# Strain Measurement in Single Crystals by 4D-ED

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## Abstract

**:**

## 1. Introduction

## 2. Instruments and Materials

^{TM}nanomanipulator.

_{solubility}~4 × 10

^{20}cm

^{−3}), so that a non-equilibrium technique, such as nanosecond laser doping, is required. We employ gas immersion laser doping (for a review see [12]): a precursor gas (BCl

_{3}) is injected into an ultra-high vacuum chamber, where it saturates the chemisorption sites on the Si surface. The silicon is melted up to the desired doping depth by a XeCl 308 nm excimer laser of 25 ns pulse duration and tuneable energy. The melting of Si induces the rapid diffusion of the chemisorbed B into the liquid Si region. Due to the high recrystallization velocity (4 m/s [13]), the Si undergoes quenching, and high dopant concentrations beyond the solubility limit can be reached. As a result, a homogeneously doped box-like Si:B crystal is grown by liquid phase epitaxy on the underlying Si substrate. Multiple chemisorption/melting cycles determine the number of dopants introduced in the doped layer [14]. In this work, an Si:B layer of total thickness d = 121 nm is investigated. The thickness is determined by cleaving the doped spot in the middle and performing a KOH chemical etch at 80 °C for 45 s on the cleaved edge. The KOH etches away the underneath Si substrate, while the p-type ultra-doped Si:B acts as an etch stop. The hanging ledge is then observed in a scanning electron microscope at 90° angle to determine the thickness of the highlighted Si:B layer. The found thickness is confirmed within a 10% error by the strain analysis, as a deformation in the normal direction, induced by the B substitutional incorporation, is observed over the top 127 nm. The energy surface density necessary to melt d = 121 nm is E = 960 mJ/cm

^{2}. This energy is determined from the laser energy temporal profile, measured in an integrating sphere with a nanosecond photodiode. The energy is calibrated through the comparison, in situ, with the reference melting energy of an Si monocrystalline (100) undoped substrate placed near the doped samples [15]. The described doping process is repeated 300 times, in order to reach an active concentration n

_{B}= 3.06 × 10

^{21}cm

^{−3}(6.1 at.%). The active concentration is determined by Hall measurements, with a Hall factor of 0.75 [16]. The total B concentration C

_{B}(which includes both electrically active—i.e., substitutional—and inactive B atoms), C

_{B}= 4 × 10

^{21}cm

^{−3}, is known from a series of secondary ion mass spectroscopy (SIMS) measurements on Si:B samples with a varied number of doping cycles (1 to 700) and thickness (65 nm to 176 nm). As expected, C

_{B}increases linearly with the number of doping cycles: indeed, the amount of B introduced at each cycle is determined by the constant number of chemisorption sites, which are saturated before the laser melting. The activation ratio (n

_{B}/C

_{B}), is in this sample 77%, highlighting the large activation achievable by nanosecond laser doping, even at the extreme doping concentrations performed. The superconducting critical temperature of the sample is T

_{c}= 0.35 K, determined by low-temperature resistance vs. temperature measurements in an adiabatic demagnetization refrigerator. The superconducting critical temperature depends on both the active B concentration and the deformation of the Si:B layer [17]. Thus, a precise determination of the deformation profile in the layer thickness is essential for a correct understanding of the establishment of superconductivity in superconducting silicon.

## 3. The Strain4DED Method

#### 3.1. Basic Principles and Workflow

_{xx}, ε

_{xy}, ε

_{yx}, ε

_{yy}and the derived quantities, shear and rotation) are determined from the positions of the lattice spots, identically to the procedure used in different variants of the HRTEM-FFT-based GPA method [3,21].

#### 3.2. Details of Operation

#### 3.2.1. Peak Finding Algorithm

#### 3.2.2. Fitting of the Lattice

#### 3.2.3. Calculating Strain Components

_{xx}and the lateral component is called ε

_{yy}. In our case the ε

_{xx}is calculated from the length of the 0N vector and ε

_{yy}is from the 0L vector. Similarly, the cross partials ε

_{xy}and ε

_{yx}measure the movement of the diffraction spot perpendicular to the direction of the g-vector. If u

_{x}and u

_{y}represent the components of the shift vector of the spots (g

_{ref}-g

_{strained}),

#### 3.2.4. Presentations of Strain Components

#### 3.2.5. Concentration of Substitutional Element

_{Si}= 0.5431 nm is taken from XRD database card #04-002-0118. Elementary B is not available in cubic form, however, an Si-1%B is found in XRD database NIMS_MatNavi card #4295421637_1_2 with a

_{Si-1%B}= 0.54166 nm. From this value, a virtual pure B cubic crystal was extrapolated resulting in a

_{solute}= a

_{B}= 0.40009 nm. It is between the extreme values for cubic B lattice values reported in the literature (from a

_{B}= 0.378 nm [22] to a

_{B}= 0.4084 nm [23]. In an isotropic crystal, the lattice constants deformed in normal and lateral directions are connected by Poisson’s ratio to the unstrained lattice constant:

## 4. Discussion

^{21}cm

^{−3}. This is compared to the concentration of substitutional element (B) in Figure 6 since it is assumed that B at substitutional sites corresponds to the active B quantity. The agreement is good, in particular in the top 90 nm, where the two values are within 7 % relative. The discrepancies may be due to the choice of a

_{B}, or to the use of the Si Poisson coefficient, which may be modified in ultra-doped Si:B. However, such curves give important information, drawing attention to a fully strained (zero lateral deformation), lower doping, and 30 nm region at the bottom of the layer. Since superconductivity only appears in relaxed layers, such observation points out that superconductivity does not extend through the whole doped layer. In-depth characterization of Si:B samples from the point of view of superconductivity, electrical properties, and applications is the topic of separate papers.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Example of selecting area and collecting 4D-ED data set: (

**a**) HRTEM image of an area of the sample; (

**b**) BF image of the same area with enhanced contrast around defects; (

**c**) STEM image from the sub-area from where the 4D-ED dataset is collected; (

**d**) Diffraction pattern recorded with the nearly parallel (α = 0.24 mrad) beam from an area of 1 nm diameter (the size of the beam in this mode). A total of 20,000 such patterns form the 4D-ED dataset.

**Figure 2.**Illustration of how different diffraction patterns can be measured within a 400 nm wide region of a Si sample. Locations are marked on (

**g**). (

**a**) The pattern recorded at location A. Three diffraction spots are marked with numerals (0 is the central spot, 1 and 2 mark the endpoints of the two shortest reciprocal lattice vectors) (

**b**) The pattern recorded at location B. Three diffraction spots are marked with characters (0 is the central spot, N and L mark normal and lateral directions from the central beam) (

**c**) The pattern recorded at location C. (

**d**) The pattern recorded at location D. (

**e**) The pattern recorded at location T, which is a position of nanotwins. (

**f**) The pattern recorded at location F, which is the position of C-Pt protecting layer deposited during FIB preparation. (

**g**) STEM HAADF image of the 400 nm wide region. Letters mark the locations where the diffraction patterns were collected.

**Figure 3.**Block diagram showing the steps of fitting the full lattice to the measured subset of spots.

**Figure 4.**Two-dimensional strain distribution in an Si:B sample. The layer is on a bulk substrate. The top surface (covered with protecting Pt during FIB cutting) is on the right side of the map. The 4D-ED dataset was recorded from the area shown in Figure 1c. Pixels in the map are black, where it was impossible to fit a grid to the corresponding pattern (e.g., protecting layer). (

**a**) Normal component. (

**b**) Lateral component.

**Figure 5.**One-dimensional strain distribution in an Si:B sample as a function of depth. (

**a**) Normal component. (

**b**) Lateral component.

**Figure 6.**One-dimensional depth distribution of B in the Si:B sample. (

**a**) Linear scale concentration (at%) to better show details. (

**b**) Logarithmic scale (atom/cm

^{3}) more usual presentation in semiconductor physics.

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**MDPI and ACS Style**

Lábár, J.L.; Pécz, B.; van Waveren, A.; Hallais, G.; Desvignes, L.; Chiodi, F.
Strain Measurement in Single Crystals by 4D-ED. *Nanomaterials* **2023**, *13*, 1007.
https://doi.org/10.3390/nano13061007

**AMA Style**

Lábár JL, Pécz B, van Waveren A, Hallais G, Desvignes L, Chiodi F.
Strain Measurement in Single Crystals by 4D-ED. *Nanomaterials*. 2023; 13(6):1007.
https://doi.org/10.3390/nano13061007

**Chicago/Turabian Style**

Lábár, János L., Béla Pécz, Aiken van Waveren, Géraldine Hallais, Léonard Desvignes, and Francesca Chiodi.
2023. "Strain Measurement in Single Crystals by 4D-ED" *Nanomaterials* 13, no. 6: 1007.
https://doi.org/10.3390/nano13061007