and between levels 2 and 3 lead to essentially different non-radiative decay rates for these two pairs of levels. As a result, population inversion in each QD and lasing become possible. Another scheme of a cascade laser utilizing two levels in each QD is operative for strong couplings between QDs (Fig. 4). Obviously, this scheme is applicable for QDs with both equidistant and non-equidistant energy spectra. A large tunnelling matrix element D
w LO  /t 21 between resonant states 1(0) and 2(0) leads to strong mixing and splitting of states. The resulting spectrum can be adjusted by the applied voltage V so that

e 2 - e1 =

D 2 + [e dV /N ]2
w LO ,

e1 - e 2¢ = eV /N - D 2 + [e dV /N ]2 > w LO ,

(6)

where dV = V - V0 stands for detuning from the resonance, as previously. As in the former case, levels 1 and 2 are in thermal equilibrium, provided by a relatively large relaxation time t
t 21 for the lasing transition 1 Æ 2 ¢ . Introducing the total occupancy of a dot, n = n1 + n2 , and using Eq. (2) we obtain the inversion, d = n1 - n2 , and the current through the chain, J = et -1 n1 (1 - n2¢ ) , in the form d = coth


w LO
w LO - coth 2 - n(2 - n) = Jt (1 - e -
w LO /T )/e . 2T 2T

(7)

1 0.8 2 1

2(0)

1(0)

0.6

d 0.4

2¢

0.2

1¢

0.5

Fig. 4 Two-level scheme of QD cascade laser implying strong splitting/mixing of the resonant levels in adjacent QDs.

1

n

1.5

2

Fig. 5 Inversion vs. dot occupancy for the scheme of Fig. 4 (Eq. (7)). From top to bottom: T = {0, 100, 200, 300, 400, 500} K.

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

990

I. A. Dmitriev and R. A. Suris: Quantum cascade lasers based on quantum dot superlattice

One sees that a strong inversion occurs in a very broad temperature range (Fig. 5). Maximum inversion corresponds to a dot occupancy n = 1, d (n = 1) = tanh (
w LO / 4T ) . The current through a chain is related to the inversion in the same way as previously [Eq. (5)].

4

Threshold characteristics

We turn now to the threshold characteristics of a cascade laser with an active medium made of coupled QDs. The laser structure is illustrated in Fig. 1. The QD array is placed near the wall of the cavity, where the transverse electric field amplitude E x ( x ) for the TH mode of the cavity has a maximum. We suppose here that the (homogeneous) line width of the lasing transition is determined by the fast transfer between levels 1 and 2, t ^-1 ∼ t 21-1 . The inhomogeneous line width induced by fluctuations in QD size and position in our model is smaller than t 21-1. The opposite case will be considered elsewhere. The peak modal gain at the lasing threshold is given by the condition

G th = Gx pae -1/ 2 r | r32 |2 Q d th /a = b .

(8)

Here a is the fine-structure constant, e the permittivity of the medium, r the sheet density of QD chains, r32 the dipole moment of the lasing transition, a the period of a QD chain, Q = wt ^ the Q-factor of the lasing transition, w the transition frequency, and b the total loss in the cavity. The transverse optical confinement factor can be estimated as Gx = 4 e Na/l , where N is the number of QDs in a chain (the number of cascades), l the operating wavelength, and we implied that the QD array occupies a small fraction of the l/ 2 cavity only (see Fig. 1). Equation (8) allows us to determine the inversion d th required for generation. In order to estimate the threshold current density jth = r J (the factor of two for spin is included in r ), we use Eqs. (5) and (7) and obtain jth =

e b l (1 - e -
w LO /T )-1 . 2 t | r32 | N Q 4 pa

(9)

We numerically estimate the laser characteristics at the threshold. First of all one should provide a moderately high transition rate t 21-1 to prevent the negative effects of an inevitable technological dispersion in QD sizes and positions. This requires a homogeneous gain line width of at least a few meV or t ^ ∼ t 21 smaller than ps . In order to reduce the non-radiative decay rate t -1 we choose the lasing frequency
w = e 3 - e 2 = 2.5
w LO = 90 meV , which corresponds to the operation wavelength l = 10 µm. Such a choice (see, for example, Ref. [6] and references therein) allows us to expect a reduction by a few orders of magnitude of t -1 as compared to t 21-1, say, t ∼ 100 ps. Taking the number of cacades N = 10 , a = 100 Å, r 32 = 20 Å, r -1/ 2 = 200 Å, and total losses b of 10 cm–1 typical for QW CLs, we obtain – the threshold inversion d th ∼ 0.03 that is ten times less than the achievable value at T = 300 K (see Fig. 3); – the threshold current density jth ∼ 10 A/cm -2 is two orders of magnitude less than typical values for QW cascade lasers; – Equation (9) gives the characteristic temperature T0 (T ) ∫ (∂ ln jth /∂T )-1 = (T 2 /
w LO ) (e
w LO /T - 1) , which is as high as ∼ 600 K at room temperature (we neglect a weak T-dependence of t ,t ^ ). We also stress that, in contrast to a conventional QW superlattice [7], here one can choose the operating voltage region (as marked by dots in the inset in Fig. 2) in such a way that the differential static conductivity of the QD array is positive, ∂j /∂V > 0 , and the whole structure is stable with respect to formation of strong DC field domains. In conclusion, we have developed a theory of the threshold characteristics of a novel quantum cascade laser with an active medium made of a regular array of quantum dots. Our schemes require a relatively high transition rate between quantum dot levels near the resonance with optical phonons with respect to © 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Original Paper

phys. stat. sol. (a) 202, No. 6 (2005) / www.pss-a.com

991

this rate for higher separation of the levels. Under such conditions, a strong inversion is achievable at room temperature and the threshold current density is significantly lower than in quantum well cascade lasers [2, 3]. Acknowledgements This work was supported by RFBR (grant 02-02-17610), by the Leading scientific schools support program (grant 2160.2003.2), by RAS (program “Low-dimensional quantum structures”), and by the Russian Ministry of Science (program “Solid state nanostructures”). One of the authors (R.A.S.) greatly appreciates the support by the A. von Humboldt Foundation.

References [1] [2] [3] [4] [5]

R. F. Kazarinov, R. A. Suris, Sov. Phys. Semicond. 5, 707 (1971). J. Faist et al., Science 264, 553 (1994). F. Capasso et al., Phys. Today 55, 34 (2002). R. A. Suris, NATO ASI Ser. E 323, 197 (1996). Our schemes differ considerably from other proposed schemes for QD cascade lasers [N. S. Wingreen, C. A. Stafford, IEEE J. Quantum Electron. 33, 1170 (1997); C.-F. Hsu et al., IEEE J. Sel. Top. Quantum Electron. 6, 491 (2000)], where active QDs are separated by passive current-carrying regions (injectors) similar to those used in QW cascade lasers. [6] L. Jacak et al., Phys. Rev. B 67, 035303 (2003). [7] R. A. Suris, Sov. Phys. Semicond. 7, 1030 (1974); Sov. Phys. Semicond. 7, 1035 (1974).

© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim