Modelling of Realistic Intermediate band solar cells : Materials Focus J. M. Rorison, C. Broderick, Q Wang, W. Xiong Department of Electrical and Electronic Engineering, University of Bristol, Merchant Ventures Building, Woodland Road, BS8 1UB, Bristol, UK

Outline Introduction to the Intermediate Band Solar Cell (IBSC) Ideal IBSC model: approximations and efficiency results Possible candidates: Quantum wells and dots, Impurity Bands and Highly Mismatched Alloys Examination of Approximations: Controlling of IB Position: material choices, alloying and tuning carrier losses Possible QD alloys Future Work

Introduction to IBSC IBSC [1] [2] Left graph, solar spectrum in energy domain. Right graph, optimise Efficiency by compromising between BG and J. (small bandgap → small output voltage → large possible current density → optimised power in certain BG) P-i-N junction, e.g. GaAs p-i-n junction (only between VB and CB, photon energy larger than BG 1.43eV as BLUE) Inserting IB to extend the photon absorption with energy less than BG as GREEN and RED. P-i-N Solar Cells IBSC [1] G.F. Brown and J. Wu, “Third generation photovoltaics,” Laser & Photonics, Review vol. 3, Jul. 2009, pp. 394-405. [2]. Luque A, Martí A. Advanced materials (Deerfield Beach, Fla.). 2010;22(2):160-74.

Ideal IBSC model Ideal Conditions: [1] VB, CB and IB only linked optically -No carrier can be extracted from the IB Detailed Balance Approach: Balance population VB↔IB and IB↔CB** No carrier loss in any level -Non-radiative transitions → forbidden. non-overlapping absorption Carrier mobility → high The cell is thick enough → full absorption. IB have no broadening (IC1) only absorption & radiative recombination terms (IC2) Once carrier goes into CB, it would be swept out (IC3) absorption coefficient have their bandwidths for transitions between different bands (IC4) the number of carriers in IB should keep un-change in steady state (IC5) Full absorption, means one photon one electron (IC6) single level IB, not a narrow band [1] A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Physical Review Letters, vol. 78, no. 26, pp. 5014–5017, 1997 [1] A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Physical Review Letters, vol. 78, no. 26, pp. 5014–5017, 1997.

Ideal IBSC model Based on Photon Flux Density function: No carrier change in IB level: Electrons can only extract from CB: : [1] [1] A. Luque and A. Marti, “Increasing the efficiency of ideal solar cells by photon induced transitions at intermediate levels,” Physical Review Letters, vol. 78, no. 26, pp. 5014–5017, 1997.

Ideal IBSC Efficiency results: Optimised Efficiency(Red) for ideal IB position (Green) for Different BG (Blue) No Non-radiation 1.95eV 63.24% 1.24eV 0.71eV Conduction Band BG IB E2 E1 IB BG E2 E1 Bandgap (BG) IB Valence Band The peak efficiencies are same efficiency at the same bandgap which is 1.95eV. The IB position changed, but two sub-bandgaps (0.71eV and 1.24eV) were still keeping the same. Therefore, according to the case which considers the radiative recombination only to calculate the efficiency, can conclude that the max efficiency 63.24% occur at bandgap 1.95eV and the IB position can be 0.71eV difference to either VB or CB. Intrinsic region

Examination of Non-Ideal Position of IB The efficiency is reduced substantially if the position of the IB is not ideal More important as the bandgap increases VERY IMPORTANT +5% of Eg ideal -5% of Eg

Quantum Well and Quantum Dot Criteria: Intermediate Band can only be coupled optically to CB and VB → Quantum Wells (QWs) with 1D confinement and 2D continuous good → Quantum Dots (QDs) are ideal candidates as 3D confinement. In(Ga)As/GaAs (initially studied) [1] Valence Band Conduction Band - + ħω Intermediate Band FI Different material → different bandgap → different IB position . We only want one IB. so Chosen materials, no QD levels associate to valance band. Need high density of uniform QD to give highest DOS. We will exam the broadening due to the inhomogeneity (due to non-ideal), homogeneous (due to temperature effect (gives e-e e-p scattering)) [1] A. Luque, A. Martí, and C. Stanley, “Understanding intermediate-band solar cells,” Nature Photonics, vol. 6, no. 3, pp. 146–152, Feb. 2012.

Quantum Dot Systems Investigated so far So far efficiencies <20% Questions: is it position of IB? separation of IB from CB or VB? carrier losses (from IB)? not enough states in IB/ not good absorption other problems- carrier dynamics? Popescu and Zunger Phys Rev B

Impurity Band Criteria: Intermediate Band can only be coupled optically to CB and VB → high density of impurity energy levels are ideal candidate. GaN: Mn3+ [1] Different material → different bandgap → different IB position . We only want one IB. so Chosen materials, no QD levels associate to valance band. Need high density of uniform QD to give highest DOS. We will exam the broadening due to the inhomogeneity (due to non-ideal), homogeneous (due to temperature effect (gives e-e e-p scattering)) Optical absorption of GaN:Mn, GaN:Mn:Si, and AlN:Mn with Mn ~ 1020 cm-3. A, the onset ~1.8 eV in GaN is emission of holes from Mn3+ acceptors to the VB. B ~1.5 eV to the internal spin-allowed 5E!5T transition of the deep neutral Mn3+ state. [1] T. Graf, M. Gjukic, M. S. Brandt, M. Stutzmann, and O. Ambacher, “The Mn[sup 3+/2+] acceptor level in group III nitrides,” Appl. Phys. Lett., vol. 81, no. 27, pp. 5159–5161, 2002.

Highly mis-matched Alloys Ga(In)As:N / GaAsP:N Ga(In)N: Bi The conduction band density of states is split between E+ and E- So have transitions from VB and IB->CB -adv: have good density of states -> Good absorption

N-> 2 fold degenerate at EN means (8+2)=10 -band Hamiltonian

Quantum Well Band lineups for III-V alloys and HMAs We present three material systems here: GaAs/Al0.2 Ga0.8As, QW width 8nm GaNAs/AlGaAs , with 4.7% Nitrogen QW width 8nm GaBiAs/AlGaAs, with 8% Bismuth QW width 8nm The band lineups left show the band offsets for the lowest conduction band (CB), light hole and heavy hole (LH and HH) bands. The zero energy is taken at the valence band maximum of the unstrained host matrix.

Band structure derived using 8 or 10 band k·p model We plot conduction and valence sub bands energy E as a function of kII for three different quantum well material systems. Zero energy is taken at the valence band maximum. In system B, the GaNAs QW is under tensile strain, so there are no HH-like bound states exist. In system C, the GaBiAs QW is under compressive strain (much more HH-like states).

The optical absorption for each of the QW material systems plotted as a function of photon wavelength (for a QW width). We calculate the absorption for sheet carrier densities of 0.01, 1, 2, 3, 4, 5*10^12 cm-2. These results are calculated by two kinds of models, one is set up for binary miscible alloy and the other one is for non-miscible alloy material system. Figure on the left shows the TE (solid lines) polarized absorption v.s. photon wavelength for the GaAs QW, and figure on the right shows the absorption trends against the wavelength on TM polarized (dashed lines)

Figure B shows the TM absorption (dashed lines) for the GaNAs QW, figure C shows the TE absorption (solid lines) for the GaBiAs QW. As Nitrogen or Bismuth replace a small amount of the group V in a III-V compound, the energy gap is reduced and the absorption is depended on the bandgap.

Carrier Loss in IBSC Loss (L) in model Included in ideal model (recycling the photons) Important for low carrier density, Not for concentrator SC Important for high carrier density We lump Defect and Auger together

Carrier Loss in IBSC model Carrier loss from the CB or VB reduces the overall carriers collected and the efficiency proportionally Carrier loss from the IB upsets the detailed balance so is different Intrinsic region Conduction Band Valence Band Eg IB E2 E1 LCB LIB LVB

Ideal IB with carrier losses included E1, IB position (eV) losses Case A Intrinsic region C B V B Eg IB E2 E1 Case B Increased losses from IB reduce efficiency-strongest effect for large Eg Optimised IB position lowering leads to better efficiency Case B suffers less than Case A and has larger shift in optimised IB position

Why? Integration range in blackbody radiation function αCI αIV αCV Case A αCI αIV αCV E2 E1 Eg Case B αIV αCI αCV E1 E2 Eg

Case A: It is similar to practical systems considered . InAs QD GaAs-based QD-IBSC: Eg =1.43eV, losses are not so important. Mn-dopped GaN system: Eg = 3.42eV, losses are important. System 1: GaAs Eg = 1.43eV Ideal model: E1 = 0.95eV E2 = 0.48eV efficiency = 59.91% Real: E1=1.213 eV E2=0.212 eV Efficiency=45.6% Intrinsic region C B V B Eg IB E2 E1 System 2: GaN Eg = 3.42eV Ideal model: E1 = 1.99eV E2 = 0.43eV efficiency = 49.57% Mn-doped GaN: E1 = 1.80eV E2 = 0.62eV efficiency = 21.36%

GaN system with losses included: with & without IB position shift GaN with band gap 3.42eV Dashed lines-loss model with ideal IB Solid line-loss model with IB moving Ideal IB position with no-losses is 1.99eV above VB, gives 49.57% efficiency. (Solid black line) When Losses added efficiency drops. When IB loss=70%, Original IB position (1.99eV) gives 20.39% efficiency; (red dashed line) New IB position (1.8eV) gives 23.41% efficiency.(solid green line) This New IB 1.8eV matches the Mn level in GaN system. IB=1.8eV IB=1.99eV

Potential of Alloying: Mn doped in In1-XGaXN Unstrained InGaN

Introduction of strain on energy bands Band offset ratio: ∆ 𝑬 𝑪 :∆ 𝑬 𝑽 =𝟔𝟎:𝟒𝟎 𝜺≡ 𝒂 𝒏𝒂𝒕𝒊𝒗𝒆 − 𝒂 𝒔𝒖𝒃 𝒂 𝒏𝒂𝒕𝒊𝒗𝒆 Mn Level is not affected by strain 𝐻=(−𝑎)∙2 𝐶 11 − 𝐶 12 𝐶 11 𝜀 𝑆=(−𝑏)∙ 𝐶 11 + 2𝐶 12 𝐶 11 𝜀 𝛿= 1 2 ∆ 1−2 𝑆 ∆ +9 𝑆 ∆ 2 1 2 −(1− 𝑆 ∆ ) ≈ 2 𝑆 2 ∆ Material Lattice Constant   Deformation Potentials (eV) Elastic Moduli 𝟏𝟎 𝟏𝟏 𝐝𝐲𝐧 𝐜𝐦 𝟐 Spin-obit split-off energy a (Å) 𝐚 𝐜 𝐚 𝐯 b 𝐂 𝟏𝟏 𝐂 𝟏𝟐 ∆ (eV) InN 4.98 1.85 1.5 1.2 187 125 0.003 GaN 4.5 2.2 5.2 293 159 0.02 ∆ 𝐸 𝐶𝐵 = 𝐻 𝐶 ∆ 𝐸 𝐻𝐻 = 𝐻 𝑉 −𝑆 ∆ 𝐸 𝐿𝐻 = 𝐻 𝑉 +𝑆−𝛿 ∆ 𝐸 𝑆𝑂 = 𝐻 𝑉 +𝑆+𝛿

1 3 2 Unstrained InGaN (1) Strained InGaN / GaN (2) Compressive: 0 ~ 9.64% Strained InGaN / InN (3) Tensile: 0 ~ -10.67% 3 2

Summary so far IBSC offer good efficiency only if IB is at ideal position IBSC possibilities: QWs, QDs, Impurity Band, HMAs Tuning possible with host matrix composition, strain, and carrier losses need band structures, density of states need to introduce realistic absorption band profiles and allow overlap NOW FOCUS ON QDOTs

Realistic QD-IB Solar Cell model Size and shape fluctuation → inhomogeneous broadening (Temperature Independent) The QD density of state (DOS) can be modelled by Gaussian distribution. There is also fundamental linewidth broadening → homogeneous broadening (Temperature Dependent) This is modelled by Lorentzian distribution.

Methodology: Replace single IB level → discretized DoS of broadened IB band; Include broadening (inhomogeneous and homogeneous) to weight the contribution of the different-IB Solar Cell cases. Calculate out the efficiencies of QD-IBSC with different broadening.

Explanation of homogeneous broadening Same Inhomogeneous Broadening High Temp Low Temp To be absorbed, “narrow linewidth photon” must overlap with QD distribution: At low T, the QD linewidth is narrow, so photon overlap with only 1 QD group. At High T, the QD linewidth is wide, so photon overlap with a number of QD groups. E E At High T, the QD linewidth is wide, photon overlap with a number of QD groups. To be absorbed, ‘narrow linewidth photon’ must overlap with QD distribution. At low T, the QD linewidth is narrow, so photon overlap with only 1 QD group.

Thank you !