## Abstract

We quantitatively determined the photocarrier diffusion length in intrinsic Ge nanowires (NWs) using scanning photocurrent microscopy. Specifically, the spatial mapping of one-dimensional decay in the photocurrent along the Ge NWs under the scanning laser beam (λ= 532 nm) was analyzed in a one-dimensional diffusion rate equation to extract the diffusion length of ~4-5 μm. We further attempt to determine the photocarrier lifetime under a finite bias across the Ge NWs, and discuss the role of surface scattering.

© 2011 OSA

## 1. Introduction

Diffusion of photogenerated carriers in semiconductors is a basis to understand the operation of photodiodes and photovoltaic cells, and the length scale of carrier diffusion often determines their performances [1]. Semiconductor nanowires (NWs) can serve as the model platform to test the characteristics of one-dimensional photocarrier dynamics, such as the photogeneration of electron-hole (*e*-*h*) pairs, their drift or diffusion transport, and the photocarrier collection into the output signal [1]. Here we report a direct observation of photocarrier diffusion in intrinsic Ge NWs [2, 3], by a scanning photocurrent microscopy (SPM) technique [3–9], where one-dimensional decay of diffusion current can be visually mapped out at the nanometer scale. We find the diffusion length in intrinsic Ge NWs to be on the order of 4-5 μm at a zero bias across the NWs. We also discuss a method to determine the photocarrier lifetime under a finite bias.

## 2. Experimental results

Intrinsic Ge NWs were grown by a Au catalystic chemical vapor deposition (CVD) method using 10%-diluted GeH_{4} precursors in H_{2} in a quartz tube furnace. The Ge NWs with the mean diameter of 100 nm were transferred onto a 100-nm-thick SiO_{2}/p^{+}-Si substrate, where the *p*-doped Si is used as a gate (G) electrode. Two Ni/Au electrodes as a source (S) and drain (D) were thermally deposited onto a Ge NW using electron beam lithography and liftoff to form a NW field-effect transistor (FET). Figure 1(a)
shows a schematic SPM setup for an intrinsic Ge NW FET consisting of the NW conduction channel, S, D, and G, where a chopped laser (λ= 532 nm) with a beam diameter of approximately 500 nm is scanning over the active area. The chopper frequency was 10 kHz, which was also synchronized with those of a lock-in amplifier (for a photocurrent) and a photodiode (for a reflectance image). When an incident laser beam is injected onto the substrate, a part of the laser beam is absorbed at the beam spot and the other is retro-reflected through a 50:50 beam splitter. The former generates *e*-*h* pairs according to the absorption coefficient of the Ge NW in a given geometry, resulting in a photocurrent (*I*
_{ph}) between the S and D. The latter can be collected by surface reflectance at the photodiode to map the morphology of a given area. Figure 1(b) shows the representative two-terminal dark conductance *G* = *I*/*V*
_{SD} as a function of the gate voltage, *V*
_{G}, and it shows the typical *p*-type semiconductor characteristics, where the conductance decreases (increases) with positively (negatively) increasing *V*
_{G}. The FET hole mobility *μ*
_{h} of the Ge NW was estimated to be 30cm^{2}/Vs at 300 K using a cylinder-on-plate capacitor model [10]. The inset of Fig. 1(b) shows linear *I*-*V*
_{SD} characteristics as a function of *V*
_{G}, indicating the Ge NW channel is ohmic-contacted. This is prerequisite to observe photocarrier diffusion processes in a quantitative manner, otherwise the local band bending at the Schottky contacts may cause unwanted photocarrier motions, such as local drift [5]. Figure 2(a)
shows the SPM images of the *I*
_{ph} at *V*
_{SD} = 35 mV, 15 mV, 0 V, −15 mV, and −35 mV, where the corresponding reflectance images of the electrodes were overlapped to mark the junction positions of the D and S at x_{1} and x_{2} of the NW (dashed line) axial coordinate x. Figure 2(b) shows the 3-dimensional *I*
_{ph} profile at *V*
_{SD} = 0 V with a schematic SPM setup for visual clarity. Upon the photogeneration of electrons (*e*) and holes (*h*) at a point of the NW, they are subject to migrate within the NW in a diffusive (at the zero bias) or drift (at a finite bias) manner. In a steady state, a portion of the photocarriers survives from the *e-h* recombination and are collected at either the S or D, resulting in the corresponding *I*
_{ph} of the either positive or negative sign. Note that the *I*
_{ph} is a negative (positive) value, when holes, which is the majority carrier, are collected at a positively (negatively) biased electrode, as shown in Fig. 2(a) and (b). Figure 2(c) shows the NW-axial profile of the *I*
_{ph} as a function of the coordinate x for various *V*
_{SD}. Note that the *I*
_{ph} exhibits small dips at the x_{1} and x_{2} due to higher reflectance from metallic surfaces. We emphasize here that the significant *I*
_{ph} intensity is detected not only within the NW channel between the electrodes, but also outside the channel, where it exponentially decays over the distance. This observation is clearly marked from the case of the photocarrier drift, where the *I*
_{ph} would be detected and decayed solely within the channel. In general, under the high photocarrier injection condition, an asymmetric nonequilibrium carrier (electrons and holes) motion results in the finite internal field due to the local space charge accumulation, giving arise to local drift. Nevertheless, we note that the ratio of *I*
_{ph}/*I*
_{dark} in our case is only 10%, suggesting that the number of the photogenerated nonequilibrium excess carriers is marginal. Thereby we restrict ourselves upon the quantitative model fit into our observations solely in the one-dimensional diffusion limit.

## 3. Discussion

The one-dimensional diffusion rate equation for the channel length of *L* of the *p*-type (*n*-type) photogenerated carrier density *δn*
_{h(e)} can be expressed as

_{h(e)}is the hole (electron) diffusion constant,

*E*=

*V*

_{SD}/

*L*is the electric field between the S and D, and

*τ*

_{h(e)}is the lifetime of a hole (electron). $\text{g}\text{=}{\text{g}}_{\text{0}}(\text{x})\delta (\text{x}-{\text{x}}_{\text{0}})$is the photogeneration rate at a point x

_{0}, where

*δ*is the Dirac delta function. The equilibrium diffusion photocurrent

*I*

_{ph}can be defined as

*I*

_{S(D)}is the photocurrent collected at the S (D) electrode.

*I*

_{h1(2)}and

*I*

_{e1(2)}indicate the hole and electron currents at the x

_{1(2)}. Notably, the (-) sign at the S in the ohmic Ge NW FET at

*V*

_{SD}= 0 V indicates that holes are diffused into the S electrode as shown in Fig. 2, as aforementioned.

When holes are the majority carriers, Eq. (2) can be approximately expressed as

Figure 3(a)
shows the *I*
_{ph} at *V*
_{SD} = 0, where the circles and solid line indicate the experiment data and the fitting curve by Eq. (3). We summarize the axial *I*
_{ph} profile into three features along the three different regions, as marked in I, II and III: (I) a linear slope within the S-D channel, (II) a single exponential decay in the right side of the x_{2}, and (III) strong damping near the NW ends. First, we find a reasonable fit with a single exponent in the region (II), where the diffusion decay exponentially over x, and therein we extracted *I*
_{h0} = 85±5 nA and ${\lambda}_{\text{h}}\text{=}\sqrt{{\text{D}}_{\text{h}}{\tau}_{h}}=4.5\pm 0.2$ μm. In the region (I) with the channel length *L* (~3 μm) < λ_{h} (= 4 ~5 μm), the *I*
_{ph} can be expressed as with the first order approximation, *i.e.,* the *I*
_{ph} is linear with respect to x within the *L*. The strong suppression of the *I*
_{ph} in the region (III) probably results from the back-scattering of the photocarriers at the NW ends. We further discuss the case, where the minority photocarriers are included, and Eq. (3) can be expanded as,

From the fitting *I*
_{h0} = 95±3 nA, λ_{h} = 4.4±0.5 μm, *I*
_{e0} = 10±2 nA, and λ_{e} = 3.5±2.4 μm are obtained, and the electron concentration in the total *I*
_{ph} is estimated to be approximately 10%. Conclusively, the majority photocarriers in the Ge NW are holes, which is consistent with that of Ge NW FET as shown in Fig. 1(b) and the λ_{h} is about 4 ~5 μm. For a finite bias (*V*
_{SD} ≠ 0 V), the logarithmic-scale photocurrent derivative with respect to x, d(ln(*I*
_{ph}))/dx can be expressed by

_{h}=

*k*Tμ

_{h}/

*e*was employed, where

*k*and

*e*are the Boltzmann constant and the electron charge. From the fitting λ

_{h}/(μ

_{h}

*τ*

_{h}) = 12.5±0.2 mV/μm are obtained as shown in Fig. 3(b), the τ

_{h}and D

_{h}are estimated to be ~100 ns and ~1.6 cm

^{2}/s, respectively. The hole mean free path,

*l*

_{h}can be estimated to be 10 ~30 nm in our intrinsic Ge NWs by D

_{h}~v

_{F}

*l*

_{h}[11], where the Fermi velocity, v

_{F}~10

^{6}m/s. Therefore, the Ge NW with 100 nm in diameter can be considered as a quasi one-dimensional wire as discussed the above.

Earlier reports by surface photovoltage [12] and γ radiation [13] methods measured the carrier life time in bulk Ge crystals to be on the order of 0.1 ms and 1 ms, which are much longer than ~100 ns in our Ge NWs. In parallel, the diffusion length of the hole in bulk Ge is estimated to be 0.13 mm ~40 mm with the diffusion constant of 1.6 cm^{2}/s. Large difference in the diffusion length between the three-dimensional bulk Ge and the one-dimensional Ge NW in our study is presumably due to the significant contribution from intrinsic dimensional confinement effects and surface scattering in Ge NWs. Given that *l*
_{h} is estimated to be 10-30 nm, the surface scattering in the 100-nm-diamter Ge NW may be dominant for the hole relaxation. In fact, it is well known that the Ge surface states have been categorized into fast and slow surface states for their carrier lifetime; the former resides at Ge/GeO_{x} interface giving rise to a lifetime of the order of ps to ns [14] and the latter in the GeO_{x} with the lifetime of the order of 1 s to 100 s [15–19]. The lifetime of ~100 ns in the Ge NW in this study is shorter than that of the bulk Ge slow surface states and longer than that of bulk Ge fast surface states.

## 4. Conclusions

We have investigated the photocarrier motions in an intrinsic Ge NW in the diffusion limit using scanning photocurrent microscopy. Within the framework of the one-dimensional diffusion, the photocarrier diffusion length of intrinsic Ge NWs is estimated to be on the order of 4-5 μm. We also derived the lifetime of the photocarriers under a finite bias across the NW, from which the average carrier drift length per unit electrical field, lifetime-mobility product (μ_{h}
*τ*
_{h}), is estimated to be 300 μm^{2}/V.

## Acknowledgments

Y.-S. Shin and D. Lee are equally contributed for this work. This work was supported by the Basic Science Research Program through the NRF (2010–0017853), the Nano Original & Fundamental Technology R&D Program (2010–0019195), the Priority Research Centers Program through the NRF (2010-0029711), the MEST-AFOSR NBIT (No. K20716000006- 07A0400-00610), the Mid-career Researcher Program (2010-0027627), and the WCU program (R31-2008-000-10059-0).

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