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Improvement in intersubband optical absorption and the effects of device parameter variations in quantum wells with an applied electric field E. K ASAPOGLU , H. S ARI , Y. E RGÜN Cumhuriyet University, Physics Department, 58140 Sivas, Turkey I. S OKMEN ˙ Dokuz Eylül University, Physics Department, Izmir, Turkey (Received 5 August 1999) The intersubband optical absorption in symmetric and asymmetric, single and coupled, double GaAs/Ga1−x Alx As quantum wells is calculated. The results have been obtained in the presence of a uniform electric field as a function of the potential symmetry, size of the quantum well, and coupling parameter of the wells. In coupled double quantum wells we obtain a large Stark effect that can be used to fabricate tuneable photodetectors. We show that the effect of an applied electric field on the intersubband optical absorption is similar to changes in the dimensions of the structure. This behaviour in the intersubband optical absorption for different wells and barrier geometries can be used to study these systems in regions of interest, without the need for the growth of many different samples. c 1999 Academic Press

Key words: intersubband optical absorption, asymmetric double quantum well, Stark effect.

1. Introduction In recent years, semiconductor growth technology has advanced to such a point that it is now possible to grow well-controlled thin-layered semiconductor structures. Because of the possibility for novel devices, the optical properties of the quasi-two-dimensional electron gas in a semiconductor structure has been studied both theoretically and experimentally, and many new Ga1−x Alx As/GaAs quantum-well photodetectors based on intersubband absorption has been proposed to replace the conventional detectors [1–8]. Intersubband absorption in quantum wells have been proposed or demonstrated experimentally to be very useful for farinfrared detectors [1–4, 9–12], electro-optical modulators [13, 14], and infrared lasers [15]. The absorption peak of the interband transition in quantum wells shifts to the long wavelength when an electric field is applied, known as the quantum-confined Stark effect. The quantum-confined Stark effect in quantum wells is more significant than the Franz–Keldysh effect in bulk semiconductor materials due to the localization of the particles in the quantum wells [13]. The position of the absorption peak of the optical intersubband transition in quantum wells shifts to the short wavelength under an applied electric field [14, 15], and the forbidden transitions when the field is zero becomes allowable for the nonzero electric field. The intersubband Stark shift is small and cannot 0749–6036/99/120395 + 10 $30.00/0

c 1999 Academic Press

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be used in optical modulation applications, because the relative energies of the sublevels in a single well do not change much in an electric field [5]. In asymmetric quantum-well structures, the Stark shifts and changes in the absorption coefficients were predicted theoretically and confirmed experimentally to be larger than the changes that occur in conventional square QWs [16–20]. In this study we investigate the effect of the symmetry of the structure on the intersubband optical transitions in an external electric field. These transitions are predicted to have a very large dipole and narrow bandwidth, and are of practical interest as tuneable optical devices.

2. Method In the presence of optical radiation, the interaction Hamiltonian between the crystal electron and the radiation is given by e e H 0 = − Ap = − A0 [exp[−iqr − iwt] + cc]ˆεp, (1) m0 2m 0 where A is the vector potential, εˆ is the polarization vector, q is the wavevector for incoming optical radiation, e is the magnitude of the charge of the electron, m 0 is the free space electron mass, and p is the momentum vector of the electron crystal. For a given interaction potential H 0 , the transition rate is given by [21] 2π |h9 f |H 0 |9i i|2 δ(E f − E i − ~ω), (2) ~ where E i and E f are the energies of the electron in the initial and final state, respectively, and ω is the angular frequency of the incident photon. By neglecting interaction between the electrons in the well, wavefunctions for the initial state 9i and after absorption the final state 9 f can be written as Wfi =

9i = Uc (r)ζi (r) = A−1/2 Uc (r) exp[ikt rt ]8i (z) 9 f = Uc0 (r)ζ f (r) = A−1/2 Uc0 (r) exp[ik0t rt ]8 f (z),

(3)

where A is the area of the well, kt and are the (x − y) plane wavevectors of electrons in the initial and final states, respectively, rt is position vector in the (x − y) plane, and Uc and Uc0 are cell periodic functions near the conduction-band extremum. The envelope functions 8i (z) and 8 f (z) satisfy the following Schrödinger equation for an electron in the QW subject to a uniform electric field perpendicular to the well: k0t

−

~2 ∂ 2 8(z) + V (z)8(z) + eF z8(z) = E8(z), 2m ∗ ∂z 2

(4)

where V (z) is given by,

VL z < −b 0 −b < z < b VR z > b. In eqn (4), m ∗ and F denote the effective mass of an electron and electric field, respectively. We have solved Schrödinger’s equation by using quasi-bound criteria [22]. In general, the matrix element of the photon absorption process can be approximately written as [1] Z Z Z Z 0 ∗ 0 ∗ ∗ ∼ h9 f |H |9i i = Uc0 H Uc dτ ζi ζ f dτ + Uc0 Uc dτ ζi H 0 ζ f dτ. (5) V (z) =

(

V

V

For the case of direct intersubband transitions, the second term of the above expression gives zero, because of the fact that the Hamiltonian used to obtain 9i and 9 f is Hermitian. For the intersubband case we are interested in e A0 h9 f |H 0 |9i i ≈ hζ f |H 0 |ζi i = − hζ f |eiqr εˆ p|ζi i 2m 0

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397

e A0 εˆ hζ f |p|ζi i (6) 2m 0 e A0 =− (E i − E f )ˆεhζ f |r|ζi i, 2i~ where the cell periodic function part has been taken care of as in Ref. [1] and we have used the dipole approximation in eqn (6) which is certainly valid for the long-wavelength radiation. The absorption constant α in the well is defined as, ≈−

α=

~ω × number of transitions per unit well volume and time . incident energy flux

By calculating the total transition rate and taking into account the line broadening [21], we obtain conservation of momentum in the plane of the well and α as α=

X X µcm e kB T e2 (cos2 2)|Mfi |2 ⊗ 2 m 2 Ln ω π~ r 0 i f (z) E F − E (z) EF − Ei (0/2) f ln 1 + exp 1 + exp kB T kB T (~ω − E fi )2 + (0/2)2

with the matrix element Mfi =

Z m 0 (E i(z) − E (z) f ) i~

+∞ −∞

8∗f (z)z8i (z)dz,

(7)

(8)

(z) (z) where E fi = E i(z) − E (z) f ve E i and E f denote the quantized energy levels for the initial and final states, respectively, µ is the permeability, c is the speed of the light in free space, kB is Boltzmann’s constant, T is the temperature, 2 is the angle between the polarization vector εˆ and the normal to the quantum well, n r is the refractive index, E F is the Fermi energy which depends on the density of electrons in the well, and 0 is the linewidth. For refractive index we use the first-order Sellmeier equation: λ2 n 2r = A + B 2 , (9) λ − C2

with A = 8.950, B = 2.054, C 2 = 0.390 for T = 298 K [23], E F = 6.49 meV which corresponds to about n e = 1.6 × 1017 cm−3 electrons in the well, 2 = 0, and 0 = 10 meV from the experimental results [4].

3. Results and conclusion In this study, for numerical calculations we have taken m e = 0.067 m 0 , ε0 = 12.5, aB = 165.6 Å, R = 3.47 meV, and VLe = 228 meV. These parameters are suitable for Ga1−x Alx /GaAs heterostructures with an Al concentration x ∼ = 0.3. We have assumed the conduction band discontinuity to be 56% of the total bandgap difference, which has recently been suggested as a more appropriate value [18]. We define the β and σ parameters in order to describe the ratio of the left well width L 1 to the right well width L 2 and the ratio of the right potential VR(e,h) to the left potential e.g. β =

L1 L2 ,

σ =

(e)

VR

(e)

VL

.

The variation of the absorption coefficient α is given as a function of the symmetry parameter σ in Fig. 2A. The absorption peak for (1–2) intersubband optical absorption is not very sensitive to the symmetry parameter σ , but for the (2–3) intersubband transition, the absorption peak increases in magnitude, and the position of the peak decreases in energy with decreasing σ , because electrons in the higher subbands are mostly localized at the lower side of the well, and the overlap between subbands becomes too large. As seen in Fig. 2A, the forbidden transitions (1–3, 2–4) when σ = 1 become allowable for σ 6= 1 because the parity which prohibits

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Superlattices and Microstructures, Vol. 26, No. 6,1999

3 VLe

VRe 2 1

hω

z –b

0

b

Fig. 1. Potential energy profile for a finite quantum well subject to an electric field F (dashed line) in the presence of incoming radiation with angular frequency ~ω.

transitions (1–3, 2–4) no longer exists when σ 6= 1. Thus we can say that the effect of the symmetry of the structure is similar to that of the external electric field. Thus in the asymmetric structures we have more subband transitions. The relation of the eigenenergy difference 1E between subbands as a function of the applied electric field is shown in Fig. 2B. As expected, a large variation of 1E arises due to the applied field. For small electric field values the variation of 1E is sensitive to the symmetry parameter β. However, for large electric field values the Stark effect 1E depends on the variation of the symmetry parameter σ . In particular, for small electric field values 1E decreases as σ decreases, since in the asymmetric QW structures the higher subband energies, which are more energetic with respect to the ground state, can easily penetrate into the barriers thus their energies decreases as σ decreases. This effect is also seen in Fig. 2A where the position of the absorption peak becomes smaller in energy as σ decreases. The relation between the absorption coefficient and the applied electric field for the single QW well with a well width of 200 Å and a well depth of 228 meV is given in Fig. 3A. One can easily see that the absorption peak for intersubband optical absorption increases in energy with an increasing electric field over a wide range of electric fields (from a zero field to 100 kV cm−1 ), because for increasing electric fields the energy of the ground state decreases monotonically, while those of the higher subband states increase slightly then decrease monotonically and those of the higher subband states increase slightly then decrease slowly [13]. It is well known that, for exciton absorption, the absorption peak decreases in energy with increasing electric field, because both the ground states of electrons and holes decrease. Also, it is seen that the absorption peak increases in magnitude with increasing electric field, since the electric field pushes the electrons to the same side of the well for all states. This increases the overlap function between different subbands. Here we should point out that for the exciton absorption, an increasing electric field causes further separation of electrons and holes in the well, thus the absorption peak decreases in magnitude as the electric field increases. In Fig. 3B, the relation between the absorption coefficient and the applied electric field for the single QW with σ = 0.3 is given. By comparing the absorption peaks of Fig. 3A with those of Fig. 3B one can see that in the single QW structure the absorption coefficient is not very sensitive to the symmetry parameter σ .

Superlattices and Microstructures, Vol. 26, No. 6,1999 8000

399

A L = 200Å 1–2

2–3 σ = 0.5 σ = 0.8

6000

α (cm)–1

σ=1

4000

σ = 0.3 σ = 0.5

2000

1–3 (× 103) σ = 0.5

σ = 0.8 σ=1

σ = 0.8 0 40

0

80

120

Photon energy (meV) 160 B L = 200 Å 1–3

E (meV)

120

σ=1 80

σ = 0.8

σ = 0.5 σ=1 40

1–2 σ = 0.8 2–3

σ = 0.5 σ=1

σ = 0.8

σ = 0.3

σ = 0.5

0 0

20

40 F

60

80

100

(kV/cm–1)

Fig. 2. A, The symmetry parameter σ dependence of the variation of the absorption coefficient as a function of the incident photon energy. B, The electric field dependence of intersubband (1E i−f ) transitions in a single QW with L = 200 Å width for different σ values.

400

Superlattices and Microstructures, Vol. 26, No. 6,1999 16000 A

F (kV/cm–1) L = 200 Å σ=1

F = 100

α (cm)–1

12000

8000 F = 50 F = 30 F=0 F = 10 4000

0 40

0

80

120

Photon energy (meV) 16000 B

F (kV/cm–1) L = 200 Å σ = 0.3

F = 100

α (cm)–1

12000

8000 F=0

F = 50 F = 30

F = 10 4000

0 0

40

80

120

Photon energy (meV) Fig. 3. A, The variation of the absorption coefficient α as a function of the applied electric field for σ = 1. B, The variation of the absorption coefficient α as a function of the applied electric field for σ = 0.3.

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401

20000

F = 100 F (kV/cm–1) σ=1 β = 1.5

16000

β=1

α (cm)–1

12000

F = 50

8000

F=0 4000

F = 30 F = 10

0 0

40

80

120

Photon energy (meV) Fig. 4. The variation of the absorption coefficient as a function of the incident photon energy in symmetric (bold) and asymmetric (dashed line) DQWs for several electric field values.

Figure 4 illustrates the variation of the absorption coefficient as a function of the applied electric field for the symmetric and asymmetric coupled QW structure. The dashed lines show the absorption coefficient of the asymmetric coupled QW structure. The resonant absorption energy is sensitive to the applied electric field as expected. For the symmetric case, by increasing the electric field, the absorption coefficient decreases slightly then increases rapidly while that of the asymmetric coupled QW increases monotonically. We can explain this behaviour as follows; the electric field does not push the electron in the ground state to the left-hand side well at all, but pushes the first excited state to the right-hand well then pushes it to the left-hand side well. In asymmetric coupled double QW structures, the ground and the higher states are mostly localized in the left-hand side well, thus under the electric field this confinement increases, and the absorption coefficient is larger with respect to the symmetric coupled double QW structure. Under the electric field, the enhancement of the absorption coefficient in the asymmetric structure (β = 1.5) is larger than that of the symmetric structure (β = 1), thus β can be used as a tuneable parameter of the structure. In coupled double QW structures, the reduction absorption coefficient at zero electric field also has the advantage of diminishing the on-state transmission loss. However, the magnitude of the absorption coefficient is comparable to that for a conventional single square QW. In Fig. 5, the variation of the absorption coefficient is given as a function of the incident photon energy for several σ values. As seen in this figure, the transitions between higher subbands are very sensitive to the parameter σ . In this structure for (2–3) transitions, the parameter σ can be used as a tuneable parameter. By comparing this figure with Fig. 2A we see that all transitions in the single quantum well are sensitive to the parameter σ , but here only the (2–3) transition is sensitive to the parameter σ .

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Superlattices and Microstructures, Vol. 26, No. 6,1999

6000 Lef= 250 Å

2–3 σ = 0.5

σ = 0.8 σ=1

4000 α (cm)–1

σ = 0.3

1–2

2000

σ = 0.8 σ = 0.5 σ = 0.3

σ=1

σ = 0.3

σ = 0.8

1–3

σ = 0.5

0 0

20

40

60

80

100

Photon energy (meV) Fig. 5. The variation of the absorption coefficient as a function of the photon energy for different σ values.

The variation of the energy difference between the first and second subband 1E 1−2 is given in Fig. 6A as a function of the parameter β for several σ values at F = 50 kV cm−1 and L b = 10 Å. For coupled DQWs, the energy difference 1E 1−2 increases with parameter β and afterwards converges to a constant when the parameter β is further increased. Since, by increasing the parameter β, the subbands tends to localize in the left-hand side well, thus the energy levels increase, and for further large β values we have a single QW. Also, in coupled DQWs the energy difference 1E 1−2 decreases when σ decreases for all β values. Since for small σ values the localization of the subbands in the right-hand side well is still appreciable and a single QW regime can be observed at very large β values as the parameter σ increases (see Fig. 6A). The calculated transition energy (1E 1−2 ) is given in Fig. 6B as a function of the parameter β for different barrier widths. For small barrier width values, where the coupling between adjacent wells is dominant, firstly 1E increases as β increases and when β is further increased, an SQW regime becomes dominant and 1E 1−2 tends to be stable. On the other hand, for large barrier widths coupling is very weak and the variation in 1E 1−2 is not sensitive to the β parameter. If we consider this behaviour in the symmetric DQW (σ = 1) when barriers are thin enough to couple the wells, by changing the β parameter we can tune the transition energy 1E 1−2 from 55 meV to 70 meV. This case gives a new degree of freedom in the device applications. In conclusion, linear intersubband optical absorption in a single and coupled double GaAs QW has been studied under an external electric field. Various device parameters and their effects on the optical absorption have also been studied. We show that the effect of the applied electric field on the intersubband optical absorption is similar to changes in the dimensions of the structure, thus this behaviour can be used to study these systems in regions of interest, without the need for the growth of many different samples. Calculated

Superlattices and Microstructures, Vol. 26, No. 6,1999

403

A 68

σ=1

F = 50 kV/cm–1

8 σ = 0. 5 0 σ= . 3 σ = 0.

Lb= 10 Å

E (meV)

64

60

56

52 2

1

3

4

5

6

β 75 B

Lb = 40 Å Lb= 30 Å

E (meV)

70

Lb= 10 Å

65

F = 50 kV/cm–1 σ=1

60

55 2

4

6

8

10

β Fig. 6. A, The variation of the intersubband transition energy as a function of β(L 1 /L 2 ) for different σ values. B, The variation of the intersubband transition energy as a function of β(L 1 /L 2 ) for different barrier widths.

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Superlattices and Microstructures, Vol. 26, No. 6,1999

results reveal that, using a heterostructures one can tailor the energy band structure by an appropriate layer sequence which may improve the mechanism for light-induced modification of the band structure, obtaining even, enhanced, nonlinear optical properties. This result is an improvement of optoelectronic devices.

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