Direct observation of hole carrier density profiles and its light induced manipulation at the surface of Ge

We demonstrate that, by using low-energy positive muon ($\mu^+$) spin spectroscopy as a local probe technique, the profiles of free charge carriers can be directly determined in the accumulation/depletion surface regions of p- or n-type Ge wafers. The detection of free holes is accomplished by measuring the effect of the interaction of the free carriers with the $\mu^+$ probe spin on the observable muon spin polarization. By tuning the energy of the low-energy $\mu^+$ between 1 keV and 20 keV the near-surface region between 10 nm and 160 nm is probed. We find hole carrier depletion and electron accumulation in all samples with doping concentrations up to the $10^{17}$/cm$^3$ range, which is opposite to the properties of cleaved Ge surfaces. By illumination with light the hole carrier density in the depletion zone can be manipulated in a controlled way. Depending on the used light wavelength $\lambda$ this change can be persistent ($\lambda = 405, 457$ nm) or non-persistent ($\lambda = 635$ nm) at temperatures $<270$ K. This difference is attributed to the different kinetic energies of the photo-electrons. Photo-electrons generated by red light do not have sufficient energy to overcome a potential barrier at the surface to be trapped in empty surface acceptor states. Compared to standard macroscopic transport measurements our contact-less local probe technique offers the possibility of measuring carrier depth profiles and manipulation directly. This new approach may provide important microscopic information on a nanometer scale in semiconductor device studies.


I. INTRODUCTION
The characterization of semiconductor materials and devices is key for understanding and developing semiconductor technologies. Specifically, the changes and the controlled manipulation of charge carrier concentrations at semiconductor interfaces are of fundamental importance for their functionality in devices. With the tremendous growth of the field of semiconductor device physics and the advancement of experimental characterization techniques over the past decades an enormous progress has been achieved, providing new insights and improvements of semiconductor devices [1][2][3].
Usually, a combination of macroscopic transport measurements and simple modeling are used to determine charge carrier depth profiles and electric field gradients across semiconductor interfaces [1,2]. Carrier densities can be determined by Hall effect measurements, without depth resolution. The density of dopants can be determined by optical techniques (infrared spectroscopy, photoluminescence, plasma resonance, and free carrier absorption) where depth resolution is also limited. Capacitance-voltage measurements are usually used to gain information about free carrier and impurity profiles in a non-destructive way . These measurements require the manufacture of a Schottky contact, which is not always easy to build. The depth resolution of this technique is limited by the zero-bias space-charge region at the surface, by voltage breakdown at larger depths, and by the Debye limit [2]. * thomas.prokscha@psi.ch A local probe technique, capable of detecting the variation of carrier densities as a function of depth, offers the unique possibility of measuring carrier profiles and their manipulation directly. Here, we employ a beam of polarized low-energy positive muons (LE-µ + ) with tuneable energies between 1 and 20 keV, and implant the µ + at variable mean depths between 10 nm and 120 nm in commercial Ge wafers. In semiconductors and insulators the µ + stops at an intersitial site, where it can capture zero, one or two electrons to form the hydrogenlike muonium states Mu + , Mu 0 , or Mu − [4,5]. The interaction of free charge carriers with these muonium states may cause a detectable change of muon spin polarization, which can be observed by measuring the time evolution of the µ + polarization in a muon spin rotation (µSR) experiment [4][5][6]. In this way, the stopped µ + may act as "sensor" for free charge carriers. The technique of using LE-µ + in depth selective low-energy µSR (LE-µSR) has been recently applied to measure the persistent photo-induced inversion of a Ge surface layer from n-to p-type [7], and the effect of band bending on the activation energy of shallow muonium states close to the surface of commercial CdS and ZnO wafers [8]. These experiments demonstrate the capability of LE-µSR to determine quantitatively i), the charge carrier concentrations in a near-interface region in certain cases (which was in Ref. [7] a photo-induced hole carrier concentration of ∼ 1.5×10 14 cm −3 ) and ii), the electric field profile due to band bending at a semiconductor surface/interface [8].
In this paper we significantly advance the methodology to perform depths scans of carrier concentrations of n-and p-type Ge wafers. For the first time we directly explore by means of LE-µSR the hole depletion region width at the surface of p-type Ge, and we demonstrate the manipulation of the hole carrier concentration p in the depletion region by using illumination with a blue LED light source or laser (wavelengths λ = 405, 457 nm), or a red laser (λ = 635 nm). Whereas after illumination with blue light a persistent increase of hole carrier density p 10 14 cm −3 is observed in the depletion zone, i.e. filling of the depletion region, illumination with red light induces a dynamic charge carrier equilibrium with p ∼ 1 × 10 12 cm −3 at a depth of 20 nm, increasing to ∼ 4×10 12 cm −3 at a depth of 120 nm in a p-type Ge wafer with nominal p ∼ 10 15 cm −3 . At the same time, electron accumulation is found in the surface region without illumination at depths of order 100 nm. Our new method has the potential to provide new insights in charge carrier transport phenomena on a nanometer scale at semiconductor interfaces, relevant for device technology.

II. EFFECT OF FREE CHARGE CARRIERS ON MUON SPIN POLARIZATION
Implantation of a positively charged muon (µ + ) in a semiconductor or insulator normally leads to the formation of the hydrogen-like muonium (Mu) state, where the µ + may capture one or two electrons to form the neutral Mu 0 or negatively charged Mu − state, or it ends up without capturing an electron (Mu + ). Thus, analogous to hydrogen, the three charge states Mu + , Mu 0 , and Mu − can occur, depending on their formation energies and free charge carrier concentrations. The two states Mu + and Mu − , where the µ + is not coupled to an unpaired electron, are called diamagnetic states, in contrast to the paramagnetic state Mu 0 , where the hyperfine coupling with the bound unpaired electron causes an additional magnetic field on the µ + . In a transversefield muon spin rotation experiment (TF-µ + SR), where an external magnetic field B is applied perpendicular to the initial muon spin direction, the muon spin in the diamagnetic states precesses at the muon Larmor frequency ν µ = γ µ /(2π) · B (γ µ /(2π) = 135.54 MHz/T, the muon gyromagnetic ratio), whereas in Mu 0 , much higher frequencies corresponding to transitions between various hyperfine states are observed [6].
In the group IV or III-V semiconductors with cubic diamond or zinc blende crystal structure, two Mu sites are known: Mu at a bond center (Mu BC ), or Mu at the tetrahedral interstitial site (Mu T ) [4,5]. The Mu BC state is usually the donor-like configuration with positive or neutral charge (Mu + BC , Mu 0 BC ), whereas the acceptor like Mu T configuration can be in the neutral or negative charge state (Mu 0 T , Mu − T ). The electron distribution of the Mu 0 BC state is axially anisotropic, which symmetry axis along the 111 direction. The electron density is centred at the host atoms, and relatively low at the µ + site, which explains the small value of the hyperfine coupling A hf c in Ge of the order of 100 MHz of Mu 0 BC with respect to vacuum muo-nium (A hf c = 4.46 GHz). In contrast, the electron density of Mu 0 T in Ge is much larger and isotropic, with A hf c = 2.36 GHz at low temperatures T < 40 K, and linearly decreasing with temperature at T > 40 K due to the coupling of Mu T with phonons [9,10].
The sensitity of muons to free charge carriers originates from the interaction of the charged/neutral muonium states with these carriers. The interaction may lead to a change of the diamagnetic and paramagnetic fractions, or to a change of muon spin depolarization rates in the presence of cyclic charge-exchange processes, or to phase shifts in the muon spin precession signal if a neutral pre-cursor state transforms into a diamagnetic state. The low temperature charge fractions in undoped Ge are about 75% in Mu 0 T , about 10%-20% in Mu 0 BC , and less than 10% in a diamagnetic state [4,11]. In the presence of free electrons with concentration n > 10 17 cm −3 in doped samples the Mu 0 T acceptor state may capture an electron to form the diamagnetic Mu − T state, resulting in an increase of the diamagnetic fraction at low temperatures to 30% at n ∼ 2 × 10 18 cm −3 , and to 80% at n ∼ 2×10 19 cm −3 [12]. Increasing the temperature in Ge leads first to an onset of thermally activated ionization of the Mu 0 BC state at 150 K with an activation energy of about 145 meV [13], where the Mu 0 BC is completely transformed to Mu + BC at 200 K [11]. At T > 200 K thermally activated ionization of Mu 0 T → Mu − T sets in with an activation energy of about 170 meV [5,11,14], where the diamagnetic fraction reaches 100% close to room temperature.
It is the Mu 0 T state at T > 200 K that we use as a sensor for free hole carriers. In the presence of free holes a recombination of a hole h + with a Mu − T state may occur to form again the neutral state. As a consequence chargeexchange cycles between Mu 0 T and Mu − T are established, which lead to a depolarization of the TF-µSR precession signal due to fast turning on/off of the hyperfine field in the Mu 0 T state, where the depolarization rate is proportional to the hole carrier concentration p [7,11,14]. In the absence of holes the depolarization rate of the diamagnetic signal is slow, with rate λ S < 0.2 µs −1 at T < 300 K [7]. In the presence of charge-cycles, a fast component with depolarization rate λ F > λ S appears, where λ F ∝ Λ c with Λ c the hole capture rate.
We developed a Monte-Carlo simulation [14,15] for the charge-exchange cycles which allows us, in combination with a calibration measurement on a p-doped wafer with p = 10 15 cm −3 , the determination of the hole capture rate Λ c for a measured muon depolarization rate λ F [14]. The simulation is modeling the cyclic reaction where the forward reaction (Mu 0 T ionization) is described by an Arrhenius rate process with the temperature dependent ionization rate Λ i (T ) = Λ 0 exp(−E A /k B T ), Λ 0 the attempt frequency, E A the activation energy, and k B the Boltzmann constant. The reverse reaction, hole capture of Mu − T , is governed by the hole capture rate where we assume a constant hole carrier concentration p in the temperature range of the experiment, while the temperature dependence of Λ c (T ) is absorbed in the product of the hole carrier velocity v h and the hole capture cross section σ h c . It has been shown in the calibration experiment that v h σ h c ∝ T −2.2(2) (see appendix A), indicating that the temperature dependence of v h σ h c is goverend by the temperature dependence of the hole mobility (∝ T −2.3 ) [14]. Figure 1 displays the simulation results at an applied transverse field of 10 mT. The increase of the depolarization rate with increasing hole capture rate is obvious from the µSR asymmetry spectra in Fig. 1(a) and (b). Figure 1(c) demonstrates the linear relationship between hole carrier density -which determines the hole capture rate -and the emergent fast depolarization rate λ F in the presence of charge cycles. Figures 1(c) and (d) show, that for a given p, λ F decreases with increasing temperature. This is due to the exponentially increasing Mu 0 T ionization rate, which means that the muons spend less and less time in the neutral state, causing less and less dephas-ing/depolarization of the muon spin during the charge cycles. The solid lines in Fig. 1(d) are fits of the equation where ω 0 = 2π · 2150 MHz is the hyperfine coupling of Mu 0 T averaged over the temperature range 220 K -290 K [10], and x = B/B 0 with B 0 = ω 0 /(γ µ − γ e ) = 0.0764 T is the hyperfine magnetic field at the electron, with gyromagnetic ratio γ e . Equation 2 is the expression for 1/T 1 in a longitudinal field (field applied parallel to the initial muon spin direction, LF-µSR) [10,16,17]. Due to the fact that in TF-µSR the depolarization caused by nuclear dipolar fields is small in Ge compared to λ F , the TF-µSR depolarization rate λ F can be well approximated by 1/T 1 . Equation 2 will be used in the analysis of the depth dependent hole carrier profiles under illumination. The temperature dependence of λ F is mainly determined by the exponential temperature dependence of Λ i , and to lesser extent by Λ c and ω 0 . With increasing temperature, Λ i is exponentially increasing, causing the decrease of λ F .
The LE-µSR experiments were carried out at the lowenergy muons (LEM) facility at the µE4 beam line [18]  keV energies is generated by moderating a 4-MeV-µ + beam, generated by the PSI proton accelerator, in a cryogenic moderator layer of solid Ar/N 2 [21][22][23]. The moderated muons with eV energies are electrostatically accelerated up to 20 keV and transported by electrostatic elements to the sample region. The samples are glued with conductive silver on a Ag-coated sample plate made of aluminum, where the final implantation energy is adjusted by applying an electric potential up to ±12.5 kV to the sample plate. The implantation profiles in Ge of muons with energies between 4 and 18 keV are displayed in Fig. 2. To illuminate the samples, either LEDs or solid state lasers are available [7,24,25]. For illumination with blue light we used either a LED source with λ = 405 nm (Bluepoint, Hönle AG, Gräfeling, Germany) or a diode pumped solid state laser with λ = 457 nm. For red light a diode laser with λ = 635 nm was used (both lasers from DelMar Photonics Inc San Diego CA, United States). In all cases the maximum light intensity at the sample was 50 -100 mW/cm 2 .
In addition to the near-surface investigations with LEµ + , the bulk of the Ge wafers was studied at a mean depth of about 300 µm employing the two instruments DOLLY and GPS [26] which use 4-MeV-µ + . The original wafers were cut in about 1 cm 2 pieces to fit into the cryostats of the bulk instruments. In all muon experiments the magnetic field was applied parallel to the 100 direction, and transverse to the initial muon spin direction.

IV. RESULTS
A. Diamagnetic fractions at different depths and doping levels Figure 3 shows the temperature dependence of the diamagnetic fraction F D for various Ge samples with different doping levels and at different muon implantation energies, i.e. different mean depths of stopping µ + . F D is defined as the fraction of muons in a diamagnetic state, determined by the fraction of muons precessing at the muons' Larmor frequency. We begin with the description of the bulk µSR measurements of the undoped and p-and n-type Fig. 3 a). The diamagnetic fraction F D is < 20% at T < 100 K, and either slowly increasing from 5 K to 100 K for the undoped and 1p15 sample, or nearly constant for the 6n17 sample. While the origin of the slow increase below 100 K is less clear, the increase of F D between 100 K and 175 K can be attributed to the thermally activated ionization of Mu 0 BC , which is "completed" at T 175 K [11]. This appears as a flattening of F D in the 1p15 sample: here the increase of F D due to the thermally activated formation of Mu − T at T > 180 K is not observable -in contrast to the undoped and 6n17 sample -because the presence of holes drives too quickly the reverse reaction in Eq. 1. Implanting the muons much closer to the surface with an energy of 14 keV at a mean depth z 80 nm reveals considerable differences: i), F D at T < 150 K is significantly larger than in the bulk, ii), the increase of F D due to thermally activated Mu − T formation appears to begin at a lower temperature around 150 K, and iii), the most striking feature, we observe the thermally activated formation of Mu − T in the 1p15 sample. The latter can be only explained by the absence of holes at least to a depth of 120 nm, i.e. the presence of a hole depletion layer. This is supported by the observed weak depolarization rate at 220 K, which is smaller than the expected depolarization rate of ∼ 0.06 µs −1 for p ∼ 10 11 cm −3 [ Fig. 1 a)], implying p 10 11 cm −3 in the depletion layer. Additionally, the larger F D below 100 K indicates an electron accumulation in the near-surface region, with n ranging between n ∼ 10 18 cm −3 and 10 19 cm −3 [4,12]. This means that the 1p15 wafer exhibits a surface layer inversion, where hole depletion and electron accumulation are generated by band bending at the surface. The presence of free electrons in the accumulation region adds to the Mu − T formation rate due to thermal activation the electron capture rate of the process Mu To further support the interpretation of a hole depletion layer, we show in Fig. 3 b) the results of a p-type sample with an order of magnitude larger hole concentration (2p16). We also present for comparison the temperature dependencies of two n-type samples which show similar trends as the undoped sample. In the 2p16 sample at 14 keV ( z 80 nm) the thermally activated formation of Mu − T is no longer observed: F D does not increase at T > 180 K, indicating that p 10 14 cm −3 , or in other words the hole depletion layer is now significantly shifted towards the surface. This is confirmed by lowering the implantation energy to 6 keV ( z 35 nm), where we again can observe -as in the case of the 1p15 sample at 14 keV -the thermally activated formation Mu − T , which means that the hole depletion layer is still present, but with reduced width. This is supported by the fact, that F D is larger below 150 K compared to the n-type samples, indicating electron accumulation.

B. Hole carrier profile in the depletion region and its manipulation by illumination
As a first attempt to manipulate the depletion region we illuminated the sample at 220 K with blue light (λ = 405 nm) at an intensity of up to 80 mW/cm 2 . At an energy of 6 keV, F D drops within minutes to the value measured at 14 keV, indicating that the depletion region is removed, or at least significantly shifted towards the surface. After turning off the light, F D does not change, indicating the persistent change/removal of the depletion layer. This effect has been observed previously [7] and it is attributed to the trapping of photo-generated electrons in empty surface acceptor states, charging the surface negatively and thus pulling holes from the bulk and the photo-generated holes into the depletion region. On warming the sample in the dark, F D begins to increase at T 270 K, where trapped electrons from the surface acceptor states are released and move back into the bulk of the wafer where they recombine with the holes, reestablishing the hole depletion zone. The release of the electrons appears as a thermally activated process with an energy barrier of about 1.1 eV [7].
To determine the width of the depletion region in the two p-type samples we measured F D as a function of implantation energy E imp between 1 keV and 20 keV at T = 220 K, see Fig. 4. In the 1p15 sample only the slowly relaxing component F D,S is observed, indicating p 10 12 cm −3 in the entire energy/depth range. The solid lines in Fig. 4 a) are from stopping profile simulations assuming an increase of p to 3 × 10 11 cm −3 beyond 140 and 150 nm, respectively. In this case, a fast component should appear, leading to a decrease of the slowly relaxing component. No sharp drop of F D,S is observed, excluding an increase of p in this region. This implies, that the depletion width W D is larger than 160 nm, the maximum range of 20-keV µ + . The observed weak decrease of F D,S can be explained by a slowly increasing activation energy E A for Mu 0 T ionization as a function of depth: the presence of an electric field due to the band bending in the depletion zone will result in a reduced E A on approaching the surface, increasing the Mu 0 T ionization rate Λ i at a fixed temperature and there-   Fig. 1 c). d) Calculated p(z) in the Schottky and in the weak space-charge layer approximation using Eq. 3. The shaded area indicates the region where the fast component can be observed by LE-µSR, which means that p must be in the range 3 × 10 11 cm −3 and < 10 14 cm −3 in this region.
fore increasing F D,S [8]. Below 5 keV both samples exhibit F D,S 0.8. In the 2p16 sample at E imp > 5 keV, F D,S begins to drop, and the data can be well described by the solid line shown in the figure. This line is calculated assuming p < 10 12 cm −3 at depths z < 45 nm, and the emergence of holes with p > 10 12 cm −3 beyond 45 nm, which implies the appearance of a fast relaxing component F D,F . Indeed, we observe this fast component, changing as a function of implantation energy as shown in Fig. 4 b). The data can be best modeled by assuming a depth interval of [45:65] nm where p is in the range of p 10 12 cm −3 and p < 10 14 cm −3 . A larger p 10 14 cm −3 means a too fast depolarization of the µSR signal, causing a loss of the observable fraction F D,F . This explains the drop of the sum of both components as a function of E imp : if the µ + reach regions with p 10 14 cm −3 , the fast component can no longer be observed causing a reduction of F D,F as shown in Fig. 4 b). Figure 4 c) displays p as a function of implantation energy, derived from the measured λ F by scaling according to the simulation data of Fig. 1 c). It is in the expected range, but the errors are getting very large at E imp > 12 keV due to the decreasing F D,F and the relatively poor statistics of the data. Thus, no firm conclusions about the carrier profile in the depth range [45:65] nm can be drawn from this plot. Instead, we use simple modeling to calculate carrier profiles and determine the parameters of the model to obtain qualitative agreement with the experimental data. The hole carrier profile p(z) in the depletion region depends on the local band deformation V (z) at depth z as [27] where N A is the bulk acceptor density of the p-type material, and is the electrostatic potential with its bulk value φ B . Since there is actually not only depletion but inversion at the surface, we can assume that the surface potential φ s determining the band bending significantly exceeds k B T , |eφ s | k B T [27]. In this Schottky space-charge approx- imation the electrostatic potential decays quadratically from its surface value φ s into the bulk with with the depletion width W D = 2εε 0 /(eN A )φ s [1,2]. Choosing W d = 170 nm, i.e. φ s ∼ 0.3 V, and inserting in Eq. 3 gives the red curve in Fig. 4 d), resulting in a hole carrier concentration in the shaded area (z ∈ [45 : 65] nm) in the range of 10 12 -5 × 10 13 cm −3 , fairly well agreeing with the experimental p(z) in Fig. 4 c). We note, that the Schottly approximation assumes for the free carrier densities n(z) ≈ p(z) ≈ 0, and complete ionization of the acceptors. Due to the electron accumulation in the hole depletion zone, n(z) > 0, therefore increasing the negative space charge in this region. For simplicity we assumed in the calibrations n(z) < N A , giving a total space charge N A + n N A . This seems to be justified by the fact that the bulk N A values of the two p-type samples seem to be the dominant densities to explain the observed differences in the depletion width estimates at the surfaces. For comparison and illustration, we show the blue curve p(z) in the weak space-charge limit for φ s = 0.3 V (however, the weak space-charge limit is applicable for |eφ s | < k B T , which is not fulfilled here) where with the Debye length L D = εε 0 k B T /(e 2 N A ) ∼ 30 nm for the 2p16 sample at 220 K. In this case, p(z) would be in the range 5 × 10 14 -5 × 10 15 cm −3 , in contradiction to the experimental results. In order to bring p(z) into the experimentally observed range one would need to use an even larger φ s = 0.9 V (magenta curve), clearly outside the limits of the approximation. The findings of Fig. 4 can be summarized as follows. The data are well described within the Schottky approximation, implying an abrupt change of p from < 10 12 cm −3 to p > 10 13 cm −3 within about 20 nm commencing at a depth of 45 nm. After illumination with blue light at 220 K, a persistent hole accumulation with p > 10 14 cm −3 is established in the depletion region at implantation energies between 6 keV and 14 keV, corresponding to a z range of ∼ 10 nm to ∼ 120 nm. In the 1p15 sample the width of the depletion layer is estimated to ∼ 760 nm, √ 20 times larger than for the 2p16 sam-ple, since the width of the depletion layers scales with the square root of the bulk carrier concentration. We estimate the space charge density N A · W d for the two samples to ∼ 3.4 × 10 11 cm 2 for the 2p16 sample, and ∼ 7.6 × 10 10 cm −2 for the 1p15 sample. Now we turn to the manipulation of p in the wide depletion region of the 1p15 sample by illumination with red light (λ = 635 nm). Under illumination, again a fast component appears, indicating the presence of photogenerated holes in the depletion zone. However, after turning off the light the fast component disappears and the original slow component fraction F D,S is restored, meaning that the photo-generated holes in the depletion zone immediately disappear by recombination. This implies, that the photo-electrons generated by red light do not have enough energy to overcome the ∼ 1.1 eV barrier at the surface to reach the empty surface acceptor states. In this case, a dynamic equilibrium of photogenerated holes and electrons is established in the depletion region, and a quick recombination with photogenerated electrons takes place after turning off the light. As shown in Fig. 5 a), the fast component can be tracked in the temperature range from 220 K to 290 K, and from close to the surface at 4 keV ( z 25 nm) to a mean depth of 120 nm at 20 keV. This is different to illumination with blue light, where after an illumination time of 3 min with a laser (instead of the weaker LED source) at λ = 457 nm and ∼ 100 mW/cm 2 , the fast depolarization rate λ F exceeds values of 60 µs −1 , which is the maximum detectable depolarization rate within the experimental resolution of the LEM apparatus: the fast component can no longer be resolved, and it appears as a "missing" fraction in the µSR spectra. The persistent λ F > 60 µs −1 implies a persistently generated hole concentration p 10 14 cm −3 within a few minutes of illumination with blue light. In contrast, the dynamic equilibrium photo-generated hole carrier concentration with red light is significantly smaller, with p in the order of 10 12 cm −3 , see Fig. 5 d).
The increase of λ F as a function of energy in Fig. 5 a) points towards an increasing hole carrier concentration with increasing depth. For a quantitative analysis we used Eq. 2 to fit the data (solid lines in Fig. 5 a)). In the fits we fixed according to appendix A i), the exponent of the temperature dependence of Λ c to -2.2 [Λ c (T ) = Λ c (290K) · (T/290) −2.2 ], ii), the pre-factor Λ 0 in the ionization rate to 3.2 · 10 13 /s, and iii), ω 0 = 2π · 2150 MHz, which is the average value of the hyperfine coupling of Mu 0 T in the temperature range between 220 K and 290 K [10]. The value of Λ 0 is within the range of 1.2 · 10 13 /s and 6.7 · 10 13 /s of pre-factors found in Ref. [10]. The free fit parameters were Λ c (290K) and the activation energy E A , which are displayed in Figs. 5 b) and c), respectively. From Λ c the hole carrier concentration p( z ) as a function of mean depth z is calculated in, Fig. 5 d), using the re-analyzed calibration data of Ref. [14] in appendix A. In doing so we assume that the measured λ F (E imp ) at implantation energy E imp equals λ F ( z ) at the corresponding mean depth. The justification for the validity of this assumption is discussed in Appendix B. Figure 5 c) indicates a change of E A by about 10 meV under illumination between the near-surface region and a depth of about 80 nm, where it reaches its bulk value. This indicates a rather weak band bending under illumination with an electric field < 0.5 mV/nm, where E A is reduced by < 10 meV [8]. According to Fig. 5 d), the electric field is becoming too weak to change E A within errors in regions where p 2 × 10 12 cm −3 . These electric field values are confirmed by fitting Eq. 3 in the Schottky approximation to the data in Fig. 5 d). In order to obtain a good fit we added a constant offset term p 0 to Eq. 3, where we attribute the appearance of p 0 to the non-equilibrium situation under illumination. The fit yields a depletion width W d ∼ 555 nm, which is significantly smaller than the estimated W d ∼ 760 nm for the dark sample. A lowering of W d under illumination is expected due to the reduction of band bending by the partial compensation of the surface charge by the photogenerated charge carriers. Using the reduced value of W d in Eq. 4 gives an electric field of ∼ 0.5 mV/nm at a depth of 80 nm, in good agreement with the expected value estimated at the beginning of the paragraph.

V. DISCUSSION
To the best of our knowledge this is the first time that by means of a contact-less, non-destructive local probe technique the free charge carrier concentration profile p(z) has been determined directly over a depth range from close to the surface up to 160 nm. Knowing p(z) the electrostatic potential φ(z) can be calculated using Poisson's equation. A direct experimental determination of the bending of φ(z) at the surface is possible by photoemission spectroscopy. However, this method is limited to the first few nanometers at the surface, i.e. to high bulk doping levels 10 19 cm −3 , because only in this case the depletion range is comparable to the photoelectron escape depth [28]. We note, that φ(z) can be determined directly by LE-µSR in cases, where a change of Mu 0 activation energies is observable [8]. In contrast to photoemission spectroscopy the sensitivity of LE-µSR to free hole carrier concentrations is orders of magnitude larger in Ge: p(z) as low as ∼ 10 11 cm −3 can be detected by LE-µSR over a hundred times longer length scale of about 200 nm with a resolution of a few nanometers. This resolution exceeds significantly the capabilities of capacitance-voltage techniques, where due to the Debyelength limitation it is not possible to profile closer than about 1L D to the surface [2], which is ∼ 30 nm in the 2p16 sample, and ∼ 130 nm in the 1p15 sample.
The high sensitivity to holes in the Mu − T +h → Mu 0 T reaction is due to the large hole capture cross section σ h c ∼ 10 −13 cm 2 of Mu − T , where we estimate σ h c using the relation Λ c = p · v h · σ h c with Λ c = 0.15 MHz (corresponding to p ∼ 10 11 cm −3 ) as a lower detection threshold [ Fig. 1 a)], and a hole velocity v h ∼ 10 7 cm/s. We now turn to the implications of our results on the surface charge. The observed hole depletion and electron accumulation means a positively charged surface, which implies the presence of empty, positively charged donor states. The fact that the surface changes to negative charge under illumination with blue light implies the existence of empty surface acceptor states, which are persistently filled by photo-generated electrons. With red light, no persistent charging occurs, which means that the photo-generated electrons do not have enough energy to overcome the surface barrier of about 1.1 eV [7]. The observed electron accumulation even for the 6n17 sample means that surface donor states are still not filled with electrons, implying that these donor states must be located close to the conduction band -otherwise, they were filled and neutral. The surface acceptor states, filled under blue illumination, must be also similarly high in energy as the surface donor states -otherwise, they were filled as well at 6n17 doping, leading to a negative surface charge, and thus changing the band bending to remove electron accumulation. For a cleaved Ge surface without oxide layer it is well established that there exists Fermi level pinning close to the valence band, causing an upward band bending at the Ge surface with hole accumulation and electron depletion [29][30][31][32]. This is different to our commercial Ge wafers with a ∼ nanometer-thin native oxide layer. The oxide can exist in various oxidation states GeO x which may strongly affect the electronic properities/band bending of the Ge/GeO x interface [33,34]. The prevailing oxidation state for native oxide is +4 (GeO 2 ), with the presence of GeO x with x < 4 at the Ge/GeO 2 interface [35]. Assuming that the band structure at the interface is determined by the band alignment of GeO 2 with a band gap of ∼ 5.7 eV and a conduction band offset ∆E c ∼ 1 eV with respect to Ge [36], we speculate that i), the band bending in Ge at the Ge/GeO 2 interface is opposite to a cleaved Ge surface, yielding electron accumulation and hole depletion as illustrated for isotype heterojunctions in [1], and ii), the surface energy barrier is determined by ∆E c . While the details of the Ge/GeO 2 interface are important for device applications, its more detailed characterization is out of the scope of this study and remains for upcoming work. Here, our intention is to demonstrate the capability of charge carrier profiling at the surface of a semiconductor which allows getting insights also in the surface characteristics.

VI. CONCLUSIONS
In summary we have shown by means of low-energy µSR that charge carrier profiles at semiconductor interfaces can be directly studied with nanometer depth resolution, if a muonium state forms in the semiconductor that is interacting with free carriers. The sensitivity of the technique depends on the cross section of carrier cap-ture by the muonium state. In the case of Ge it is the interaction of the Mu − T state with holes at T > 200 K which is utilized for this purpose, where hole carrier concentrations can be determined in the range 10 11 − 10 15 cm −3 by the measureable effect on the muon spin depolarization rate in transverse magnetic field. This allowed us to determine the hole carrier profile and its light induced manipulation in the hole depletion/electron accumulation region at the surface of commercial Ge wafers with a thin native GeO 2 layer on top.
As an outlook the method can be applied to characterize on a microscopic level the properities of the GeO x /Ge interface which might yield new insights for technologically interesting Ge device applications. The study of pre-cleaned surfaces, where the native oxide layer has been removed, would be interesting to provide complementary quantitative information on a nanometer scale of the expected hole accumulation due the Fermi level pinning close to the valence band.
can be approximated by a stretched exponential function [37]. However, in our case the experimental data can be well fitted by a single exponential depolarization function. To test the validity of using a single exponential depolarization function, we added the µSR data of energies {4, 6,9,12,15,17,20} keV, where the average energy is 11.9 keV. The mean depth z of the sum of stopping distributions is 69.3 nm, which agrees well with the mean depth of the single 12 keV data with z = 67.5 nm. The sum spectrum is very well fitted with a exponential depolarization function with a reduced χ 2 of 0.96 with 2036 degrees of freedom, strongly supporting the single exponential function model, where λ F of the summed data agrees within statistical error with λ F of the 12 keV single energy data. The same result is obtained when choosing a different set of data, e.g. {12,15,17,20} keV, where λ F of the sum spectra with an average energy of 16 keV equals the λ F of a single energy of 16 keV. This has the important implication that the measured λ F , which is the average of λ F 's across the muon stopping profile, reflects λ F at the mean stopping depth, i.e. λ F = λ F ( z ). In other words, λ F ( z ) = c · p( z ), justifying the interpretation of the data, that the λ F for a given muon implantation energy reflects the hole carrier concentration at the corresponding mean depth z .