Non-equilibrium Green’s Function Calculation ofOptical Absorption in Nano Optoelectronic Devices

Oka Kurniawan, Ping Bai, and Er Ping LiComputational Electronics and Photonics

Institute of High Performance ComputingSingapore 138632

Email: kurniawano@ihpc.a-star.edu.sg

Abstract—The high speed of optical devices motivates theintegration between electronics and photonics. One of the mostcommon optoelectronic devices used for such integration isaphotodetector. This paper describes the formulation of an opticalabsorption inside a photodetector using the Non-equilibriumGreen’s Function (NEGF) framework. To illustrate the use oftheformulation, optical properties of a double barrier quantum wellphotodetector are simulated. The photocurrent and the spectralresponse of the photodetector are calculated and presented.We also study the effect of varying various bias voltage andintroducing non-uniformity in the effective mass. It is foundthat the peak locations do not change significantly under thesevariations.

I. I NTRODUCTION

Computational electronics with optical properties for nanodevices is gaining interest. In the past, most quantum transportcalculation focuses on the ballistic regime. With the introduc-tion of Non-equilibrium Green’s Function (NEGF), the phase-breaking phenomena can be included into the calculationthrough the self-energy matrices [1]–[5]. In most cases studiedso far, the only phase-breaking phenomena is the phononscatterings [2], [5]–[8]. Phonon scatterings cause the electrontransport to be no longer in the ballistic regime. Besidesscatterings due to the phonon, electrons are also scatteredbyphoton or light. This physical phenomena is significant foroptoelectronic devices such as photodetectors and lasers.

Very few studies deal with electrons interaction with lightwithin the NEGF framework. Theoretically, the electrons in-teraction with light is included into the computation in thesame manner as the phonon scatterings, which is through theself-energy matrices. The real problem, then, is to obtain theexpressions for these self-energy matrices. The self-energymatrices are used to take into account both optical emissionas well as optical absorption.

Recently, some results for optical absorption in carbonnanotube are obtained by two separate groups [9]–[11]. One ofthem, Stewart and Leonard, based their calculation from there-sults previously obtained by Henrickson for resonant tunnelingdiode [12]. Results obtained by Henrickson, however, requirea terminating barrier in one side of the diode to ensure no hole-current correction is present in one of the contacts (Fig. 1).Moreover, the work in [12] did not calculate the photocurrentwhen the device is under bias. In fact, applying a bias to thedevice is a common operation for photodetectors, such as the

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

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

In this work we extend the results obtained by Henricksonfor semiconductor materials, and we show that a photocurrentcan be calculated for the case under bias and without aterminating barrier. First, we will lay out the formulationforthe optical absorption calculation within the NEGF framework.Then, we will show that our calculation is able to duplicate theresults obtained by Henrickson for the case of zero bias witha terminating barrier. After that, we will further show the casewhen the device is under bias and the terminating barrier is notpresent (Fig. 2). Lastly, we studied the effect of introducing anon-uniform effective mass, which was neglected in [12].

Fig. 2. Schematic of a resonant tunneling diode underVb bias.

II. OPTICAL ABSORPTIONCALCULATION

A. NEGF Formulation

The interaction between the photon and the electrons in thedevice is included into the Non-equilibrium Green’s Function

(NEGF) calculation through the self-energy matrices. Beforewe obtain the expressions for these self-energy matrices,we first need to express the interaction Hamiltonian. Theinteraction Hamiltonian in the second quantized form is givenby

H1 =∑

rs

〈r|H1|s〉a†ras (1)

wherer ands are the site-basis eigenstates. The terma† anda are the creation and the annihilation operator. The matrixelement ofH1 is

〈r|H1|s〉 =q

m0

A · 〈r|p|s〉 (2)

wherep is the momentum operator. Now, if we use the finitedifference method to discretize the Hamiltonian, it can beshown that

H1 =∑

rs

Mrs

(

be−ıωt + b†eıωt)

(3)

where

Mrs =qh

ı2a

√

h√

µrǫr

2NωǫcIωPrs (4)

In the above expression,a is the grid spacing,µr and ǫr arethe relative permeability and relative permittivity respectively.The termN is the number of photon,ω is the frequency ofthe photon, andIω is the photon flux. Lastly,

Prs =

+1/m∗s , s = r + 1

−1/m∗s , s = r − 1

0 , else

(5)

and the other symbols have their usual meanings. Note thatm∗

s is the effective mass at the sites.Now, following the steps in [12], it can be shown that the

self-energy matrices are given by

Σ>rs(E) =

∑

pq

MrpMqs[NG>pq(E + hω)

+ (N + 1)G>pq(E − hω)] (6)

Σ<rs(E) =

∑

pq

MrpMqs[NG<pq(E − hω)

+ (N + 1)G<pq(E + hω)] (7)

and

G<(E) = GΣ<G† (8)

G>(E) = (G<)† (9)

The main calculation, then, is to obtain the retarded Green’sfunction G. The retarded Green’s function is calculated foreach energy grid from

G(E) = [ES + ıη−H0−diag(U)−Σ1−Σ2−Σph]−1 (10)

where S is the overlap matrix, and for finite differenceit is simply an identity matrixI. The term η is a small

positive number added to the energy,H0 is the unperturbedHamiltonian matrix of the device,U is a diagonal matrix thattakes into account the potential drop due to the bias. The termsΣ1 andΣ2 are the self-energy matrices for the left and rightcontacts and are given by [5]

Σi = −t exp (ika) (11)

where t = h2/(2m∗ca

2), and m∗c is the effective mass at

contact i, and a is the grid spacing. The termka can beobtained from the following dispersion relation.

E = Ec + U + 2t(1 − cos ka) (12)

Finally, Σph in (10) is the electron-photon self-energy matrixwhich elements are given by the expressions in (6) and (7).Once the Green’s functions are calculated, we can obtainthe other properties such as electron density, local density ofstates, and the photocurrent.

The expression for the current at the contacts can be writtenas

I =q

πh

∫

t(G<n−2,n−1

(E) − G<n−1,n−2

(E))dE (13)

where the site index arei = 0, 1, 2, . . . , n−2, n−1. Note thatthe current is calculated from the last two nodes at the righthand side. We have taken into account the spin degeneracy inthe above expression.

B. Numerical Calculation

The previous calculation was implemented using a C++code. The code starts by initializing the conduction bandenergy, number of grids, grid spacing, permittivity, and thepermeability. For the case when the terminating barrier ispresent, we add the height of the terminating barrier to theconduction band in the right contact. For the case when thedevice is under bias, no terminating barrier is added.

Two energy grids are initialized in the code. One is the gridfor the photon energy,hω, and the other one is the grid for theenergy integral in the Green’s function calculation (13). Thephoton energy grid spans from 0.1 to 2.5 eV. On the otherhand, 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 toobtain the Green’s functions. To do this, first, the self-energymatrices for the contacts are calculated from (11) and (12).With this, the non-interacting retarded Green’s function can becalculated from (10) with zeroΣph. Once the retarded Green’sfunction is obtained, the less-than Green’s function for thenon-interacting case is calculated from (8). We calculate allthese matrices for two energy levels. One is atE, and theother one atE − hω.

Now, we are ready to compute the photon interaction part.Before we do this, we can simplify (6) and (7) for the casewhen the only dominant process is the optical absorption. Forthis case, the expressions become simply

Σ>rs(E) =

∑

pq

MrpMqsNG>pq(E + hω) (14)

and

Σ<rs(E) =

∑

pq

MrpMqsNG<pq(E − hω) (15)

From these expressions, the self-energy matricesΣ<ph are

computed. And hence, the less-than Green’s function for theinteractingcase, i.e.G<

ph, is simply obtained from (8). In thislast calculation, we approximate the retarted Green’s functionneeded in (8) using the non-interacting case obtained earlier.

Once the interacting less-than Green’s function is obtained,thephotocurrentis calculated using (13). In most cases, how-ever, we are interested in thephotocurrent responsefunction.This photocurrent response is defined as

RI =Iph

qIω

(16)

whereIω is the photon flux at energyhω.It is important to note thatG<

ph, which is used to obtainthe photocurrent, gives the distribution of the photoexcitedelectrons and not the total electron population. The reasonis that, in order to obtain the total electrons, we need tocalculate self-consistentlyG< by iteration until convergenceis achieved. In the steps described above, however, we stop atthe first iteration. To obtain a more accurate total current ortotal electron density, we have to perform at least one moreiteration of the self-energies. This second iteration calculatesthe first-order correction toG<.

III. PHOTOCURRENT USINGNEGF

A. Verification

We first compare our calculation results with that of Hen-rickson’s [12] which uses a terminating barrier on one sideof the resonant tunneling diode. The resonant tunneling diodeis built from GaAs/GaAlAs materials (Fig. 1). Since uniformmaterial properties are used in [12], GaAs properties are usedthroughout the first simulation. We present the result for thenon-uniform material properties in a latter part of this paper.

The grid spacing is set to 0.25 nm. And using this gridspacing, we follow the same device dimension as the case in[12], i.e. barrier width of 2 nm and quantum well width of5 nm. The barrier height is set to 2 eV, which is rather largefor GaAs system. Nevertheless, our purpose is to verify theresults with that in [12], and so we will follow closely thestructure used in that work.

As mentioned in the Introduction, the case considered byHenrickson requires us to add a terminating barrier as shownat the right contact in Fig. 1. The reason for this additionis that once an electron jumps to an excited state, it leavesa hole-carrier behind. This hole-carrier can tunnel outside thebarrier and produce a current. When one calculates the currentat one terminal, the net current will be zero. To overcome this,Henrickson adds a terminating barrier. Hence, the only currentcalculated 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 occupiedby electron, the Fermi level must be set above the ground state

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 0.5 1 1.5 2 2.5

Pho

tocu

rren

t Res

pons

e, R

I (nm

2 /pho

ton)

Photon Energy (eV)

Our SimulationHenrickson’s

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

energy. By setting the Fermi level to 0.17 eV, the ground statebecomes occupied and is ready to be excited by photons.

Now, we are ready to present our calculation result incomparison to Henrickson’s. Fig. 3 shows that our calculationresult produces a reasonably accurate photocurrent for thestructure proposed in [12]. In fact, no significant differencecan be observed for the first peak. The slight differences on theother peaks are due to the difference in the band energy modelused for the Hamiltonian. In our case, we use a simple effectivemass approximation with finite difference discretization.It isknown that single band Hamiltonian, such as the one we usehere, tends to overestimate the eigenenergies [7].

B. Spectral Response under Various Bias

10-5

10-4

10-3

10-2

10-1

0 0.5 1 1.5 2 2.5

Pho

tocu

rren

t Res

pons

e, R

I (nm

2 /pho

ton)

Photon Energy (eV)

0.4

1.1

1.9

Vb = 0.05 VVb = 0.10 VVb = 0.20 V

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

After verifying our results, we now turn to the case when theresonant tunneling diode is biased and no terminating barrieris 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 biasedwith 0.05 V, 0.10 V, and 0.20 V. It can be observed that thepeak locations do not change significantly. The three peaks arelocated around 0.4 eV, 1.1 eV, and 1.9 eV respectively. How dowe obtain these peak locations? Looking at the eigenenergiesinside the well gives us the answer.

The transmission curves for the three bias voltages areshown in Fig. 5. The resonance energies for the case of 0.2V are located around 0.043, 0.470, 1.139, 1.957, and 2.785eV. Taking the differences of the energies between the lowestresonant energy and the others gives us 0.43, 1.1, and 1.9eV. These are the peak locations of the photocurrent responsein Fig. 4. Therefore, the photocurrent peaks are the resultsof electrons transition from the ground state to the excitedstates by absorbing photons. Furthermore, it can be seen thatthe resonant energies are shifted to the left for higher bias.However, the distances between these peak locations do notseem to change significantly, which explains the results in Fig.4.

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

0 0.5 1 1.5 2 2.5

Tra

nsm

issi

on

Energy (eV)

Vb = 0.05 VVb = 0.10 VVb = 0.20 V

Fig. 5. Transmission curves for various bias voltages. Though the peaksare shifted to the left, the distances between the peaks doesnot changesignificantly.

C. Effects of Non-uniform Effective Mass

In the previous results, a uniform effective mass wasused. This follows [12] which uses GaAs effective mass, i.e.0.067m0. It is interesting to study how the spectral responsewould change when we use a non-uniform effective mass.What it means is that we use the actual AlGaAs effectivemass of0.089m0 for the barrier region and the GaAs effectivemass of0.067m0 for the well region. This non-uniformity ofeffective mass can be included in the Hamiltonian and in (5).

Fig. 6 shows the two photocurrent spectral responses forboth uniform and non-uniform effective mass. It can be seenthat the photocurrent response changes imperceptibly. Thepeak locations for non-uniform effective mass remain aboutthe same as that for uniform effective mass. The results arereasonable since there is no large difference between theeffective masses of AlGaAs and GaAs. This justifies the resultsin [12] which uses uniform material parameters throughout itscalculations.

IV. CONCLUSION

In conclusion, our work presents the formulation of op-tical absorption within the Non-equilibrium Green’s Functionframework. The calculation is first verified with the previouslypublished results. We then show the photocurrent spectralresponse when the photodetector is under various bias voltage.

10-5

10-4

10-3

10-2

0 0.5 1 1.5 2 2.5

Pho

tocu

rren

t Res

pons

e, R

I (nm

2 /pho

ton)

Energy (eV)

uniformnon-uniform

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

It is observed that the photocurrent peak locations do notchange significantly under the bias considered in this work.We then studied the effect of non-uniform effective mass. Itis also found that the effect of non-uniform effective mass isnegligible for GaAs/AlGaAs system. The computer code usedin the calculation is available upon request from the author.

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