Non-equilibrium Green's Function Calculation Of Optical Absorption In Nano Optoelectronic Devices

Non-equilibrium Green’s Function Calculation of Optical Absorption in Nano Optoelectronic Devices Oka Kurniawan, Ping Bai, and Er Ping Li Computational Electronics and Photonics Institute of High Performance Computing Singapore 138632 Email: [email protected]

Abstract—The high speed of optical devices motivates the integration between electronics and photonics. One of the most common optoelectronic devices used for such integration is a photodetector. This paper describes the formulation of an optical absorption inside a photodetector using the Non-equilibrium Green’s Function (NEGF) framework. To illustrate the use of the formulation, optical properties of a double barrier quantum well photodetector are simulated. The photocurrent and the spectral response of the photodetector are calculated and presented. We also study the effect of varying various bias voltage and introducing non-uniformity in the effective mass. It is found that the peak locations do not change significantly under these variations.

one found in Quantum Well Infrared Photodetectors (QWIP) [13].

I. I NTRODUCTION Computational electronics with optical properties for nano devices is gaining interest. In the past, most quantum transport calculation focuses on the ballistic regime. With the introduction of Non-equilibrium Green’s Function (NEGF), the phasebreaking phenomena can be included into the calculation through the self-energy matrices [1]–[5]. In most cases studied so far, the only phase-breaking phenomena is the phonon scatterings [2], [5]–[8]. Phonon scatterings cause the electron transport to be no longer in the ballistic regime. Besides scatterings due to the phonon, electrons are also scattered by photon or light. This physical phenomena is significant for optoelectronic devices such as photodetectors and lasers. Very few studies deal with electrons interaction with light within the NEGF framework. Theoretically, the electrons interaction with light is included into the computation in the same manner as the phonon scatterings, which is through the self-energy matrices. The real problem, then, is to obtain the expressions for these self-energy matrices. The self-energy matrices are used to take into account both optical emission as well as optical absorption. Recently, some results for optical absorption in carbon nanotube are obtained by two separate groups [9]–[11]. One of them, Stewart and Leonard, based their calculation from the results previously obtained by Henrickson for resonant tunneling diode [12]. Results obtained by Henrickson, however, require a terminating barrier in one side of the diode to ensure no holecurrent correction is present in one of the contacts (Fig. 1). Moreover, the work in [12] did not calculate the photocurrent when the device is under bias. In fact, applying a bias to the device is a common operation for photodetectors, such as the

Fig. 1. Schematic of the resonant tunneling diode used in simulation. The collector side on the right has a terminating barrier of 0.2 eV.

In this work we extend the results obtained by Henrickson for semiconductor materials, and we show that a photocurrent can be calculated for the case under bias and without a terminating barrier. First, we will lay out the formulation for the optical absorption calculation within the NEGF framework. Then, we will show that our calculation is able to duplicate the results obtained by Henrickson for the case of zero bias with a terminating barrier. After that, we will further show the case when the device is under bias and the terminating barrier is not present (Fig. 2). Lastly, we studied the effect of introducing a non-uniform effective mass, which was neglected in [12].

Fig. 2.

Schematic of a resonant tunneling diode under Vb bias.

II. O PTICAL A BSORPTION C ALCULATION A. NEGF Formulation The interaction between the photon and the electrons in the device is included into the Non-equilibrium Green’s Function

(NEGF) calculation through the self-energy matrices. Before we obtain the expressions for these self-energy matrices, we first need to express the interaction Hamiltonian. The interaction Hamiltonian in the second quantized form is given by X H1 = hr|H 1 |sia†r as (1) rs

†

where r and s are the site-basis eigenstates. The term a and a are the creation and the annihilation operator. The matrix element of H 1 is q A · hr|p|si (2) hr|H 1 |si = m0 where p is the momentum operator. Now, if we use the finite difference method to discretize the Hamiltonian, it can be shown that X
H1 = (3) Mrs be−ıωt + b† eıωt rs

where

r √ h µr ǫr q¯ h ¯ Iω Prs (4) Mrs = ı2a 2N ωǫc In the above expression, a is the grid spacing, µr and ǫr are the relative permeability and relative permittivity respectively. The term N is the number of photon, ω is the frequency of the photon, and Iω is the photon flux. Lastly,  ∗   +1/ms , s = r + 1 −1/m∗s , s = r − 1 (5) Prs =   0 , else

and the other symbols have their usual meanings. Note that m∗s is the effective mass at the site s. Now, following the steps in [12], it can be shown that the self-energy matrices are given by Σ> rs (E) =

X

¯ ω) Mrp Mqs [N G> pq (E + h

pq

+ (N + 1)G> hω)] pq (E − ¯ X < < Σrs (E) = Mrp Mqs [N Gpq (E − ¯ hω)

(6)

pq

+ (N + 1)G< ¯ ω)] pq (E + h

(7)

G< (E) = GΣ< G†

(8)

and >

< †

G (E) = (G )

(9)

The main calculation, then, is to obtain the retarded Green’s function G. The retarded Green’s function is calculated for each energy grid from G(E) = [ES + ıη − H0 − diag(U ) − Σ1 − Σ2 − Σph ]−1 (10) where S is the overlap matrix, and for finite difference it is simply an identity matrix I. The term η is a small

positive number added to the energy, H0 is the unperturbed Hamiltonian matrix of the device, U is a diagonal matrix that takes into account the potential drop due to the bias. The terms Σ1 and Σ2 are the self-energy matrices for the left and right contacts and are given by [5]

2

Σi = −t exp (ika) /(2m∗c a2 ),

(11)

m∗c

where t = h ¯ and is the effective mass at contact i, and a is the grid spacing. The term ka can be obtained from the following dispersion relation. E = Ec + U + 2t(1 − cos ka)

(12)

Finally, Σph in (10) is the electron-photon self-energy matrix which elements are given by the expressions in (6) and (7). Once the Green’s functions are calculated, we can obtain the other properties such as electron density, local density of states, and the photocurrent. The expression for the current at the contacts can be written as Z q < t(G< (13) I= n−2,n−1 (E) − Gn−1,n−2 (E))dE π¯h where the site index are i = 0, 1, 2, . . . , n − 2, n − 1. Note that the current is calculated from the last two nodes at the right hand side. We have taken into account the spin degeneracy in the above expression. B. Numerical Calculation The previous calculation was implemented using a C++ code. The code starts by initializing the conduction band energy, number of grids, grid spacing, permittivity, and the permeability. For the case when the terminating barrier is present, we add the height of the terminating barrier to the conduction band in the right contact. For the case when the device is under bias, no terminating barrier is added. Two energy grids are initialized in the code. One is the grid for the photon energy, ¯hω, and the other one is the grid for the energy integral in the Green’s function calculation (13). The photon energy grid spans from 0.1 to 2.5 eV. On the other hand, the energy integration grid spans from -0.3 to 2.9 eV. the photocurrent response is computed or each photon energy. To compute the photocurrent response, the main task is to obtain the Green’s functions. To do this, first, the self-energy matrices for the contacts are calculated from (11) and (12). With this, the non-interacting retarded Green’s function can be calculated from (10) with zero Σph . Once the retarded Green’s function is obtained, the less-than Green’s function for the non-interacting case is calculated from (8). We calculate all these matrices for two energy levels. One is at E, and the other one at E − ¯hω. Now, we are ready to compute the photon interaction part. Before we do this, we can simplify (6) and (7) for the case when the only dominant process is the optical absorption. For this case, the expressions become simply X Σ> Mrp Mqs N G> ¯ ω) (14) rs (E) = pq (E + h pq

=

Mrp Mqs N G< pq (E

pq

−¯ hω)

(15)

From these expressions, the self-energy matrices Σ< ph are computed. And hence, the less-than Green’s function for the interacting case, i.e. G< ph , is simply obtained from (8). In this last calculation, we approximate the retarted Green’s function needed in (8) using the non-interacting case obtained earlier. Once the interacting less-than Green’s function is obtained, the photocurrent is calculated using (13). In most cases, however, we are interested in the photocurrent response function. This photocurrent response is defined as I ph RI = qIω

(16)

where Iω is the photon flux at energy ¯ hω. It is important to note that G< , which is used to obtain ph the photocurrent, gives the distribution of the photoexcited electrons and not the total electron population. The reason is that, in order to obtain the total electrons, we need to calculate self-consistently G< by iteration until convergence is achieved. In the steps described above, however, we stop at the first iteration. To obtain a more accurate total current or total electron density, we have to perform at least one more iteration of the self-energies. This second iteration calculates the first-order correction to G< . III. P HOTOCURRENT

USING

NEGF

A. Verification We first compare our calculation results with that of Henrickson’s [12] which uses a terminating barrier on one side of the resonant tunneling diode. The resonant tunneling diode is built from GaAs/GaAlAs materials (Fig. 1). Since uniform material properties are used in [12], GaAs properties are used throughout the first simulation. We present the result for the non-uniform material properties in a latter part of this paper. The grid spacing is set to 0.25 nm. And using this grid spacing, we follow the same device dimension as the case in [12], i.e. barrier width of 2 nm and quantum well width of 5 nm. The barrier height is set to 2 eV, which is rather large for GaAs system. Nevertheless, our purpose is to verify the results with that in [12], and so we will follow closely the structure used in that work. As mentioned in the Introduction, the case considered by Henrickson requires us to add a terminating barrier as shown at the right contact in Fig. 1. The reason for this addition is that once an electron jumps to an excited state, it leaves a hole-carrier behind. This hole-carrier can tunnel outside the barrier and produce a current. When one calculates the current at one terminal, the net current will be zero. To overcome this, Henrickson adds a terminating barrier. Hence, the only current calculated is the electron current due to the photoexcitation. The height for this terminating barrier is set to 0.2 eV. To ensure that the ground state inside the well is occupied by electron, the Fermi level must be set above the ground state

Photocurrent Response, RI (nm2/photon)

Σ< rs (E)

X

100

Our Simulation Henrickson’s

10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8

0

0.5

1

1.5

2

2.5

Photon Energy (eV)

Fig. 3. Comparison of photocurrent calculation with results from Henrickson for the case of zero bias. The first peak agrees considerably well.

energy. By setting the Fermi level to 0.17 eV, the ground state becomes occupied and is ready to be excited by photons. Now, we are ready to present our calculation result in comparison to Henrickson’s. Fig. 3 shows that our calculation result produces a reasonably accurate photocurrent for the structure proposed in [12]. In fact, no significant difference can be observed for the first peak. The slight differences on the other peaks are due to the difference in the band energy model used for the Hamiltonian. In our case, we use a simple effective mass approximation with finite difference discretization. It is known that single band Hamiltonian, such as the one we use here, tends to overestimate the eigenenergies [7]. B. Spectral Response under Various Bias Photocurrent Response, RI (nm2/photon)

and

10-1

Vb = 0.05 V Vb = 0.10 V Vb = 0.20 V

10-2

10-3

10-4

0.4

1.9 1.1

10-5

0

0.5

1

1.5

2

2.5

Photon Energy (eV)

Fig. 4. Photocurrent calculation under various bias voltage. Peak location does not change significantly under bias.

After verifying our results, we now turn to the case when the resonant tunneling diode is biased and no terminating barrier is present. The schematic of the problem is shown in Fig. 2. We assume that the potential drop is linear in the channel. Fig. 4 shows the spectral response when the device is biased with 0.05 V, 0.10 V, and 0.20 V. It can be observed that the peak locations do not change significantly. The three peaks are located around 0.4 eV, 1.1 eV, and 1.9 eV respectively. How do we obtain these peak locations? Looking at the eigenenergies inside the well gives us the answer.

100

Photocurrent Response, RI (nm2/photon)

The transmission curves for the three bias voltages are shown in Fig. 5. The resonance energies for the case of 0.2 V are located around 0.043, 0.470, 1.139, 1.957, and 2.785 eV. Taking the differences of the energies between the lowest resonant energy and the others gives us 0.43, 1.1, and 1.9 eV. These are the peak locations of the photocurrent response in Fig. 4. Therefore, the photocurrent peaks are the results of electrons transition from the ground state to the excited states by absorbing photons. Furthermore, it can be seen that the resonant energies are shifted to the left for higher bias. However, the distances between these peak locations do not seem to change significantly, which explains the results in Fig. 4.

10-2

uniform non-uniform

10-3

10-4

10-5

0

0.5

1

1.5

2

2.5

Energy (eV)

Fig. 6. Photocurrent comparison between uniform and non-uniform effective mass. The effect of non-unfiromity in effective mass is imperceptible for GaAs/AlGaAs system.

10-1 Transmission

10-2

It is observed that the photocurrent peak locations do not change significantly under the bias considered in this work. We then studied the effect of non-uniform effective mass. It is also found that the effect of non-uniform effective mass is negligible for GaAs/AlGaAs system. The computer code used in the calculation is available upon request from the author.

10-3 10-4 10-5 10-6 10-7

Vb = 0.05 V Vb = 0.10 V Vb = 0.20 V

10-8 10-9

0

0.5

1

1.5

2

2.5

Energy (eV)

Fig. 5. Transmission curves for various bias voltages. Though the peaks are shifted to the left, the distances between the peaks does not change significantly.

C. Effects of Non-uniform Effective Mass In the previous results, a uniform effective mass was used. This follows [12] which uses GaAs effective mass, i.e. 0.067m0. It is interesting to study how the spectral response would change when we use a non-uniform effective mass. What it means is that we use the actual AlGaAs effective mass of 0.089m0 for the barrier region and the GaAs effective mass of 0.067m0 for the well region. This non-uniformity of effective mass can be included in the Hamiltonian and in (5). Fig. 6 shows the two photocurrent spectral responses for both uniform and non-uniform effective mass. It can be seen that the photocurrent response changes imperceptibly. The peak locations for non-uniform effective mass remain about the same as that for uniform effective mass. The results are reasonable since there is no large difference between the effective masses of AlGaAs and GaAs. This justifies the results in [12] which uses uniform material parameters throughout its calculations. IV. C ONCLUSION In conclusion, our work presents the formulation of optical absorption within the Non-equilibrium Green’s Function framework. The calculation is first verified with the previously published results. We then show the photocurrent spectral response when the photodetector is under various bias voltage.

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