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Extraction of the Built-in Potential for Organic Solar Cells From Current–Voltage Characteristics Prashanth Kumar Manda , Saranya Ramaswamy, and Soumya Dutta , Member, IEEE
Abstract — The built-in potential (Vbi ) of an organic diode and solar cell is an important parameter that decides the rectification behavior of organic diodes and affects the open circuit voltage and thereby the efficiency of organic solar cells. In this paper, we propose a physics-based model and an experimental method to extract Vbi from current density–voltage (J–V) characteristics. The proposed model is developed by solving the carrier transport and the continuity equations to obtain the analytic equations for charge carrier profile and current density. The proposed method is thoroughly verified using numerical simulation results. Applicability of this method on experimental results is further validated for poly(3-hexylthiophene):phenyl-C61butyric acid methyl ester solar cells. Finally, Vbi is extracted from dark J–V characteristics of fabricated devices. Index Terms — Built-in potential, injection limited current (ILC), organic diode, organic solar cell, Schottky barrier, space charge limited current (SCLC).
I. I NTRODUCTION IGNIFICANT improvement in efficiency of organic solar cell, especially over a decade, has shown a promising direction in third generation photovoltaic research as far as performance per cost ratio is concerned. Recent studies have shown that the efficiency of organic solar cell can go beyond 10% [1], [2]. Considerable research efforts are being carried out both experimentally and theoretically to improve the device performance and to understand the device physics. In organic diodes and solar cells, organic semiconductors are typically sandwiched between two dissimilar metal electrodes [2]–[15], and the work function difference of the electrodes determines Vbi [4]–[6], which is a crucial metric for both the organic diode and solar cells because the nature of current density–voltage(J –V ) (dark and photo) characteristics depends on Vbi . Further, the internal electric field, charge
S
Manuscript received October 5, 2017; revised November 6, 2017; accepted November 9, 2017. Date of publication December 4, 2017; date of current version December 27, 2017. This work was supported in part by the Department of Science and Technology (Govt. of India), in part by Nissan, and in part by the Indian Institute of Technology Madras. The review of this paper was arranged by Editor A. G. Aberle. (Corresponding author: Soumya Dutta.) The authors are with the Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai 600036, India (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TED.2017.2773708
concentration profiles, open circuit voltage VOC , and the efficiency depend on Vbi . In case of solar cells, the maximum achievable VOC is limited by Vbi [16]. This emphasizes the importance of a theoretical and experimental understanding of extraction of Vbi . Typically, the metal–organic semiconductor contacts are Schottky type, and hence the metal contacts inject the charge into the semiconductor by thermionic emission process. The charges that get injected from the contacts cause an excess potential drop near the metal–semiconductor junctions [17]–[19]. This excess drop in potential leads to a reduction of Vbi to a lower value, say Vbi . Therefore, the different measurement techniques that are followed to measure Vbi like dark and photo capacitance measurements can give only Vbi but not Vbi . Hence, the extracted Vbi from these methods found to vary with the thickness of semiconductor [16], temperature [10], [16], and the illumination intensity [9]. Thus, the models based on capacitance measurements as reported earlier [7]–[9], [20] cannot give direct access to Vbi . Alternatively, extraction of Vbi from J –V characteristics is an efficient route incorporating inexpensive characterization setup compared with the other measurement techniques. To this end, Malliaras et al. have proposed a model for Vbi [21], which is based on compensation voltage and photocurrent density–voltage (JPH –V ) characteristics. Similarly, Torto et al. [22] have reported a method to estimate Vbi from JPH –V characteristics. The reported methods are valid only under the exposure of light. Mantri et al. [23] proposed a method for estimating Vbi from dark J –V characteristics. However, the method is empirical and does not take the effect of injected charge into account. This can lead to overestimation of Vbi . On the other hand, Kemerink et al. [19] even though proposed a model, considering the effect of injected charge carriers, the model does not provide a direct method to calculate Vbi from J –V characteristics. Moreover, the estimation of Vbi depends on the voltage that corresponds to the onset of diffusion current, which cannot be defined precisely [24]. Thus, in case of organic diode/solar cells, there are no reports that can give the analytical framework to extract Vbi directly from J –V characteristics without exposing any light. In this paper, we propose a physics-based analytical model for estimating Vbi from dark J –V characteristics, which takes the effect of injected charge into account. Our model is
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Fig. 1. (a) Schematic of equilibrium energy-band diagram of an organic diode, with organic semiconductor of thickness d and bandgap Eg , where LUMO and HOMO being the lowest unoccupied and highest occupied molecular orbitals, respectively. (b) Band diagram for LSC case for different V (TCAD results).
further extended to establish a method for extracting Vbi by using T dependency of Vbi . Finally, the method is employed to estimate Vbi of as fabricated organic solar cells based on poly(3-hexylthiophene) (P3HT):phenyl-C61-butyric acid methyl ester (PCBM), which has been considered as the model system to understand the device physics of organic solar cells.
Fig. 2. (a) Electric field profile of LSC (lines) and HSC (symbols) cases for different V. (b) Electron profile for different V√. J –V characteristics for (c) LSC case (d) (left axis: J –V, right axis: J–V characteristics) HSC case where the symbols are TCAD and solid lines are model. The parameters used for the simulation of LSC case are Eg = 1.3 eV, φ1 = 1.0 eV, Vbi = 0.7 V, μn = μp = 1 × 10−4 cm2 /Vs, NC = NV = 1 × 1019 cm−3 , ε = 3.3ε0 , d = 100 nm, and T = 300 K. HSC case is resembled by changing φ1 to 1.15 eV and keeping all other parameters the same as that of LSC. J–V characteristics for HSC case, where the symbols are TCAD and solid lines are model [see (9)].
II. S IMULATION AND A NALYTICAL R ESULTS We use the metal–insulator–metal methodology for numerical simulation [18], [25]–[28]. The numerical simulations are done using commercially available Sentaurus technology computer-aided design (TCAD) tool [29]. In numerical simulations, we consider Schottky contacts at anode and cathode with barriers φ1 (φ3 ) and φ2 (φ4 ) for electrons (holes), respectively [Fig. 1(a)]. Therefore, the carrier concentrations at the contacts are determined by thermionic emission process and are given as φ1 φ2 , n d = NC exp − n 0 = NC exp − q Vt q Vt φ3 φ4 , pd = NV exp − (1) p0 = NV exp − q Vt q Vt where Vt is the thermal voltage, n 0 ( p0 ) and n d ( pd ) are the electron (hole) concentrations at anode and cathode, respectively, and NC (NV ) is the effective density of states for electrons (holes).
A. Classification of Diodes At equilibrium (applied voltage V = 0 V), the dissimilar metal work-functions are aligned leading to band bending [Fig. 1(a)], which sets up a built-in electric field inside the device. The strength of the built-in electric field depends on Vbi , d, and the injected charge. Depending on the magnitude of the injected charge and its effect on the electric field, the diodes can be classified into two categories: 1) low space charge (LSC) case and 2) high space charge (HSC) case. The injected charge can be modified by changing NC (NV ), barrier for electrons (holes), and the temperature. In this particular study, we keep NC (NV ) unchanged and vary the barriers for
electrons (holes) and the temperature to explain LSC and HSC cases. The parameters associated with the simulation for LSC and HSC cases are given in Fig. 2. 1) Low Space Charge Case: In LSC case, the electric field due to the injected charge is very less compared with the electric field generated due to the work-function difference of metal contacts. Hence, the electric field is expected to be uniform as shown in Fig. 2(a), maintaining linear band bending inside the device as shown in Fig. 1(b). The injected carriers (from metals) undergo diffusion and drift concurrently in opposite direction to each other, as a consequence of concentration gradient and electric field, respectively. Thus, in order to model the carrier profiles and the current density, both drift and diffusion have to be considered simultaneously. The transport equation, describing electron current density, can be expressed as Jn = qnμn E + qμn Vt
∂n ∂x
(2)
where q is the electron charge, E(x) is the electric field, n(x) is the electron carrier concentration, and μn is the mobility of electron. As discussed above, E(x) is uniform and it is represented as E(x) =
− (Vbi − V ) . d
(3)
In order to arrive at analytic solution under steady-state conditions, we consider three assumptions: 1) semiconductor is intrinsic; 2) carrier mobilities (μn and hole mobility μ p ) are constant with respect to V and T ; and 3) there is no carrier generation and recombination. The last assumption modifies
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the continuity equation for electrons as ∂ Jn (x) = 0. ∂x
(4)
Using Eqs. (2)–(4), a second-order differential equation is developed for n(x) as ∂ 2 n(x) E(x) ∂n(x) = 0. + ∂x2 Vt ∂x
(5)
By employing the thermionic emission boundary condition for electrons [see (1)], an analytic solution for n(x) is obtained as Vbi−V x n d −n 0 exp VbiV−V +(n −n ) exp 0 d Vt d t n(x) = . (6) Vbi−V 1−exp Vt A similar approach can be used to obtain an analytic solution for holes. The extracted electron profiles inside the device using TCAD simulation (symbols) and (6) (solid lines) for different V are compared in Fig. 2(b) which ensures that (6) is in good agreement with the TCAD results. Further, the analytic solution for current density can be expressed using (2), (3), (6) and their hole counterparts as q(μn n 0 + μ p pd )(Vbi − V ) exp VVt − 1 . (7) J= d 1 − exp − VbiV−V t The variation of J with respect to V , based on TCAD simulation (symbols) and (7) (solid line), is displayed in Fig. 2(c), showing excellent consistency. A similar equation has been reported by different groups in the literature [5], [21], [25], [27], [30]. Jung et al. [5] arrived at a similar analytical equation for less disordered organic materials with Gaussian density of states. However, the present method is completely rest upon charge-based model with coherent device physics considering the effective density of states. According to Fig. 2(b), charge increases exponentially from anode to cathode for V < Vbi (0.7 V in particular). The exponential nature of charge along with linear variation of E(x) with V results in exponential variation of J with respect to V . On the other hand, for V > Vbi , the charge carrier profile changes significantly [Fig. 2(b)] by virtue of field reversal [Fig. 2(a)]. Charge carrier concentration increases from anode to cathode like a logistic function [Fig. 2(b)], maintaining its spatially uniform nature inside the device except near the anode–semiconductor junction. The uniform nature of both charge carrier concentration and electric field profiles leads to a linear variation of current. In LSC case, the current is typically injection limited current (ILC) since dominant part of the current is controlled by the injected charge carriers. 2) High Space Charge Case: In HSC case, the electric field due to the injected charge becomes comparable with the electric field associated with band bending, expressed by (3). Hence, the net electric field within the device becomes nonuniform. HSC case can be realized by reducing the barrier height for electrons (holes) or by increasing NC (NV ) or by increasing the thickness of the semiconductor. However, in this paper,
HSC is realized by reducing the barrier height at anode– semiconductor junction in particular. Under equilibrium, a uniform electric field is observed within the device except near the anode–semiconductor junction where injected charge is high [Fig. 2(a)]. However, for V < Vbi , the magnitude of uniform electric field is slightly less than that of LSC case. Thus, J –V characteristics maintain the same exponential nature as that of LSC case, exhibiting a reduction in builtin potential [Fig. 2(d)]. Hence, it is essential to modify the electric field in case of HSC by reducing Vbi to Vbi − φ (i.e., Vbi ) where φ accounts for the reduction in E(x) due to the injected charge. However, in case of V > Vbi , the nonlinearity in electric field profile near the anode–semiconductor junction becomes predominant upon applying voltage and spreads throughout the device differing drastically from that of LSC case. As a consequence, the electric field and the carrier concentration become interdependent, leading to nonlinear J –V characteristics. The current density varies with square of √ V (for V > Vbi ) as evidenced by a linear nature of J –V characteristics [Fig. 2(d)]. As the current is controlled by the space charge, it is space charge limited current (SCLC). For V < Vbi , in the uniform electric field region, the electric field strength can be modeled as − Vbi − V . (8) E(x) = d Using (8), (7) can be modified as q(μn n 0 + μ p pd ) Vbi − V exp VVt − 1 J= . (9) V −V d 1 − exp − biVt J –V characteristic using (9) shows a good agreement with TCAD results under V < Vbi for φ = 0.0544 V [Fig. 2(d)], where φ is obtained by fitting TCAD results with (9).
B. Extraction of Built-in Potential In LSC case, J changes its nature from exponential to linear for V > Vbi , whereas in HSC case, J changes its nature from ILC to SCLC for V > Vbi . Most of the practical organic diodes belong to HSC case. To understand more about current transition from exponential to linear or ILC to SCLC, we adopt a function G, proposed in [23], where G is defined as ∂ ln(J ) . (10) ∂ ln(V ) The variation of G with respect to V shows three distinct regions signifying three different natures of current [Fig. 3(a)]. In region-1, the variation of G can be fitted using a simple exponential function of V as [exp(V /Vt ) − 1], as represented by the solid line. In region-3, G follows a power law as (Vγ − V )m with an exponent m, where Vγ and m are fitting parameters. Transition between these two regions (1 and 3) occurs through region-2, showing a combined effect of exponential and power law. Moreover, in region-2, G–V characteristics exhibit a peak at a voltage, termed as Vα . TCAD simulation results for the variation of Vα with respect to temperature as a function of different Vbi are depicted in Fig. 3(c). It emphasizes that modeling Vα variation with Vbi G=
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Fig. 4. Variation of extracted Vbi and φ3 with respect to φ1 , for different φ2 [comparison between TCAD results (symbols) and model (solid lines)].
Fig. 3. (a) Schematic of G–V characteristics. (b) G–V characteristics for LSC and HSC cases, where the symbols are TCAD and solid lines are model [see (11)]. (c) Vα variation with temperature for different Vbi variation with respect to T for different V with (TCAD results). (d) Vbi bi φ2 = 0.45 eV, where the symbols are TCAD and solid lines are model [see (14)].
will help in determining Vbi from J –V characteristics. Using (9) and (10), a unified expression for G is developed as G=
V V + . Vbi −V V − Vbi Vt 1 − exp − Vt
(11)
Equation (11) consists of two terms that correspond to two different natures of current. The first term shows exponential nature, whereas the second term represents a power law with Vγ = Vbi and m = 1. Equation (11) shows an excellent match with the TCAD results for LSC case, where Vbi = Vbi [Fig. 3(b)]. In case of HSC, there is indeed a good agreement between (11) and TCAD results throughout regions 1 and 2. However, G–V characteristics deviate from (11) in region-3 due to the presence of SCLC, which cannot be captured by the present model. Thus, (11) is in good agreement with the TCAD results for both LSC and HSC cases throughout region-1 and region-2. Hence, (11) can be used for extracting Vα for both LSC and HSC cases. Using (11) and equating its first derivative to zero at V = Vα , we obtain
2 Vbi − Vα Vα Vbi Vt − Vbi − Vα exp − 1+ = 0. (12) Vt Vt Equation (12) can be solved numerically to get Vbi . For higher values of Vbi , the second term inside the square brackets of (12) can be neglected. Therefore, a compact analytical equation is realized for Vα as 1/2 Vα = Vbi − Vbi Vt . (13) The possible solutions for Vbi are given as
4Vα . Vbi = Vα + 0.5Vt 1 ± 1 + Vt
(14)
From Figs. 2(d) and 3(b) it is evident that Vbi is greater than Vα . However, the smaller root for Vbi is always less than Vα , which is not physical. Hence, the larger root is taken as the solution for Vbi . Using (14) and the value of Vα (extracted from G–V characteristics), Vbi can be calculated. It is evident from Fig. 3(d) that Vbi increases with the decrease in T and saturates to Vbi . The variation of Vbi with respect to T arises due to φ, which in turn depends on dominant injected charge near metal–semiconductor junction [ p0 = NV exp(−φ3 /(kT ))] and thereby on T . Upon decreasing temperature, the amount of injected charge decreases, resulting that φ tends to be zero and hence Vbi approaches to Vbi . φ can be calculated from the relation φ = Vbi − Vbi for different T and Vbi . From the variation of φ with respect to T , a semiempirical model is developed for φ (Appendix) as ⎡ ⎤ 2 q ε NV exp − φ3 kT 2kT ⎢ ⎥ (15) ln ⎣ + 1⎦ φ= q 2kT r 2 C g2 where C g = ε/d, ε is the dielectric constant, and r is a fitting parameter being independent of T . Using (15), Vbi can be written as ⎡ ⎤ 2 q ε NV exp − φ3 kT 2kT ⎢ ⎥ Vbi = Vbi − ln ⎣ + 1⎦ . (16) 2 2 q 2kT r C g Vbi and φ3 can be obtained by solving (16) self-consistently with T dependent variation of Vbi . The extracted parameters are in good agreement with TCAD results and it is validated for different combinations of φ1 and φ2 , which shows the robustness of our model [Fig. 4]. III. E FFECT OF R ECOMBINATION In this section, we discuss the applicability of the proposed extraction method for Vbi in the presence of trap-assisted [Shockley–Read–Hall (SRH)] recombination. The recombination rate (R) due to SRH recombination is defined as R=
np − n 2i τ p n + n i exp qδVEt + τn p + n i exp − qδVEt
(17)
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TABLE I E XTRACTED PARAMETERS FOR P3HT:PCBM D IODE W ITH D IFFERENT T HICKNESSES
Fig. 5. TCAD simulation results of (a) J–V and (b) G–V characteristics variation with temperature for different at T = 300K. (c) Vα and (d) Vbi carrier lifetimes with Vbi = 0.7 V and φ2 = 0.4 eV.
where τn , τ p are the lifetimes of carriers, and δ E is trap depth. The carrier lifetimes τn and τ p are related to carrier capture coefficient (for electrons Cn , for holes C p ) and the total trap density Nt as τn/ p =
1 Cn/ p Nt
.
(18)
The simulations are done with Cn = C p and by considering the traps to be located in the middle of the bandgap (δ E = 0 eV). The obtained results are compared with ideal device (without recombination). It is noted that the effect of SRH recombination is more prominent in reverse bias and low forward bias regime as shown in Fig. 5, where the current due to SRH recombination is termed as leakage current. The leakage current increases with decrease in τ [see (17)], which is reflected in J –V characteristics, as shown in Fig. 5(a). The parameter G is calculated using (10) and is plotted against V , as shown in Fig. 5(b). A minimal change in Vα with τ is noted at T = 300 K. However, with increase in T , R also increases, resulting in deviation of Vα compared with the ideal device as shown in Fig. 5(c). Correspondingly, Vbi also varies with τ [Fig. 5(d)]. Nevertheless, it is to be noted that for τ > 10−6 s, the extracted Vbi is in close match with ideal device characteristics. According to the reported values of Cn and Nt , the calculated value of τ for organic device is more than 10−6 s [31]–[33]. Hence, the proposed method can be applied to the experimental results, obtained from the organic solar cells/diodes under consideration. IV. E XPERIMENTAL R ESULTS In order to validate our model, Vbi is extracted from experimental results of the organic solar cell, fabricated in our laboratory. Organic solar cells consisting of P3HT:PCBM as active material with aluminum as cathode and indium tin oxide/poly(3,4-ethylenedioxythiophene):polystyrene sulfonate as anode were fabricated inside a nitrogen glove box.
Since P3HT:PCBM-based devices are sensitive to atmosphere with fast degradation [34] and can lead to error in analysis, the devices were encapsulated in nitrogen glove box prior to characterize in a vacuum probe station. The devices were tested repeatedly almost after each experiment to confirm the stability and reproducibility. In order to validate the versatility of our model, we used three different thicknesses, 173 nm (Device A), 154 nm (Device B), and 106 nm (Device C) of P3HT:PCBM, resulting from the spin speed of 850, 1000, and 1500 r/min, respectively. Dark J –V characteristics of Device A as a function of temperature are plotted in semilogarithmic scale and linear scale, as shown in Fig. 6(a). It is observed that the forward current increases with increase in T as expected.
A. Model Validation for Experimental Results To check whether the experimental results follow the proposed model, we have to analyze the ln(J ) variation with 1/Vt for different V . For 0 < V < Vbi , by considering μn n 0 > μ p pd and using (1) and (9), ln(J ) can be written as qμn NC Vbi − V S + (19) ln(J ) = ln d Vt where S=
V − φ1 . η
(20)
According to the proposed model, ln(J ) varies linearly with 1/Vt with a slope S being dependent on V . According to (20), S varies linearly with V having a slope (1/η) equal to one and the intercept gives the value of one of the barrier potential (φ1 ). The experimental variation of ln(J ) for Device A (symbols) is shown in Fig. 6(b) and it confirms the linear variation of ln(J ) with 1/Vt for different applied voltages. Hence, the experimental results are fitted with linear variation to find S and the intercept and this study is extended for Devices B and C. The variation of S with V is shown in Fig. 6(d). One can note from Fig. 6(d) that for 0.54 < V < 0.65, S varies linearly with V . Moreover, the linear fit in that particular regime gives a slope that is nearly equal to one (Table I), which is in consistent with the proposed model. Hence, this confirms the applicability of the proposed model on these experimental results. In addition, we extract one of the barrier potentials φ1 (or φ4 ) for different devices, which is nearly equal to 0.8 eV as tabulated in Table I. Using (10), we obtain G–V plot for Device A, as shown in Fig. 6(c). Vα is extracted from the peak position of G–V plot for different temperatures and shown
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Fig. 6. Experimental results of Device A. (a) dark J–V characteristics. (b) ln (J) variation with 1/Vt for different V, where symbols are experimental and solid lines are linear fit to the experimental data. (c) G–V variation with T. (d) Variation of S with V, where symbols are experimental and solid lines are linear fit to the experimental data, (e) Experimental variation of Vα with T. (f)–(h) Experimental variation of Vbi with T for devices with different P3HT:PCBM thicknesses, where symbols are experimental and the solid lines are model.
in Fig. 6(c) for different devices. Subsequently, Vbi , calculated using (12) for Devices A–C, are shown in Fig. 6(f)–(h) respectively, with symbols. As explained earlier, by solving (16) self-consistently, Vbi and φ2/3 are extracted and presented in Table I. It is important to note that the extracted values of Vbi are almost the same for different thicknesses. Hence, Vbi obtained using the present model is independent of thickness, as expected. Moreover, the extracted values of Vbi are in consistent with the reported values for P3HT:PCBM device [16], which validates our model and ensures the method of extracting Vbi from J –V characteristics of organic diode or solar cell.
with
x0 =
2εkT . q 2 p0
(22)
The above equation is derived for a semi-infinite semiconductor. As our study involves semiconductor with finite thickness, the potential drop (φ) is taken between metal– semiconductor interface and a finite position (x = d/r ) inside the semiconductor, up to which the effect of injected charge is predominant. Hence, φ can be given as d 2kT ln +1 (23) φ= q r x0 where r is a fitting parameter (r >1).
V. C ONCLUSION In summary, the effect of injected charge on the internal electric field and the nature of J –V characteristics is discussed. We developed analytic models for injected charge profile, J –V characteristics, and Vbi . Vbi is estimated using temperature-dependent variation of Vbi . The estimated values of Vbi are in good agreement with TCAD results. The value of Vbi extracted for P3HT:PCBM-based solar cells using the proposed method is in close match with the reported value in the literature. Moreover, the extracted value of Vbi is independent of the thickness of P3HT:PCBM showing the reliability of the present model. A PPENDIX F ORMULATION OF B AND B ENDING PARAMETER According to [35], the potential profile inside a semiconductor with an hole injecting contact (placed at x = 0) can be written as 2kT ψ(x) = ln(x + x 0 ) (21) q
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Prashanth Kumar Manda received the B.Tech. degree in electronics and communication engineering from Jawaharlal Nehru Technological University, Hyderabad, India, in 2011. He is currently pursuing the M.S. and Ph.D. degrees with the Department of Electrical Engineering, IIT Madras, Chennai, India. His current research interests include the fabrication and modeling of organic solar cells and MIS capacitors.
Saranya Ramaswamy received the B.E. degree in electronics and communication engineering and the M.Tech. degree in nanotechnology from Anna University, Chennai, India, in 2012 and 2014, respectively. From 2014 to 2017, she was a Research Assistant with the Department of Electrical Engineering, IIT Madras, Chennai. Currently, she is a Research Assistant with the Indian Institute of Science, Bengaluru, India.
Soumya Dutta (M’14) received the Ph.D. degree from the Jawaharlal Nehru Centre for Advanced Scientific Research, Bengaluru, India, in 2006. He is currently an Assistant Professor with the Department of Electrical Engineering, IIT Madras, Chennai, India. His current research interests include organic and perovskite semiconductors-based solar cells, thin-film transistors, photo-transistor, polymer-based SAW devices, and graphene-based NEMS devices.