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Intersubband electro-optic modulators for near and mid infrared applications
Superlattices and Microstructures, Vol. 23, No. 5, 1998
Intersubband electro-optic modulators for near and mid infrared applications R. Kapon, A. Segev, A. Sa’ar Deprtment of Applied Physics, The Fredi and Nadine Herrmann School of Applied Science, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
(Received 15 July 1996) Optical transitions between conduction subbands of asymmetrical quantum-well structures can produce large second-order optical nonlinearities. In particular, it is possible to engineer asymmetrical quantum-well structures to have very large electro-optic (EO) coefficients. In this paper we study three alternative approaches to design and fabricate intersubband EO modulators, i.e. the Stark modulator, the quantum-interference (QI) modulator, and the carrier-density (CD) modulator. We show that near a given resonance the Stark modulator has the largest EO coefficient. However, the linear intersubband absorption limits the usability of this modulator to a very short device length. Far from resonance we found that the CD and the QI modulators are more efficient. Furthermore, we found that these modulators can be utilized for near-infrared modulation in the 1.3–1.5 µm spectral range. These modulators are almost unaffected by the linear intersubband absorption so that their efficiency can be very high. We present preliminary experimental results that demonstrate the operation of the QI modulator. c 1998 Academic Press Limited
Key words: modulators, electro-optics, intersubband transitions, quantum wells.
1. Introduction Over the past few years there has been an ongoing research effort to develop semiconductor high-speed electro-optic (EO) modulators that can monolithically be integrated with other optical and electronic devices on the same chip. One class of semiconductor EO modulators that has recently been proposed is based on intersubband transitions (ISBTs) in quantum wells (QWs) [1– 6]. These EO modulators can be divided into three classes according to the physical mechanism that generates the EO effect: the Stark, carrier-density (CD) and the quantum-interference (QI) modulators. The Stark modulator relies on the linear dc Stark effect in QWs (i.e. a linear shift of the energy levels under the application of an external dc electric field [1–3]). The QI modulator is based on a coherent interference among the QW envelope states [4] while the CD modulator is related to a modulation of the CD in the QW levels [3, 5–6]. Recently we have found that under most circumstances the EO Stark susceptibility dominates near a given intersubband resonance [3–4]. However, taking into account the linear intersubband absorption we have shown that the Stark modulator cannot provide high-modulation efficiency. In addition, we investigated a coupled QW (CWQ) structure for which the QI effect dominates over the Stark effect. In this paper we extend the idea of intersubband EO modulation for off-resonance operation. We show .
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that far from resonance the QI and the CD susceptibilities dominate over the Stark susceptibility. Taking into account the linear intersubband absorption we derive a figure of merit and conclude that efficient near-infrared EO modulators can be realized. Furthermore, we have designed a modulator that operates at 1.5 µm and is suitable for optical communication applications. Finally, we present preliminary experimental results that demonstrate the operation of the QI modulator at the mid-infrared wavelengths.
2. Definition of the intersubband electro-optic susceptibilities Our approach to derive the EO susceptibilities of asymmetric QWs is very similar to that of [4]. The influence of the external dc electric field on the QW is taken into account by letting the QW energies, envelope wavefunctions and CDs depend on the externally applied dc electric field across the QW. Expanding the linear intersubband susceptibility [7] in a power series of the dc electric field, F, and keeping only linear terms in the expansion yields the following expression for the second-order EO susceptibilities. X ∂χi j ∂ωi j (1a) χ S(2) (ω) = ∂ωi j ∂ F F=0 i, j X ∂χi j ∂z i j ∂χi j ∂z ji (1b) + χ Q(2)I (ω) = ∂z i j ∂ F ∂z ji ∂ F F=0 i, j X ∂χi j ∂ N j ∂χi j ∂ Ni (2) + (1c) χC D M (ω) = ∂ Nj ∂ F ∂ Ni ∂ F F=0 i, j .
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where χi j (ω, F) =
z i j (F)z ji (F) e2 [N j (F) − Ni (F)] ε0 h¯ [ω − ωi j (F) − i0i j ]
χi j is the linear susceptibility associated with the i → j ISBT [7], z i j = h9i |z|9 j i is the exchange electric dipole-matrix element (divided by e), ωi j = (εi − ε j )/h¯ is the transition frequency, 9i and Ni are the envelope wavefunction and the density of carriers associated with the ith subband respectively, 0i j is the transition linewidth and the summation is over all confined subbands of the QW conduction band. In the following we shall show that: .
1. The first term corresponds to the contribution of the linear Stark effect to the EO susceptibility. 2. The second term corresponds to induced changes in the electric dipole matrix elements that arise from interference between the QW envelope states. 3. The third term originates from changes in the population of the subbands in the presence of a dc electric field. In the following we shall derive the EO susceptibility for a QW with at least three confined subbands. The necessity for a third confined subband will be clarified below. 2.1. The linear Stark effect The linear Stark effect is related to a linear shift of the subband energies under the application of an external dc electric field. Taking the interaction Hamiltonian between the dc electric field and the QW to be of the form Hint = −eF z and using first-order perturbation theory one obtains, εi (F) = εi(0) − eF z ii . This together with eqn (1a) and the expression for the linear susceptibility yields the EO susceptibility for a three-level system as follows: N1 e3 X (z 1 j )2 (z j j − z 11 ) (2) χ S(2) (ω) = ε0 h¯ 2 j6=1 (ω − ω j1 − i0 j1 )2
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where the summation is over all excited subbands. To maximize the Stark susceptibility one has to maximize the product of z 12 j (z j j −z 11 ), and to have many carriers in the ground subband. Figure 1A shows an asymmetricstep QW, based on the GaAs/AlGaAs system, that is designed to maximize the Stark susceptibility. The real part (which is related to electrorefraction (ER)) and the imaginary part (related to electroabsorption (EA)) of the Stark susceptibility are shown in Fig. 2A near the first resonance frequency. We see that the real part of the Stark susceptibility (ER) has a maximum at the resonance (where the linear intersubband absorption also has a maximum) while the imaginary part (EA) is zero at the resonance. For the structure shown in Fig. 1A (h¯ ω21 ≈ 119 meV) we obtain: χ S,max (ω = ω21 ) ∼ = 0.375 cm kV−1 where we have taken 021 = 7.5 meV and 18 −3 N1 = 10 cm . 2.2. Quantum interference modulation The origin of the QI susceptibility is a coherent envelope state mixing (i.e. interference) due to the external dc electric field. Using the same interaction Hamiltonian between the QW and the external dc electric field, the first-order correction to the dipole-matrix elements is given by: " # X eF z i j 1 1 (z j j − z ii ) + z ik z k j + (3) h9i |z|9 j i = z i j − ωi j ωik ω jk h¯ k6=i, j and the expression for the QI susceptibility of a three-level system is given by: (z 13 )2 (z 33 − z 11 ) 2N1 e3 (z 12 )2 (z 22 − z 11 ) (2) + χ Q I (ω) = ε0 h¯ 2 ω21 (ω − ω21 − i021 ) ω31 (ω − ω31 − i031 ) z 12 z 23 z 31 z 12 z 23 z 31 + + 121 (ω − ω21 − i021 ) 131 (ω − ω31 − i031 )
(4)
with 1 1 1 = + 121 ω31 ω32
and
1 1 1 = − . 131 ω21 ω32
Notice that the first two terms in eqn (4) are related to interference among the envelope states involved in the optical transition (Type I QI). The dipole-matrix elements that appear in these terms are identical to those of the Stark modulator. The last two terms in eqn (4) are related to quantum interference with a third subband in the QW (Type II QI) that is not directly involved in the optical transition and are proportional to the product of z 12 z 23 z 31 . It should be noted that, close to a given resonance, the Stark susceptibility always dominates over the Type I QI susceptibility. However, it is possible to design a QW structure where the Stark susceptibility is close to zero while the Type II QI susceptibility differs from zero [4]. A QW structure optimized for Type I QI modulation is shown in Fig. 1B. In Fig. 2B we show the ER and the EA coefficients of the QI susceptibility associated with the structure shown in Fig. 1B. Notice that, unlike the Stark susceptibility, the QI–ER coefficient is zero at the resonance (where the linear absorption has a maximum) while the QI– EA coefficient has a maximum at the resonance. Notice also that, far from resonance the QI–ER coefficient decreases more slowly than the Stark and the QI–EA susceptibilities. The implications of this result on the performance of the modulator will be discussed below. .
2.3. Carrier-density modulation The operation of this modulator is based on a modulation of the number of free carriers in each subband involved in the optical transition. We shall assume that the population of the subbands reaches thermal equilibrium at each stage of the modulation process. Taking the total number of carriers to be conserved and
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A
Potential (meV)
300 200 100 0 –200 –150 –100 –50 0 50 Position (Å) 400
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Potential (meV)
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Potential (meV)
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Fig. 1. The energy-band diagram of the QW structures used to derive the EO susceptibilities. Shown are the energy subbands and the square of the envelope wavefunctions. A, The structure used to derive the Stark modulator. B, Type I quantum interference modulator. C, Carrier-density modulator. .
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Stark susceptibility (cm kV–1)
0.4
A Im (χ) Re (χ)
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–0.2 –5
–4
–3
–2
–1 0 1 ω – ω21/021
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Re (χ)
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–4
–3
–2
–1 0 1 ω – ω21/021
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3
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Fig. 2. The real (solid) and the imaginary (dashed) parts of the second-order EO susceptibility for the various type of EO modulators. A, The Stark modulator. B, The quantum interference and the carrier-density modulators. .
expanding the energy differences between the Fermi energy and the energy of the ith subband in a power series keeping only linear terms one obtains [3]: .
P 1ε F =
i
g2D f (εi )1εi(1) P = h1ε(1) i = −eFhzi g2D f (εi )
(5)
i
where 1ε F and 1εi(1) are the first-order corrections to the Fermi energy and the ith energy level, respectively. The symbol h· · ·i denotes thermodynamic averaging over all subbands using a statistical weight function that is equal to the Fermi–Dirac distribution function, f (εi ), times the two-dimensional density of states, g2D . Hence, the first-order correction to the Fermi energy is given by a simple averaging of the Stark shift of all conduction subbands with a weight function that takes into account the thermal population of the subbands. We can now use this expression for the Fermi energy to derive the CD susceptibility of a three-level system
0.5
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CD susceptibility × 102 (cm kV–1)
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Thermal factor, f12
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Fig. 3. The thermal factor, f 12 , and the CD susceptibility at the resonance versus the Fermi level location (normalized to the energy separation between the lowest two subbands) for various temperatures. .
as follows: χC(2)D (ω) = −
e3 g2D (z 12 )2 (z 13 )2 (z 23 )2 f 12 (z 22 − z 11 ) 2 + − ε0 h¯ (ω − ω21 − i021 ) (ω − ω31 − i031 ) (ω − ω32 − i032 )
(6)
where we have defined 1/ f 12 = 1/ f (ε1 ) + 1/ f (ε2 ). Notice that the CD susceptibility is proportional to the thermal factor f 12 . Therefore, an efficient operation of this modulator requires a significant population of the second subband. The most efficient modulation is achieved when f (ε1 ) ≈ f (ε2 ) ≈ 1, so that f 12 ≈ 12 . This situation is schematically illustrated in Fig. 3 where f 12 and χC D M , of the structure shown in Fig. 1C, are plotted against the Fermi-level location. Notice also that the CD susceptibility is proportional to (z 22 − z 11 ). This means that the origin of the CD modulation is the dc Stark shift of populated subbands which leads to carrier transfer among the subbands. The CD susceptibility has a similar spectral dependence to that of the QI susceptibility. Hence, the ER and EA coefficients of the CD susceptibility follow the spectral behaviour shown in Fig. 2B.
3. Electro-optical modulation We would now like to discuss some practical aspects of EO modulation based on the three effects studied above. The maximum achievable field-induced retardation is related to the real part of the EO susceptibility, χ20 . On the other hand the (linear) absorption that limits the propagation length along the crystal, is proportional to the imaginary part of the linear susceptibility, χ100 . This suggests the following definition of a figure of merit (FOM) for the performance of the intersubband EO modulator [3]: 10max χ20 = (7) FOM ≡ F χ100 .
where the FOM measures the maximum achievable field-induced phase retardation per unit dc electric field. Notice also that 1/FOM has the units of electric field and can be interpreted as the required dc electric field for achieving a phase retardation of 1. In the derivation of eqn (7) we have taken the length of the modulator (along the propagation direction) to be equal to L e f f = 1/α1 ∝ 1/χ100 (where α1 is the linear intersubband absorption). Using the expression for the Stark susceptibility (eqn (2)) we find the figure of merit for the Stark
Superlattices and Microstructures, Vol. 23, No. 5, 1998 modulator to be:
0 χ e z 22 − z 11 FOM = S00 ∼ = h¯ χ1 021
1089
(8)
∼ where we have assumed ER modulation. For the structure shown in Fig. 1A our calculations give, z 22 − z 11 = ˚ Taking 021 ∼ ˚ and z 21 ∼ 32 A = 20 A. = 7.5 meV, we find FOM ∼ = (26 kV cm−1 )−1 . This value of a dc electric field can be achieved in MQW structures. However, it should be emphasized that the effective propagation length of the optical beam is limited to about L e f f ∼ = 1.3 µm. Hence, the linear intersubband absorption at the resonance limits the applicability of the Stark modulator to a very short device length. Similar values for the FOM and the effective length are obtained for the EA Stark modulator. A simple estimate of the FOM and the effective length of a properly designed QI modulator shows that near a resonance the required dc electric field is relatively high (as compared with the Stark modulator). Therefore, let us consider off-resonance operation of this modulator which is more attractive for EO applications. As an example let us consider the QW structure shown in Fig. 1B for which the Type I QI is larger. The FOM for this structure is given by: 0 χ Q I,I e z 22 − z 11 ω − ω21 (9) FOM = 00 ∼ =2 h¯ χ1 ω21 021 ˚ z 21 = 17 A ˚ where we have assumed that ω − ω21 021 . For this structure we obtain, z 22 − z 11 = 17 A, ∼ ∼ and h¯ ω21 = 167 meV. Let us take ω = 5ω21 = 835 meV (i.e. a modulator that operates at a wavelength of λ = 1.5 µm); we obtain FOM = (5.6 kV cm−1 )−1 . Notice that, while the dc electric field is significantly weaker than the dc electric field needed for the operation of the Stark modulator, the effective propagation length is longer, L e f f ∼ = 1500 µm. We conclude that moderate dc electric fields and reasonably long waveguide modulators that operate at 1.5 µm can be realized on the basis of the QI effect. Such a modulator is expected to be useful for many device applications. For the CD modulator we will consider off-resonance operation again since the Stark effect always dominates near the resonance. In this case: 0 χ e f 12 (z 22 − z 11 ) ω − ω21 (10) FOM = C D00M ∼ =2 χ1 ε F − ε1 021 where we took ε F − ε1 ∼ = N1 /g2D . We showed earlier that efficient operation of the modulator requires that f 12 ∼ = 12 , and, therefore, the Fermi level should be located near the second subband. This means that the energy separation between the first two subbands must be small, of the order of 25 meV (at room temperature). For modulators that could operate at the shorter wavelengths it is necessary to use other intersubband transitions such as the 1→3 transition. A structure that fulfils these requirements is schematically shown in Fig. 1C. ˚ z 13 = 11 A, ˚ z 23 = 17.3 A ˚ and For this structure we have calculated ω31 ∼ = 210 meV, z 22 − z 11 = 49.7 A, ∼ ∼ ∼ f 12 = 0.25 (where we took ε F − ε1 = 20 meV). Hence, off-resonance operation at ω = 4ω31 = 840 meV yields FOM ∼ = (583 V cm−1 )−1 and L e f f ∼ = 1600 µm. We would = e f 12 (z 22 − z 11 )(ω − ω31 )/((ε F − ε1 )031 ) ∼ like to point out that in this structure the CD effect is larger than that of the QI effect. Therefore, the CD effect will dominate over the QI effect at this spectral range.
4. Experimental The aim of our experimental work is to demonstrate the (near resonance) operation of the (Type II) modulator. In this case one should have z 22 − z 11 ∼ = 0 (no Stark effect, see eqn (2)) while z 13 6= 0. The sample used for this experiment consists of 25 periods of modulation-doped coupled QWs. A single period consists ˚ GaAs narrow QW. Each ˚ GaAs wide QW followed by a 15 A ˚ Al0.4 Ga0.6 As thin barrier and a 20 A of a 60 A ˚ are Si doped to ˚ Al0.4 Ga0.6 As barrier where the central 25 A period is separated from the other by a 400 A 5 × 1018 cm−3 .
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1T/T (× 10–3)
2.5 2.0 1.5 1.0 0.5 0.0 110
120
130
140
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160
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Photon energy (meV) Fig. 4. Electric-field-induced intersubband absorbance for two applied voltages as indicated. The squares and the triangles show the results of laser set-up measurements while the solid and the dashed lines show the results of FTIR measurements. .
Figure 4 shows the dc electric-field-induced absorption of this sample. A square voltage waveform, at a frequency of 1.5 KHz, was applied to modulate the absorption of the sample and a lock-in detection scheme was utilized to record the electric-field-induced absorption as measured using a step-scan FTIR spectrometer. In another experiment we measured the dc-induced phase retardation of a polarized CO2 laser beam as a function of the laser photon energy. The results of this experiment are shown in Fig. 5. Both experiments provide clear evidence that our EO modulator is dominated by the QI effect. The zero-induced retardation and the maximum-induced absorption at the resonance are related to Type II QI. Notice also that the CD effect is practically zero in our structure since the dc electric field cannot induce carrier transfer to the second subband that is 135 meV above the ground state. The largest induced phase retardation was measured in the 115–120 meV spectral range, whereas the linear absorption has a maximum around 135 meV. This type of spectral behaviour can be understood only as originating from a QI–EO modulator.
5. Conclusion In conclusion, we have shown that the EO susceptibility due to intersubband transitions in asymmetric QWs originates from three different types of physical mechanisms: the linear Stark effect, QI of the envelope states, and modulation of the CD in each subband. We also showed that induced retardation due to QI and CD modulation is out of phase with the linear absorption. This result leads to the conclusion that the latter are best suited for off-resonance operation. Indeed, calculations of the FOM for the various types of modulators show that QI and CD modulators may well serve in the 1.3–2 µm spectral range, yielding high retardation at moderate dc electric fields that can easily be achieved in QW structures. Finally, we provided experimental evidence to the near resonance operation of the Type II QI modulator based on a CQW structure where the QI effect dominates over the Stark effect.
References [1] }D. A. Holm and H. F. Taylor, IEEE J. Quantum Electron. 25, 2266 (1989). [2] }A. Harwit, R. Fernnandez, W. D. Eades, D. K. Kinell, and R. L. Whitney, Superlattices Microstruct. 19, 39 (1996). .
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1.0 Phase retardation (× 10–3)
0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1.0 105 110 115 120 125 130 135 140 145 150 Photon energy (meV) Fig. 5. DC-electric-field-induced phase retardation versus the photon energy (¥). The solid line is the fit of the experimental data to the quantum interference lineshape function. .
[3] }A. Sa’ar and R. Kapon IEEE J. Quantum Electron. 33, 1517 (1997). [4] }A. Segev, A. Sa’ar, Y. Oiknine-Schlesinger, and E. Ehrenfreund, Superlattices Microstruct. 19, 47 (1996). [5] }V. Berger, E. Dupont, D. Delacourt, B. Vinter, N. Vodjdani, and M. Papuchon, Appl. Phys. Lett. 61, 2072 (1992). [6] }E. Dupont, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 63, 2514 (1993). [7] }D. Kaufman, A. Sa’ar, and N. Kuze, Appl. Phys. Lett. 64, 2543 (1994). .
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Intersubband electro-optic modulators for near and mid infrared applications R. Kapon, A. Segev, A. Sa’ar Deprtment of Applied Physics, The Fredi and Nadine Herrmann School of Applied Science, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel
(Received 15 July 1996) Optical transitions between conduction subbands of asymmetrical quantum-well structures can produce large second-order optical nonlinearities. In particular, it is possible to engineer asymmetrical quantum-well structures to have very large electro-optic (EO) coefficients. In this paper we study three alternative approaches to design and fabricate intersubband EO modulators, i.e. the Stark modulator, the quantum-interference (QI) modulator, and the carrier-density (CD) modulator. We show that near a given resonance the Stark modulator has the largest EO coefficient. However, the linear intersubband absorption limits the usability of this modulator to a very short device length. Far from resonance we found that the CD and the QI modulators are more efficient. Furthermore, we found that these modulators can be utilized for near-infrared modulation in the 1.3–1.5 µm spectral range. These modulators are almost unaffected by the linear intersubband absorption so that their efficiency can be very high. We present preliminary experimental results that demonstrate the operation of the QI modulator. c 1998 Academic Press Limited
Key words: modulators, electro-optics, intersubband transitions, quantum wells.
1. Introduction Over the past few years there has been an ongoing research effort to develop semiconductor high-speed electro-optic (EO) modulators that can monolithically be integrated with other optical and electronic devices on the same chip. One class of semiconductor EO modulators that has recently been proposed is based on intersubband transitions (ISBTs) in quantum wells (QWs) [1– 6]. These EO modulators can be divided into three classes according to the physical mechanism that generates the EO effect: the Stark, carrier-density (CD) and the quantum-interference (QI) modulators. The Stark modulator relies on the linear dc Stark effect in QWs (i.e. a linear shift of the energy levels under the application of an external dc electric field [1–3]). The QI modulator is based on a coherent interference among the QW envelope states [4] while the CD modulator is related to a modulation of the CD in the QW levels [3, 5–6]. Recently we have found that under most circumstances the EO Stark susceptibility dominates near a given intersubband resonance [3–4]. However, taking into account the linear intersubband absorption we have shown that the Stark modulator cannot provide high-modulation efficiency. In addition, we investigated a coupled QW (CWQ) structure for which the QI effect dominates over the Stark effect. In this paper we extend the idea of intersubband EO modulation for off-resonance operation. We show .
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that far from resonance the QI and the CD susceptibilities dominate over the Stark susceptibility. Taking into account the linear intersubband absorption we derive a figure of merit and conclude that efficient near-infrared EO modulators can be realized. Furthermore, we have designed a modulator that operates at 1.5 µm and is suitable for optical communication applications. Finally, we present preliminary experimental results that demonstrate the operation of the QI modulator at the mid-infrared wavelengths.
2. Definition of the intersubband electro-optic susceptibilities Our approach to derive the EO susceptibilities of asymmetric QWs is very similar to that of [4]. The influence of the external dc electric field on the QW is taken into account by letting the QW energies, envelope wavefunctions and CDs depend on the externally applied dc electric field across the QW. Expanding the linear intersubband susceptibility [7] in a power series of the dc electric field, F, and keeping only linear terms in the expansion yields the following expression for the second-order EO susceptibilities. X ∂χi j ∂ωi j (1a) χ S(2) (ω) = ∂ωi j ∂ F F=0 i, j X ∂χi j ∂z i j ∂χi j ∂z ji (1b) + χ Q(2)I (ω) = ∂z i j ∂ F ∂z ji ∂ F F=0 i, j X ∂χi j ∂ N j ∂χi j ∂ Ni (2) + (1c) χC D M (ω) = ∂ Nj ∂ F ∂ Ni ∂ F F=0 i, j .
.
where χi j (ω, F) =
z i j (F)z ji (F) e2 [N j (F) − Ni (F)] ε0 h¯ [ω − ωi j (F) − i0i j ]
χi j is the linear susceptibility associated with the i → j ISBT [7], z i j = h9i |z|9 j i is the exchange electric dipole-matrix element (divided by e), ωi j = (εi − ε j )/h¯ is the transition frequency, 9i and Ni are the envelope wavefunction and the density of carriers associated with the ith subband respectively, 0i j is the transition linewidth and the summation is over all confined subbands of the QW conduction band. In the following we shall show that: .
1. The first term corresponds to the contribution of the linear Stark effect to the EO susceptibility. 2. The second term corresponds to induced changes in the electric dipole matrix elements that arise from interference between the QW envelope states. 3. The third term originates from changes in the population of the subbands in the presence of a dc electric field. In the following we shall derive the EO susceptibility for a QW with at least three confined subbands. The necessity for a third confined subband will be clarified below. 2.1. The linear Stark effect The linear Stark effect is related to a linear shift of the subband energies under the application of an external dc electric field. Taking the interaction Hamiltonian between the dc electric field and the QW to be of the form Hint = −eF z and using first-order perturbation theory one obtains, εi (F) = εi(0) − eF z ii . This together with eqn (1a) and the expression for the linear susceptibility yields the EO susceptibility for a three-level system as follows: N1 e3 X (z 1 j )2 (z j j − z 11 ) (2) χ S(2) (ω) = ε0 h¯ 2 j6=1 (ω − ω j1 − i0 j1 )2
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where the summation is over all excited subbands. To maximize the Stark susceptibility one has to maximize the product of z 12 j (z j j −z 11 ), and to have many carriers in the ground subband. Figure 1A shows an asymmetricstep QW, based on the GaAs/AlGaAs system, that is designed to maximize the Stark susceptibility. The real part (which is related to electrorefraction (ER)) and the imaginary part (related to electroabsorption (EA)) of the Stark susceptibility are shown in Fig. 2A near the first resonance frequency. We see that the real part of the Stark susceptibility (ER) has a maximum at the resonance (where the linear intersubband absorption also has a maximum) while the imaginary part (EA) is zero at the resonance. For the structure shown in Fig. 1A (h¯ ω21 ≈ 119 meV) we obtain: χ S,max (ω = ω21 ) ∼ = 0.375 cm kV−1 where we have taken 021 = 7.5 meV and 18 −3 N1 = 10 cm . 2.2. Quantum interference modulation The origin of the QI susceptibility is a coherent envelope state mixing (i.e. interference) due to the external dc electric field. Using the same interaction Hamiltonian between the QW and the external dc electric field, the first-order correction to the dipole-matrix elements is given by: " # X eF z i j 1 1 (z j j − z ii ) + z ik z k j + (3) h9i |z|9 j i = z i j − ωi j ωik ω jk h¯ k6=i, j and the expression for the QI susceptibility of a three-level system is given by: (z 13 )2 (z 33 − z 11 ) 2N1 e3 (z 12 )2 (z 22 − z 11 ) (2) + χ Q I (ω) = ε0 h¯ 2 ω21 (ω − ω21 − i021 ) ω31 (ω − ω31 − i031 ) z 12 z 23 z 31 z 12 z 23 z 31 + + 121 (ω − ω21 − i021 ) 131 (ω − ω31 − i031 )
(4)
with 1 1 1 = + 121 ω31 ω32
and
1 1 1 = − . 131 ω21 ω32
Notice that the first two terms in eqn (4) are related to interference among the envelope states involved in the optical transition (Type I QI). The dipole-matrix elements that appear in these terms are identical to those of the Stark modulator. The last two terms in eqn (4) are related to quantum interference with a third subband in the QW (Type II QI) that is not directly involved in the optical transition and are proportional to the product of z 12 z 23 z 31 . It should be noted that, close to a given resonance, the Stark susceptibility always dominates over the Type I QI susceptibility. However, it is possible to design a QW structure where the Stark susceptibility is close to zero while the Type II QI susceptibility differs from zero [4]. A QW structure optimized for Type I QI modulation is shown in Fig. 1B. In Fig. 2B we show the ER and the EA coefficients of the QI susceptibility associated with the structure shown in Fig. 1B. Notice that, unlike the Stark susceptibility, the QI–ER coefficient is zero at the resonance (where the linear absorption has a maximum) while the QI– EA coefficient has a maximum at the resonance. Notice also that, far from resonance the QI–ER coefficient decreases more slowly than the Stark and the QI–EA susceptibilities. The implications of this result on the performance of the modulator will be discussed below. .
2.3. Carrier-density modulation The operation of this modulator is based on a modulation of the number of free carriers in each subband involved in the optical transition. We shall assume that the population of the subbands reaches thermal equilibrium at each stage of the modulation process. Taking the total number of carriers to be conserved and
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Superlattices and Microstructures, Vol. 23, No. 5, 1998 400
A
Potential (meV)
300 200 100 0 –200 –150 –100 –50 0 50 Position (Å) 400
100
150
200
B
Potential (meV)
300 200
100
0 –100 –80 –60 –40 –20 0 20 40 60 80 100 Position (Å) 400
C
Potential (meV)
300
200 100
0 –120 –90 –60 –30 0 30 Position (Å)
60
90
120
Fig. 1. The energy-band diagram of the QW structures used to derive the EO susceptibilities. Shown are the energy subbands and the square of the envelope wavefunctions. A, The structure used to derive the Stark modulator. B, Type I quantum interference modulator. C, Carrier-density modulator. .
Superlattices and Microstructures, Vol. 23, No. 5, 1998
1087
Stark susceptibility (cm kV–1)
0.4
A Im (χ) Re (χ)
0.2
0.0
–0.2 –5
–4
–3
–2
–1 0 1 ω – ω21/021
2
3
QI susceptibility × 103 (cm kV–1)
1.5
4
5
B Im (χ)
1.0
Re (χ)
0.5 0.0 –0.5 –5
–4
–3
–2
–1 0 1 ω – ω21/021
2
3
4
5
Fig. 2. The real (solid) and the imaginary (dashed) parts of the second-order EO susceptibility for the various type of EO modulators. A, The Stark modulator. B, The quantum interference and the carrier-density modulators. .
expanding the energy differences between the Fermi energy and the energy of the ith subband in a power series keeping only linear terms one obtains [3]: .
P 1ε F =
i
g2D f (εi )1εi(1) P = h1ε(1) i = −eFhzi g2D f (εi )
(5)
i
where 1ε F and 1εi(1) are the first-order corrections to the Fermi energy and the ith energy level, respectively. The symbol h· · ·i denotes thermodynamic averaging over all subbands using a statistical weight function that is equal to the Fermi–Dirac distribution function, f (εi ), times the two-dimensional density of states, g2D . Hence, the first-order correction to the Fermi energy is given by a simple averaging of the Stark shift of all conduction subbands with a weight function that takes into account the thermal population of the subbands. We can now use this expression for the Fermi energy to derive the CD susceptibility of a three-level system
0.5
2.1
0.4
1.7
0.3
1.3 (ε2 – ε1)/kB T = 1 (ε2 – ε1)/kB T = 5 (ε2 – ε1)/kB T = 10
0.2 0.1 0.0 0.0
0.5
1.0 1.5 2.0 (εf – ε1)/(ε2 – ε1)
2.5
0.8 0.4
CD susceptibility × 102 (cm kV–1)
Superlattices and Microstructures, Vol. 23, No. 5, 1998
Thermal factor, f12
1088
0.0 3.0
Fig. 3. The thermal factor, f 12 , and the CD susceptibility at the resonance versus the Fermi level location (normalized to the energy separation between the lowest two subbands) for various temperatures. .
as follows: χC(2)D (ω) = −
e3 g2D (z 12 )2 (z 13 )2 (z 23 )2 f 12 (z 22 − z 11 ) 2 + − ε0 h¯ (ω − ω21 − i021 ) (ω − ω31 − i031 ) (ω − ω32 − i032 )
(6)
where we have defined 1/ f 12 = 1/ f (ε1 ) + 1/ f (ε2 ). Notice that the CD susceptibility is proportional to the thermal factor f 12 . Therefore, an efficient operation of this modulator requires a significant population of the second subband. The most efficient modulation is achieved when f (ε1 ) ≈ f (ε2 ) ≈ 1, so that f 12 ≈ 12 . This situation is schematically illustrated in Fig. 3 where f 12 and χC D M , of the structure shown in Fig. 1C, are plotted against the Fermi-level location. Notice also that the CD susceptibility is proportional to (z 22 − z 11 ). This means that the origin of the CD modulation is the dc Stark shift of populated subbands which leads to carrier transfer among the subbands. The CD susceptibility has a similar spectral dependence to that of the QI susceptibility. Hence, the ER and EA coefficients of the CD susceptibility follow the spectral behaviour shown in Fig. 2B.
3. Electro-optical modulation We would now like to discuss some practical aspects of EO modulation based on the three effects studied above. The maximum achievable field-induced retardation is related to the real part of the EO susceptibility, χ20 . On the other hand the (linear) absorption that limits the propagation length along the crystal, is proportional to the imaginary part of the linear susceptibility, χ100 . This suggests the following definition of a figure of merit (FOM) for the performance of the intersubband EO modulator [3]: 10max χ20 = (7) FOM ≡ F χ100 .
where the FOM measures the maximum achievable field-induced phase retardation per unit dc electric field. Notice also that 1/FOM has the units of electric field and can be interpreted as the required dc electric field for achieving a phase retardation of 1. In the derivation of eqn (7) we have taken the length of the modulator (along the propagation direction) to be equal to L e f f = 1/α1 ∝ 1/χ100 (where α1 is the linear intersubband absorption). Using the expression for the Stark susceptibility (eqn (2)) we find the figure of merit for the Stark
Superlattices and Microstructures, Vol. 23, No. 5, 1998 modulator to be:
0 χ e z 22 − z 11 FOM = S00 ∼ = h¯ χ1 021
1089
(8)
∼ where we have assumed ER modulation. For the structure shown in Fig. 1A our calculations give, z 22 − z 11 = ˚ Taking 021 ∼ ˚ and z 21 ∼ 32 A = 20 A. = 7.5 meV, we find FOM ∼ = (26 kV cm−1 )−1 . This value of a dc electric field can be achieved in MQW structures. However, it should be emphasized that the effective propagation length of the optical beam is limited to about L e f f ∼ = 1.3 µm. Hence, the linear intersubband absorption at the resonance limits the applicability of the Stark modulator to a very short device length. Similar values for the FOM and the effective length are obtained for the EA Stark modulator. A simple estimate of the FOM and the effective length of a properly designed QI modulator shows that near a resonance the required dc electric field is relatively high (as compared with the Stark modulator). Therefore, let us consider off-resonance operation of this modulator which is more attractive for EO applications. As an example let us consider the QW structure shown in Fig. 1B for which the Type I QI is larger. The FOM for this structure is given by: 0 χ Q I,I e z 22 − z 11 ω − ω21 (9) FOM = 00 ∼ =2 h¯ χ1 ω21 021 ˚ z 21 = 17 A ˚ where we have assumed that ω − ω21 021 . For this structure we obtain, z 22 − z 11 = 17 A, ∼ ∼ and h¯ ω21 = 167 meV. Let us take ω = 5ω21 = 835 meV (i.e. a modulator that operates at a wavelength of λ = 1.5 µm); we obtain FOM = (5.6 kV cm−1 )−1 . Notice that, while the dc electric field is significantly weaker than the dc electric field needed for the operation of the Stark modulator, the effective propagation length is longer, L e f f ∼ = 1500 µm. We conclude that moderate dc electric fields and reasonably long waveguide modulators that operate at 1.5 µm can be realized on the basis of the QI effect. Such a modulator is expected to be useful for many device applications. For the CD modulator we will consider off-resonance operation again since the Stark effect always dominates near the resonance. In this case: 0 χ e f 12 (z 22 − z 11 ) ω − ω21 (10) FOM = C D00M ∼ =2 χ1 ε F − ε1 021 where we took ε F − ε1 ∼ = N1 /g2D . We showed earlier that efficient operation of the modulator requires that f 12 ∼ = 12 , and, therefore, the Fermi level should be located near the second subband. This means that the energy separation between the first two subbands must be small, of the order of 25 meV (at room temperature). For modulators that could operate at the shorter wavelengths it is necessary to use other intersubband transitions such as the 1→3 transition. A structure that fulfils these requirements is schematically shown in Fig. 1C. ˚ z 13 = 11 A, ˚ z 23 = 17.3 A ˚ and For this structure we have calculated ω31 ∼ = 210 meV, z 22 − z 11 = 49.7 A, ∼ ∼ ∼ f 12 = 0.25 (where we took ε F − ε1 = 20 meV). Hence, off-resonance operation at ω = 4ω31 = 840 meV yields FOM ∼ = (583 V cm−1 )−1 and L e f f ∼ = 1600 µm. We would = e f 12 (z 22 − z 11 )(ω − ω31 )/((ε F − ε1 )031 ) ∼ like to point out that in this structure the CD effect is larger than that of the QI effect. Therefore, the CD effect will dominate over the QI effect at this spectral range.
4. Experimental The aim of our experimental work is to demonstrate the (near resonance) operation of the (Type II) modulator. In this case one should have z 22 − z 11 ∼ = 0 (no Stark effect, see eqn (2)) while z 13 6= 0. The sample used for this experiment consists of 25 periods of modulation-doped coupled QWs. A single period consists ˚ GaAs narrow QW. Each ˚ GaAs wide QW followed by a 15 A ˚ Al0.4 Ga0.6 As thin barrier and a 20 A of a 60 A ˚ are Si doped to ˚ Al0.4 Ga0.6 As barrier where the central 25 A period is separated from the other by a 400 A 5 × 1018 cm−3 .
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Superlattices and Microstructures, Vol. 23, No. 5, 1998 3.0 1.2 V (FTIR) 0.6 V (FTIR) 1.2 V (LASER) 0.6 V (LASER)
1T/T (× 10–3)
2.5 2.0 1.5 1.0 0.5 0.0 110
120
130
140
150
160
170
180
Photon energy (meV) Fig. 4. Electric-field-induced intersubband absorbance for two applied voltages as indicated. The squares and the triangles show the results of laser set-up measurements while the solid and the dashed lines show the results of FTIR measurements. .
Figure 4 shows the dc electric-field-induced absorption of this sample. A square voltage waveform, at a frequency of 1.5 KHz, was applied to modulate the absorption of the sample and a lock-in detection scheme was utilized to record the electric-field-induced absorption as measured using a step-scan FTIR spectrometer. In another experiment we measured the dc-induced phase retardation of a polarized CO2 laser beam as a function of the laser photon energy. The results of this experiment are shown in Fig. 5. Both experiments provide clear evidence that our EO modulator is dominated by the QI effect. The zero-induced retardation and the maximum-induced absorption at the resonance are related to Type II QI. Notice also that the CD effect is practically zero in our structure since the dc electric field cannot induce carrier transfer to the second subband that is 135 meV above the ground state. The largest induced phase retardation was measured in the 115–120 meV spectral range, whereas the linear absorption has a maximum around 135 meV. This type of spectral behaviour can be understood only as originating from a QI–EO modulator.
5. Conclusion In conclusion, we have shown that the EO susceptibility due to intersubband transitions in asymmetric QWs originates from three different types of physical mechanisms: the linear Stark effect, QI of the envelope states, and modulation of the CD in each subband. We also showed that induced retardation due to QI and CD modulation is out of phase with the linear absorption. This result leads to the conclusion that the latter are best suited for off-resonance operation. Indeed, calculations of the FOM for the various types of modulators show that QI and CD modulators may well serve in the 1.3–2 µm spectral range, yielding high retardation at moderate dc electric fields that can easily be achieved in QW structures. Finally, we provided experimental evidence to the near resonance operation of the Type II QI modulator based on a CQW structure where the QI effect dominates over the Stark effect.
References [1] }D. A. Holm and H. F. Taylor, IEEE J. Quantum Electron. 25, 2266 (1989). [2] }A. Harwit, R. Fernnandez, W. D. Eades, D. K. Kinell, and R. L. Whitney, Superlattices Microstruct. 19, 39 (1996). .
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1.0 Phase retardation (× 10–3)
0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1.0 105 110 115 120 125 130 135 140 145 150 Photon energy (meV) Fig. 5. DC-electric-field-induced phase retardation versus the photon energy (¥). The solid line is the fit of the experimental data to the quantum interference lineshape function. .
[3] }A. Sa’ar and R. Kapon IEEE J. Quantum Electron. 33, 1517 (1997). [4] }A. Segev, A. Sa’ar, Y. Oiknine-Schlesinger, and E. Ehrenfreund, Superlattices Microstruct. 19, 47 (1996). [5] }V. Berger, E. Dupont, D. Delacourt, B. Vinter, N. Vodjdani, and M. Papuchon, Appl. Phys. Lett. 61, 2072 (1992). [6] }E. Dupont, D. Delacourt, and M. Papuchon, Appl. Phys. Lett. 63, 2514 (1993). [7] }D. Kaufman, A. Sa’ar, and N. Kuze, Appl. Phys. Lett. 64, 2543 (1994). .
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