Influence of Quantum Dot Dimensions in a DWELL Photodetector on Absorption Co-efficient Soumitra Roy Joy*a, Golam Md. Imran Hossaina, Tonmoy Kumar Bhowmicka and Farseem Mannan Mohammedya a

Department of Electrical and Electronic Engineering, Bangladesh University of Engineering and Technology, Dhaka 100, Bangladesh. *e-mail: [email protected]

Abstract— The absorption co-efficient of symmetric conical shaped InAs/InGaAs quantum dots-in-a-well (DWELL) infrared photo-detector has been calculated through NonEquilibrium Green’s Function (NEGF) approach to explain their relative magnitude resulting from various quantum dot dimensions. Kinetic equation of Green’s function is solved by using method of finite difference and wave-function is calculated from greens function and source term. All possible transitions from bound level of quantum dot to any higher continuum level existing in either quantum well or barrier are taken into account. Trace of peak of absorption co-efficient with gradual change of height to base ratio of quantum dot is simulated under various polarization of incident light. Keywords- dipole moment; absorption co-efficient; quantum dot dimension; DWELL QDIP; NEGF

I.

INTRODUCTION

Quantum Dot Infrared Photodetector (QDIP) has opened a new arena in the research field on mid and long wavelength infrared photodetectors. The three-dimensional confinement for electrons in the quantum-dot structure, and hence its artificial-atom like behaviour has attracted the attention of both theoreticians and experimentalists of late. Although common photodetectors in this wavelength range use mercury cadmium telluride (MCT) because of their superiority over typical QD or Quantum Well Infrared Photodetector (QWIP) in terms of responsivity and spectral detectivity; the inherent difficulties in epitaxial growth of mercury based compound significantly obstacle the mass production of large area focal plane arrays (FPAs) [1]. The potential candidature of QDIPs as IR detectors is strengthened further due to the fact that they provide additional degrees of confinement, leading to some significant advantages such as: QDIPs are sensitive to normal-incidence IR radiation, which is forbidden in n type QWIPs because of polarization selection rules; QDIPs exhibit comparatively long effective carrier lifetimes (~100s of picoseconds), and QDIPs exhibit low dark current [2]-[4]. Ideally, QDIPs should show improved performance characteristics such as high responsivity, high detectivity, and high operating temperatures. The Quantum Dot-in-a-Well (DWELL) detector represents a hybrid between a conventional quantum well

infrared photodetector (QWIP) and a quantum dot infrared photodetector (QDIP). In DWELL, the active region consists of InAs quantum dots embedded in an InGaAs quantum well. One promising aspect of DWELL detectors is that, they, unlike QWIPs and like QDIPs, display normal incidence operation without gratings or opto-couplers and hence simplify optical configuration in many applications. They also demonstrate reproducible ‘‘dial-in recipes’’ for control over the operating wavelength, like QWIPs [2]. Moreover, the DWELL detectors also manifest biastunability and multi-color operation in the mid wave infrared (MWIR, 3–5 µm), long wave infrared (LWIR, 8–12 µm) and very long wave infrared (VLWIR, >14 µm) regimes. Photodetectors operating in the mid-infrared wave have already made their niche in various applications in medical and environmental sensing, thermal imaging, night vision cameras, and missile tracking and recognition, mine detection and remote-sensing. In this paper, absorption co-efficient of symmetric DWELL QDIP is obtained by varying QD dimensions each time, using the NEGF method. The photocurrent response of QDIP can be understood from the nature of absorption coefficient. A numerical technique based on the method of finite differences is used to solve the kinetic equation of Green’s function. Self-energy is calculated in a recursive manner. The estimated absorption co-efficient counts on transition from all bound to continuum levels. A monotonic red shift of dipole moment peak and a gradual improvement in magnitude of absorption co-efficient with incremental rise of quantum dot height to base ratio are theoretically verified within a certain range of observation. II.

THEORETICAL MODELING OF DWELL QDIP STRUCTURE

Ten-period active regions of 6nm In0.15Ga0.85As, 2.4 ML of InAs, 6 nm In0.15Ga0.85As, and 50 nm GaAs constitute the DWELL detector (in Molecular Beam Epitaxy (MBE) process), as reported in [3], which is shown in Fig. 1(a). The QD material InAs is deposited over the substrate. Due to the lattice mismatch between deposited material and substrate, cumulative strain builds up. After reaching a marginal

thickness (2.4 ML) limit, the two-dimensional growth breaks into a three-dimensional one and dislocation free QD islands begin to grow. This uncapped array of QDs is later covered by InGaAs capping layer of comparable thickness. A thick layer of GaAs is then deposited over the capping layer and the exposed surface is smoothened. We have modeled the DWELL device for numerical simulation as shown in Fig. 1(b). 60% of the band gap difference between InAs and GaAs is counted as the conduction band offset [5], [6]. The band offsets calculated are 477 meV between InAs and In0.15Ga0.85As; 93 meV between In0.15Ga0.85As and GaAs. The conduction band edge of the In0.15Ga0.85As is selected as reference energy level. To calculate the effective masses in different materials, two binary values are linearly interpolated. The effective masses of GaAs, InAs and In0.15Ga0.85As, in terms of electron’s mass, stand out to be 0.067, 0.027 and 0.061 respectively. The entire device is thought of as an assembly of numerous identical cylinders. Every cylinder holds one quantum dot at its center. To calculate Hamiltonian of the device, a cross section along the cylinder axis is taken. Then it is disintegrated into a large number of equally spaced grids. The finite difference method is used to solve the differential equation governing Green’s function. The retarded Green’s function of the system is defined as [7]





r

It gives all the eigenstates and the corresponding eigenvalues. HD is the Hamiltonian matrix of the isolated cylinder cross section that contains the quantum dot. III.

The estimated dot density of the quantum dot photodetector under our analysis is 5 x 1010cm2. Two nearmost dots are separated by about 60nm on average [3]. Due to this sufficient separation, in our simplified model of the quantum dot photo-detector, the wavefunction of one quantum dot can be justifiably considered as decoupled from the rest of the dots. A quantum dot is modeled so as to be surrounded by semi-infinite contact composed of InGaAs, GaAs layer, and InAs wetting layer. The contact can be imagined as being a continuation of cylinder radius. The axis of conical quantum dot and the vertical z axis of the cylinder coincide. The property of translational invariance of the contact can be maneuvered for obtaining a more precise result by modeling quantum dot photo detectors in the above mentioned fashion. The Hamiltonian matrix for the device, which is tridiagonal and Hermitian, is formed by finite difference method and is given by [9], 0  x1 1 0        1 x2 2    Hop   0      Nx1    0   Nx1 x( Nx)   

r

E  H op   ( E ) G ( x, x '; y , y '; E )   ( x  x ') ( y  y ') (1)

In the above equation, E is the total energy of electron,

 r is the self energy and Hop, the Hamiltonian operator of 

2

  V ( x, y )

(2)

2 m( x , y )

From the above equation, the wavefunction can be related to the Green’s function by the relation: 1

r

 c   EI  H D    S  GD S

 x (i)

ty1 U y1  2tx( i )  2ty1  ty1 U y 2  2tx( i )  2ty2  ty2     0 

(3) ( i )

If we write the self energy as,



r



f

 m

m

m

(4)

(7)

Here

the system, is given by [8] H op  .

NUMERICAL CALCULATION

 tx1, i  0      0

 0







0 tx2 , i 

0 ty2

ty( Ny 1)

0

      ty( Ny 1)  U y ( Ny )  2tx( i )  2ty ( Ny )  0

0 

      txNy , i  

m

The self energy term is

then the excitation S can be expressed as

S

( f

* n

 fn )  m

(5)

m

It gives the contribution from all propagating modes at the same total energy E. The bound state wavefunctions in the quantum dots are calculated by solving the eigenvalue problem:

 EI  H  D

b

0

(6)

 g c  0  0  0 r         0  

   

0   0            gc  

Here, tx is the coupling energy between adjacent grid points along x direction, and is given by [10],

(8)

tx 



2 *

2mx a

(9)

2

Here, gc is the retarded green’s function of a unit cell of the contact, and is solved from the recursive relation [8] 

1

gc  ((E i)I  Hc gc )

(10)

The transitional dipole moment or simply ‘dipole moment’, (usually denoted by µnm) for a transition between an initial state i, and a final state f, is the electric dipole moment associated with the transition between the two states.

 fi   e   f *r i dr  e  f r  i

(11)

The absorption co-efficient α (1/cm) in the crystal is the ratio of number of photons absorbed per unit volume to number of photons injected per unit area in each second. .  2 2  (  )  eˆ.ba  ( Ea  Eb   )( f a  fb ) (12)  nr c 0 V k k When the scattering relaxation is included, the delta function may be replaced by a Lorentzian function with a linewidth Γ. a

b

 ( Eb  E a   ) 

IV.

 / (2 ) 2

( Eb  E a   )  (  / 2)

2

(13)

RESULTS AND DISCUSSION

The reported heterostructure in [11], cited as sample B (AlGaAs/InGaAs/InAs), when compared to sample A (AlGaAs/GaAs/InAlGaAs/InAs), has exhibited a significant improvement in photocurrent response for both S-plane (in plane) and P plane (45 degree to growth plane) polarized light incidence. The authors Jiayi Shao et. al. like to attribute this improvement to the improved height to base ratio (H/B) of quantum dot in sample B than in sample A. In order to find out whether this improvement is predominantly caused by height to base ratio improvement of QD or by some other reasons (i.e. by the compositional difference brought to sample B, in comparison with sample A), we have chosen a heterostructure reported in [3]. In this specimen, the self-assembled growth in the Stranski– Krastanov (SK) growth mode allows quantum dots to have height of 6.5nm and base of 11nm on 2.4 ML wetting layer. In theoretical simulation of absorption co-efficient, we have changed the base dimension of quantum dot, keeping its height fixed at 6.5nm. Notably, in simulation, the base diameter of QD has always been kept larger than the height, in accordance with standard growth technique [12], [13]. In this paper, we are primarily interested in verifying the experimental result of improvement of magnitude of

spectral response due to H to B ratio improvement of QD. However, this result can be intuitively expected from the fact that low height to base aspect ratio of QDs leads to loss of discrete quantum confinement level. Hence the “artificial atom”-like behavior of QDs gets limited, which reduces the carrier lifetime, and consequently causes lower operating temperature and higher dark current [13]. A theoretical analysis of transitional dipole moment for standard quantum dot dimensions cited in [3] with 6.5nm height and 11nm base yields two prominent peaks. However, not all the peaks should necessarily exist in experimental result of photocurrent response, since Fermi’s Golden Rule will eliminate the possibility of any transition from even to even or odd to odd ψ function. This calculated transitional dipole moment is later used to estimate absorption co-efficient at 77K. We assumed that inhomogeneous broadening (Lorentzian broadening) is dominant in absorption phenomenon. The graph of absorption co-efficient exhibits a smoothened and broadened version of the graph for dipole moment. A relative comparison among the graphs of absorption co-efficient of QDIP for various height to base ratio of QD is shown in Fig. 2(a) and Fig. 2(b). Fig. 2(a), applicable for S polarized incident light, indicates that, the peak of absorption co-efficient (and henceforth, spectral response) is vividly highest among all H/B values taken into account. However, it does not experience a monotonic fall in maxima as H/B value declines gradually. Fig. 2(b), applicable for P polarized incident light, gives us a rather straight-forward notion. It indicates a clear improvement in absorption co-efficient with the rise of height to base ratio of QD. However, in both case of polarization, the maximum (peak) of absorption co-efficient invariably shifts towards red end with gradual rise in height to base ratio of QD. Fig. 3(a) and Fig. 3(b) indicate the change of relative magnitude of maxima of absorption co-efficient with H/B variation, for S and P polarization respectively. For S polarized incident light, the curve in Fig. 3(a), as we proceed from right to left, falls with decreasing height to base ratio. However, it shows a dip at 11 nm base dimension. Above this dimension, absorption peak tends to rise again. In our numerical simulation, we only increased the length of base gradually, keeping the height constant (and thus increased QD volume). We think, at some stage, QD volume comes to play more than QD height to base ratio in determining absorption peak. For P polarized light, the graph in Fig. 3(b) has shown a rather close proximity to linearity within the range of our simulated value. However, it remains to explore whether changes in both QD height and base simultaneously, and beyond the observed range, has any further quantum effect or not. V.

CONCLUTION

In this paper, we have pointed out the interesting phenomenon of how the quantum dot dimensions exert

impact in determining DWELL’s absorption co-efficient. This analysis will eventually shed some light on the relative change of magnitude of the photocurrent response in conjunction with change of QD dimensions. The InAs/InGaAs/GaAs DWELL structure is taken and its quantum dot base is varied while its height is kept fixed at standard value (6.5nm). In each case, absorption co-efficient of the device is calculated resulting from both S and P polarized incident light. The experimentally found fact of responsivity improvement with quantum dot height to base ratio improvement also matches with our numerically simulated result within a certain range of observation. The parabolic variation of absorption maximum with change in QD height to base ratio for S polarized light can likely be attributed to the race between two possible factors: QD volume and QD dimension ratio. However, whether absorption maximum will change in a similar parabolic way for P polarized light as well beyond our observed range or will retain its linear shape remains a matter of further investigation.

[2] [3] [4]

[5]

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[10] [11]

ACKNOWLEDGMENT The authors gratefully thank to Mr. Sishir Bhowmick and Md. Redwan Noor Sajjad for their excellent assistance with formalism. Funding from the BUET’s Research Office (CASR meeting no 218, agenda 97, dated 6 June 2009) is also greatly acknowledged.

[12]

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Sidorov Yu G, Dvoretsky S A, Yakushev M V, Mikhailov N N, Varavin V S and Liberman V I 1997 Thin Solid Films 306 253

GaAs 6 nm

InGaAs

InGaAs InAs

2.4 ML

InGaAs GaAs 30 nm

Figure 1: (a) Cross-section schematic of a 10 layer InAs/InGaAs quantum dot in a well detector, (b) Schematic of the cross section along cylinder axis of the device used for the theoretical model

Figure 2(a): Comparison of absorption co-efficient of different hypothetical QDIPs having various height to base ratios. Incident light is S polarized (normal incidence). The green curve is for QD grown in standard SK growth method (height=6.5nm, base=11nm)

Figure 2(b): Comparison of absorption co-efficient of different hypothetical QDIPs having various Height to Base ratio. Incident Light is P polarized (45 degree incidence to growth plane). The green curve is for QD grown in standard SK growth method (height=6.5nm, base=11nm)

Figure 3(a): Relative change of absorption peak with dot dimension change. Absorption peak found for 7nm QD base is taken as unity, when incident light is S polarized.

Figure 3(b): Relative change of absorption peak with dot dimension change. Absorption peak found for 7nm QD base is taken as unity, when incident light is P polarized.