Signal-to-noise ratio of topological insulator based photodetectors Xiao Zhang,1 Jing Wang,2 and Shou-Cheng Zhang3 1Department of Electrical Engineering, Stanford University, CA 94305, USA 2Department of Physics, Tsinghua University, Beijing 100084, China 3Department of Physics, McCullough Building, Stanford University, Stanford, CA 94305-4045 The performance of a photodetector relies on the in- For 3 dimensional photodetecting materials like trinsic properties of the photodetecting material. The Hg Cd Te, η =1 e−αt and R = ρ l are determined 1−x x − wt material has to have a suitable band gap, large absorption by the absorption coefficient α, resistivity ρ = 1 , qni(µh+µe) strength and high resistivity. We use the simplest kind of length l, width w and thickness t of the photodetector photodetector-“photoresistor” as an example to analyze ,whereas ni is the intrinsic carrier density, µh and µe are the performance of a photodetector. A photoresistor is mobilities for electrons and holes, thus usually made of a semiconductor. If light pumping on 2qηPA the device is of energy higher than the band gap, pho- Iph Eph tons absorbed by the semiconductor give bound electrons SNR = = (3) σ 4kB T In l ∆f enough energy to jump into the conduction band. The ρ wt resulting free electron and its hole partner conduct elec- q 1.26 tricity, thereby lowering resistance. So the figure of merit Its maximum signal-to-noise ratio is at t = α , whereas of this system is determined by the signal-to-noise ratio η = 70%. At this point, SNR, the signal is the photocurrent Iph and the major α source of noise is the thermal noise (Johnson noise)whose SNR = C 0.62 √ρα = C 0.62 (4) × × × qni(µh + µe) standard deviation is σIn at finite temperature T when r the voltage bias V kB T , whereas k is the Boltzmann , in which the intrinsic carrier density ≪ q B 2qηPA E constant. Iph = , where a factor of 2 represent kBT 3 3 − g Eph n = 2( ) 2 m 2 e 2kB T (5) an electron-hole pair, q is the electron charge, η is the i 2π~2 3D quantum efficiency, P is the incident light power density, m3D = √memp (6) A = l w (Fig.3(a) of maintext) is the detecting area and 3 3 2 2 2 × mp = (m + m ) 3 (7) Eph is the incident photon energy. On the other hand the hl hh noise current can be measured as its standard deviation . Eg is the band gap and me, mhl and mhh are effective 4kB T σIn = R ∆f, whereas ∆f is the bandwidth of the masses for electron, light and heavy holes. We assume the detector,q inversely proportional to its integrate time and carriers have the same relaxation time τ for transport, finally R is the resistance. Then the signal-to-noise ratio then the mobility of electrons is can be determined by qτ µe = (8) 2qηPA me Iph Eph SNR = = (1) and to calculate the mobility of holes, we have to average σIn 4kB T R ∆f over the concentration of both light and heavy holes q nhl nhh mhl 3 qτ mhh 3 qτ Here only the quantum efficiency η and the resistance R µh = µhl + µhh = ( ) 2 + ( ) 2 (9) are related to the intrinsic properties of the photodetect- ni ni mp mhl mp mhh ing material. η is determined by the absorption strength We know that α is proportional to the 3D joint density and R is determined by the resistivity ρ. The higher ab- of states g3D Eph Eg. Thus sorption and the larger resistivity the material has, the ∝ − Eg better it is to be used as a photodetector. p α 1 2k T In the following, we absorb all material independent S/N (Eph Eg) 2 e B (10) ∝ ni ∝ − quantities of Eq. (1) into r q It has a maximum at Eg = Eph kBT . Now we just need 2qηPA − Eph to determine me, mhl and mhh as a function of Eg (and C = (2) 1–3 4kB T thus Eph). We have for Hg1−xCdxTe , l ∆f w E +0.3eV 5 10−4T q x = g − × (11) to facilitate further analysis. Next, we represent the ma- 1.91 10−3T terial dependent part in SNR as a function of the pho- − Ep = (18+3x)eV (12) ton energy. We assume that the photon energy Eph is 2 Ep −1 much larger than the thermal energy kBT to allow suf- mhl me = m0(1 + ) (13) ficient absorption, especially, we use the non-degenerate ≈ 3 | Eg | assumption Eph =6kBT in semiconductor physics. mhh =0.4m0 (14) 2 −1 , m0 is the free electron mass and α 100cm . Now tectors. So η = β, a dimensionless absorption strength ≈ l despite the material independent parameter C, the only as we calculated and R = ρ w , here ρ is the 2 dimensional variable in SNR is Eph. resistivity. We have the signal-to-noise ratio We do the same analysis for 3D and 2D TI photode- 2qηPA I E 1 SNR = ph = ph = C β√ρ = C β (15) σIn 4kB T × × sqni(µh + µe) R ∆f q Next we need to consider the cases of 3D and 2D TI, and also respectively. First we consider the 3D case, with the two dimen- |E| sional density of states g2D = 2 2πA˜2 +∞ E dE (k T )2π qτ B µ = (23) ni = 2 E/k T = 2 (16) e ˜ 1+ e B ˜ me Z0 2πA2 24A2 qτ qτ µ = µ = (17) µh = (24) e h m∗ mh k T ~2 m∗ B (18) ≈ 2 A˜2 Now we have all the parameters as a function of Eph. Then we consider the 2D∗ case, the two dimensional The SNR is maximum at E = E . Now we have m ph g density of states g2D = 2π~2 , especially all the material dependent parameters again, as a func- tion of Eph. For 2D TI, we use the parameters by fit- 1 1 1 4 ˜ ˚ = ~2 + ~2 (19) ting the experiment and get A2 = 1.98eV A, and m Eg · e 2 ˜ ˜ 2(A˜ +2B˜ M˜ ) 2D2 for 2, 3, 4 and 5QL, M0 = 0.125, 0.07, 0.035, 0.02eV , 2 2 0 2 B˜2 = 4.24, 3.3, 22.6, 61.71eV A˚ and D˜2 = 1 1 1 − 2− · = ~2 ~2 (20) 13.59, 5.76, 3.8, 1.11eV A˚ . m Eg − h 2 ˜ · 2(A˜2 +2B˜2M˜0) 2D2 , we can have Finally we can compute ratio of the SNR for 3D and 2D TI over bulk HgxCd1−xTe photodetectors, which are m = m m (21) SNR(3DT I) SNR(2DT I) 2D √ e h and as a function of E SNR(HgxCd − T e) SNR(HgxCd − T e) kBTm2D − g 1 x 1 x n = e 2kB T (22) E . The results are plotted in the main text. i 2π~2 ph 1 Dornhaus, R. & Nimtz, G. The Properties and Applica- 3 Capper, P. Properties of Narrow Gap Cadmium-Based tions of the Hg1−xCdxTe Alloy System (Springer Tracts in Compounds (London: INSPEC, 1994). Modern Physics 78, 1976). 4 Zhang, Y. et al. Crossover of three-dimensional topological 2 Wiley, J. D. & Dexter, R. N. Helicons and nonresonant insulator of Bi2Se3 to the two-dimensional limit. arXiv: cyclotron absorption in semiconductors. ii. Hg1−xCdxTe. cond-mat/0911.3706 (2009). Phys. Rev. 181, 1181 (1969)..