Surface artificial nanostructure state density measurement method based on two-step deconvolution algorithm

Abstract

The invention discloses a surface artificial nanostructure state density measurement method based on a two-step deconvolution algorithm. The method comprises the following steps: firstly, selecting a reference feature; performing scanning tunneling spectrum measurement by using a needle point of the scanning tunneling microscope in a constant-height mode to obtain a reference differential conductivity spectrum; solving a first optimization problem containing a regularization item by using the reference state density of the reference feature and the measured reference differential conductivity spectrum to obtain an effective tip-related tunneling function; keeping the measurement conditions unchanged, and performing scanning tunnel spectrum measurement above the to-be-measured sample to obtain a sample differential conductivity spectrum; and by using the effective tip-related tunneling function and the sample differential conductivity spectrum, through solving a second optimization problem containing a regularization item, performing inversion to obtain the state density of the sample to be measured. The method is wide in application range, simple in measurement step and capable of adapting to a complex needle point structure.

Description

Technical Field

[0001] This invention relates to a method for measuring the density of states, and more particularly to a method for measuring the density of states of artificial nanostructures on surfaces based on a two-step deconvolution algorithm. Background Technology

[0002] Density of states (DOS) is a key physical quantity connecting microscopic quantum states with macroscopic properties of materials, and is essential for understanding novel physical phenomena, designing advanced devices, and optimizing material performance. Currently, scanning tunneling microscopy (STM) and its derived scanning tunneling spectroscopy (STS) techniques are the main means of probing the local density of states (LDOS) on the surface of materials, achieving atomic-level spatial resolution.

[0003] However, a core challenge in scanning tunneling spectroscopy is the direct measurement of the differential conductance spectrum (DCT). ） It is not directly equivalent to the local density of states (LDOS) of the sample. The differential conductivity spectrum is actually the sample's local density of states, or tip density of states (LDOS). ) and tunneling probability ( The convolution result of the three elements. The tip is one of the most critical components of a scanning tunneling microscope (STM). STM requires the formation of a tunneling junction between the tip, vacuum, and sample. The signal in the scanning tunneling spectrum originates from the current generated by the quantum tunneling effect between the tip and the sample. Its tunneling probability mainly depends on the electronic structure of the tip, the distance between the tip and the sample, and the electronic structure of the sample. Since the electronic structure of the tip is usually unknown and may change during the experiment, and the tunneling probability is affected by various factors such as the distance between the tip and the sample and the work function, the tunneling process results in the measured differential conductivity spectrum containing not only the electronic structure information of the sample itself but also additional information about the electronic structure of the tip and the environment. Therefore, how to isolate the unknown influence of the tip and the environment from the measured differential conductivity spectrum and accurately extract the true density of states of the sample is an urgent problem to be solved.

[0004] To address the aforementioned problems, existing solutions mainly fall into three categories: model fitting, methods utilizing known surface states, and variable distance measurement. However, all have limitations. Model fitting assumes that the density of states of the tip and the sample have specific analytical forms (such as constants or specific functions), but this method is only effective when the electronic structure of the tip is extremely simple or known, limiting its applicability. Methods utilizing known surface states (which can be considered constants) use known two-dimensional electron gas surface states for calibration, but this method typically ignores the tunneling probability effect of the tip-sample distance and is limited by the two-dimensional electron gas surface states. Variable distance measurement involves measuring multiple sets of spectra at different tip heights and solving them self-consistently, but this method significantly increases experimental complexity and time costs. Furthermore, establishing the equations requires specific tunneling model assumptions (such as the WKB approximation or the trapezoidal tunneling barrier approximation), which may lead to inaccurate results.

[0005] Therefore, there is an urgent need in this field for a high-fidelity density of states recovery method that is widely applicable, has simple measurement steps, and can adapt to complex needle tip structures. Summary of the Invention

[0006] To overcome the shortcomings of existing methods for measuring the true density of states of samples, which are only applicable to simple needle-tip structures, have complex measurement steps, and have limited applicability, this invention provides a method for measuring the density of states of artificial nanostructures on the surface based on a two-step deconvolution algorithm, which can recover the true density of states of a sample from scanning tunneling spectral data with high fidelity.

[0007] The present invention solves the above-mentioned technical problems through the following technical solution: The first aspect of this invention discloses a method for measuring the density of artificial nanostructure states on a surface based on a two-step deconvolution algorithm, comprising: Select a reference feature with a known density of states within a preset energy window; Above the reference feature, a scanning tunneling microscope is used to perform scanning tunneling spectrum measurement in constant height mode within the preset energy window to obtain the reference differential conductivity spectrum. Using the reference density of states of the aforementioned reference feature and the measured reference differential conductance spectrum, an effective tip-related tunneling function is obtained by solving a first optimization problem that includes a regularization term. Keeping the measurement conditions unchanged, scanning tunneling spectrum measurement is performed above the sample under the preset energy window and constant height mode to obtain the sample differential conductivity spectrum. The measurement conditions include needle tip structure and tunneling conditions. Using the effective tip-correlated tunneling function and the sample differential conductivity spectrum, the density of states of the sample under test within the preset energy window is obtained by solving a second optimization problem including a regularization term. The first optimization problem and the second optimization problem are both constructed based on the same physical model describing the convolution relationship between the differential conductance spectrum and the density of states; the effective tip-related tunneling function is the product of the tip density of states and the tunneling probability containing the tip structure.

[0008] Furthermore, the reference feature is selected from any of the following: a noble metal surface with a known surface state density; a two-dimensional electron gas system with a known surface state density other than a noble metal surface; a single isolated atom adsorbed on an atomically flat platform; or a superconductor surface with a known state density near the superconducting bandgap.

[0009] Furthermore, the physical models upon which the first and second optimization problems are based correspond to the following convolution integral equation:

[0010] In the formula, For reference characteristic R or the differential conductivity spectrum of the sample S to be tested, For elementary charge, For bias voltage, For the reference characteristic R or the density of states of the sample S, This is an effective tip-related tunneling function.

[0011] Furthermore, the first optimization problem is solved by minimizing the following first loss function value. Determine the effective tip-related tunneling function :

[0012] In the formula, The discretized vector of the reference differential conductance spectrum; It is based on the reference density of states And a convolution operator matrix pre-constructed based on the convolution integral equation; This is the first regularization parameter; It is a forward difference matrix.

[0013] Furthermore, the second optimization problem is to minimize the following second loss function value. To determine the density of states of the sample to be tested ( ):

[0014] In the formula, This is the discretized vector of the sample's differential conductance spectrum; It is based on the above And a convolution operator matrix pre-constructed based on the convolution integral equation; This is the second regularization parameter; It is a forward difference matrix.

[0015] Furthermore, the first and second optimization problems are solved using an iterative optimization algorithm, which is selected from any one of L-BFGS-B, BFGS, or Newton-CG.

[0016] Furthermore, after obtaining the differential conductivity spectrum of the sample but before inverting the density of states of the sample to be tested, the constant height mode scanning tunneling spectrum measurement is performed again on the reference feature surface to obtain a new reference differential conductivity spectrum. This new spectrum is then compared with the initially measured reference differential conductivity spectrum to verify that the electronic structure of the tip has not changed. After this verification, the inversion of the density of states of the sample to be tested is then performed. If the electronic structure of the tip has changed, the measurement is restarted from measuring the reference differential conductivity spectrum.

[0017] Furthermore, the preset energy window is centered on the Fermi level. The energy window range.

[0018] Furthermore, in the constant height mode, the tunnel junction spacing is 0.1~1nm.

[0019] The second aspect of the present invention discloses an electronic device, including a processor and a memory; the processor is used to call a computer program stored in the memory to execute the surface artificial nanostructure state density measurement method based on a two-step deconvolution algorithm as described in any one of the first aspects of the present invention.

[0020] This invention is based on a two-step deconvolution algorithm. First, a scanning tunneling spectrum measurement is performed on the reference feature to obtain the reference differential conductance spectrum. Then, using the known reference density of states of the reference feature and the measured reference differential conductance spectrum, an effective tip-correlated tunneling function is obtained by solving a first optimization problem including a regularization term. Next, keeping the measurement conditions unchanged, a scanning tunneling spectrum measurement is performed on the sample to obtain the sample differential conductance spectrum. Finally, using the effective tip-correlated tunneling function and the sample differential conductance spectrum, a second optimization problem including a regularization term is solved to obtain the density of states of the sample within the preset energy window.

[0021] Compared with the prior art, the present invention has the following beneficial effects: (1) By solving the effective tip-related tunneling function, the present invention does not require prior knowledge of the electronic structure of the tip. Even if the tip structure is complex (such as spectral distortion caused by adsorption of impurities), as long as the electronic structure of the tip is stable and consistent, the effective tip-related tunneling function can be effectively determined by reference features and the effects of spectral distortion introduced by the tip can be deducted.

[0022] (2) Based on the same measurement conditions, namely the same tip structure and tunneling conditions, this invention only needs to be measured in the standard constant-height mode, without the need for time-consuming and error-prone variable distance (z-spectroscopy) measurement. It is fully compatible with the existing standard scanning tunneling microscope experimental procedures, and the experimental operation is simple and saves a lot of experimental time.

[0023] (3) This invention can effectively remove the artifacts caused by the needle tip and recover the high-fidelity sample local state density through a two-step deconvolution algorithm. Furthermore, experimental verification shows that this method can reveal the subtle physical features (such as resonance state modulation caused by quantum confinement effect and inversion effect) that are masked by the needle tip effect in the original differential conductivity spectrum. It has high resolution and is basically consistent with the theoretical results without additional parameter fitting.

[0024] (4) The method of the present invention is not limited to a specific material system. The reference features can be selected in various forms such as surface states and isolated atoms. It is applicable to the study of various artificial nanostructures and surface quantum phenomena and has a wide range of applications.

[0025] (5) Through a large number of experiments, this invention has proven that even when the same sample is measured using needle tips with different structures, the density of states curves obtained after processing by this method are highly consistent, with very high stability and repeatability, which greatly improves the reliability of scanning tunneling spectrum data and the comparability of experimental data from different batches. Attached Figure Description

[0026] Figure 1 This is a schematic diagram of the process for measuring the density of artificial nanostructures on the surface based on a two-step deconvolution algorithm, according to the present invention.

[0027] Figure 2 This diagram illustrates the deconvolution results of quantum well states measured within quantum fences of different sizes constructed from iron atoms on the Ag(111) surface. (a) shows the broad Ag(111) platform surface used for acquiring scanning tunneling spectra; (b) shows the surface state scanning tunneling spectra measured on the Ag(111) platform surface using three different tips (blue, orange, and purple); (c) shows the deconvolution results of the three scanning tunneling spectra in (b), where the black reference line represents the step function state density of the Ag(111) surface (its specific value is calculated using a broadened arctangent function combined with bulk state contributions); (d), (g), and (j) respectively show the morphology of circular fences with radii of 3.2 nm, 5.1 nm, and 9.2 nm constructed on the broad Ag(111) platform using atomic manipulation techniques; (e), (h), and (k) respectively show the morphology of the three different tips (blue, orange, and purple, ... Figure 2(All spectral lines of the same color in Figure (i) represent the results of measurements using a tip with the same structure) are scan tunneling spectra measured at the center of the surface quantum fence; (f), (i), and (l) show the corresponding deconvolution results for each case in corresponding colors. Taking Figures (e) and (f) as examples, the horizontal axis of Figure (e) represents the bias voltage corresponding to the measurement, in volts, and the vertical axis represents the measured differential conductance value, in nanosimens; the horizontal axis of Figure (f) represents the bias voltage, and the vertical axis represents the local density of states, which is a dimensionless value. The black solid line in Figure (f) represents the theoretical density of states calculated using the multiple scattering Green's function method.

[0028] Figure 3 This is a schematic diagram of the deconvolution local density of states (DSO) of the scanning tunneling spectrum measured on an iron-adsorbed atom at the center of a quantum fence on an Ag(111) surface. (a) and (c) show the scanning tunneling spectra measured using two different tips (blue and orange) on an isolated iron-adsorbed atom and an iron-adsorbed atom at the center of a circular fence with a radius of 5.1 nm, respectively. (b) and (d) show the corresponding deconvolution results in corresponding colors. In (b), the black reference line represents the DSO of the isolated iron-adsorbed atom, and in (d), the solid black line represents the theoretical DSO calculated using the multiple scattering Green's function method. Detailed Implementation

[0029] Before introducing the present invention, the relevant terms used in this invention are explained as follows: In this invention, "sample density of states" refers to the local density of states of the sample. In this invention, "tip structure" includes both the electronic structure and the physical structure of the tip. The electronic structure of the tip refers to the electronic properties that affect the tunneling process, mainly including but not limited to the tip's density of states, the shape of the tunneling barrier, and the influence of electron orbitals on the tunneling probability; the physical structure of the tip refers to its geometric shape and dimensional parameters, such as the macroscopic shape of the tip, the radius of curvature of the tip, and the aspect ratio. In this invention, "deconvolution" and "deconvolution" have the same meaning.

[0030] This invention proposes a two-step deconvolution algorithm for measuring the density of states of artificial nanostructures on the surface, which is used to recover the true local density of states (LDOS) of a sample from scanning tunneling spectral data when the tip density of states (DOS) is unknown.

[0031] It should be noted that the present invention is applicable to the effective tip-related tunneling function under the small bias approximation, where small bias refers to a preset energy window near the Fermi level, for example, the energy range of -0.3eV to 0.3eV.

[0032] This invention first selects a known ideal local density of states of a reference feature or a previously measured high-quality differential conductivity spectrum as a reference benchmark. Then, it combines this with the differential conductivity spectrum of the reference feature measured by scanning tunneling spectroscopy under the current experimental conditions to obtain an equivalent tip-correlated tunneling function. Using the tip-correlated tunneling function and the differential conductivity spectrum measured on the sample by the tip of the same scanning tunneling microscope, the density of states of the sample can be further deduced.

[0033] This invention does not address the needle tip state density ( ) and tunneling probability ( Instead of solving them separately, it is defined as a combined "effective tip-related tunneling function" ( ). ) Density of needle tip states ( ) and tunneling probability containing information about complex needle tip structure The product of.

[0034] The differential conductivity spectrum of the sample under test contains physical features with a definite energy scale, such as the width of the superconducting band gap and the full width at half maximum (FWHM) of the impurity resonance peak. The energy scale of these physical features should be significantly greater than the inherent energy resolution limit of the scanning tunneling spectroscopy (STS) measurement system. The inherent energy resolution limit of the STS measurement system is primarily determined by the experimental temperature. Determined thermal expansion scale and the AC modulation voltage applied by lock-in amplification measurement technology The broadening effect introduced by the RMS value determines the measurement process. Under these conditions, the measurement process can be accurately modeled as a convolution process, thereby enabling the two-step deconvolution algorithm of this invention to accurately restore the intrinsic state density of the sample. Correspondingly, the measurement equation of the differential conductivity spectrum can be simplified to a convolution integral equation.

[0035] Based on this, the present invention proposes a method for measuring the state density of artificial nanostructures on the surface based on a two-step deconvolution algorithm, which can recover the state density of the sample from the scanning tunneling spectrum. The method mainly includes the following two parts: Part 1: Determining the Tip-Related Tunneling Function Step 1: Select a reference feature.

[0036] Select a state with a known density of states ( Alternatively, a reference characteristic of a known differential conductivity spectrum. The known differential conductivity spectrum is a high-quality, reliable differential conductivity spectrum obtained from existing literature or previous experiments. This spectrum is less affected by the tip state density and tunneling probability, and therefore can be equated to the sample state density, serving as a reference characteristic.

[0037] Specifically, the reference feature can be selected from the following four types: 1) a noble metal surface with a known surface state density (e.g., the step function type state density of an Ag(111) surface); 2) a two-dimensional electron gas (2DEG) system with a known surface state density other than a noble metal surface; 3) a single isolated atom adsorbed on an atomically flat platform (e.g., an isolated Fe atom on an Ag(111) surface); 4) a superconductor surface with a known state density near the superconducting bandgap.

[0038] In some specific embodiments, the reference standard is often selected from the differential conductivity spectrum measured at a wide step on the Ag(111) surface, or from the differential conductivity spectrum of isolated impurity iron atoms measured on a wide step on the Ag(111) surface. Ideally, such features possess a well-defined local density of states. .

[0039] Step 2: Obtain the reference spectrum.

[0040] Using the tip of a scanning tunneling microscope (STM), a constant-height mode scanning tunneling spectrum (STS) measurement was performed above a reference feature to obtain the reference differential conductivity spectrum. ).

[0041] The scanning tunneling spectroscopy (STS) measurement mainly includes the following procedures: First, establish the tunnel junction and set the tunneling conditions. Position the tip above a reference feature (such as a clean metal substrate surface or a specific known adsorbed atom), adjust the STM's feedback loop, and set the specific tunneling conditions.

[0042] The tunneling conditions mainly include the preset tunnel junction bias voltage and tunneling current. The setting of the tunneling conditions determines the distance between the tip of the needle and the reference feature surface in the vertical direction (Z direction) (i.e., the tunnel junction spacing), and it is necessary to ensure that the measurement is within the characteristic energy range of the reference feature or a specific interaction distance.

[0043] Then, the feedback loop is locked to enter constant-height mode. After the tunneling conditions described above are met, the STM feedback loop is temporarily shut down. At this point, the tip position in the vertical direction (Z direction) is physically locked and no longer automatically adjusts with changes in the local density of states or morphological fluctuations on the sample surface, thus entering constant-height mode. Finally, scanning tunneling spectroscopy (STS) measurements are performed in constant-height mode to obtain the differential conductivity spectrum data of the reference feature, i.e., the reference differential conductivity spectrum (…). ).

[0044] Understandably, the main purpose of using the constant height mode is to eliminate the influence of tip height changes on the differential conductance signal during the measurement of a single differential conductance spectral line, thereby maintaining a constant tunnel junction spacing (d). In constant height mode, the tunnel junction spacing is typically maintained within the range of 0.1~1nm (including the endpoint value). Measurements within this range can avoid direct contact between the tip and the sample due to excessively small spacing, and also avoid the problem of excessively weak differential conductance signals due to excessively large spacing.

[0045] Step 3: Deconvolution and Solve .

[0046] Construct an optimization problem and utilize known... and measured Solve for the tip-related tunneling function .

[0047] In this step, the first step is to define an effective tip-related tunneling function ( In physics, Density of needle tip states ( ) and tunneling probability containing complex needle-like structures The product of, i.e.: (1) In the formula, For energy, This refers to the effective distance between the needle tip and the sample.

[0048] Understandably, under the small bias approximation, the tunneling probability... Energy in Both the bias voltage V and the voltage V can represent numerical values ​​of energy. For example, applying a bias voltage of V[V] corresponds to... ,exist Unit of The unit of V is V, and the two are numerically equal. Therefore, T can be approximated as... Its physical essence is through effective distance The effect of the bias voltage V is equivalently absorbed, thus allowing the measurement process of the differential conductance spectrum to be described by the convolution integral equation.

[0049] Under this approximation, the differential conductivity spectrum and the tunneling probability ( ) and the density of states of the reference feature or the sample to be tested Based on the small bias approximation model, the following convolution relationship is satisfied: (2) In the formula, For reference characteristics or the differential conductivity spectrum of the sample to be tested, For elementary charge, For bias voltage, For reference characteristics or the density of states of the sample to be tested, This is the effective tip-related tunneling function. W can be set as the reference feature R or the sample S to be tested.

[0050] Based on the known reference density of states ( ) and the measured reference differential conductance spectrum ( By solving the above convolution integral equation (2), the optimal effective tip-related tunneling function can be obtained. ).

[0051] It should be noted that in the convolution integral equation (2), if ,at this time and The measured differential conductance spectrum and local density of states, representing the reference characteristics respectively, can be used to derive the effective tip-related tunneling function through the convolution integral equation (2). .like ,at this time, and The measured differential conductivity spectrum and local density of states of the sample under test are represented by the following parameters: When obtained The convolution integral equation (2) can be further used to calculate the local density of states of the sample under test. .

[0052] This solution process can be achieved by constructing and solving an optimization problem that includes a regularization term to handle the ill-conditioned characteristics of the inversion. Specifically, it can be determined by minimizing the following loss function. When the loss function value When it is at its minimum, the corresponding This is the required function.

[0053] Understandably, to achieve numerical solutions, the continuous convolution integral equation (2) needs to be discretized. Accordingly, the loss function formula is as follows: (3) In the formula, The value of the loss function; The discretized vector of the reference differential conductance spectrum contains The sampling points are uniformly distributed and of an odd number, covering a symmetrical measurement bias range [-W, W], for example, -0.3 to 0.3 V. The value depends on the specific measurement requirements and is not explicitly limited, but is generally between 50 and 200. It is based on the reference density of states And based on the convolution integral equation (2), a convolution operator matrix is ​​pre-constructed; The regularization parameter is usually... ; for OK, The forward difference matrix of a column is defined as a numerical matrix in which the elements on the lower quasi-diagonal are all 1, the elements on the upper quasi-diagonal are all -1, and the other elements are all 0.

[0054] In the formula for this loss function, For data fitting terms; This is a regularization term (Tikhonov-Phillips) that can be used to ensure the smoothness of the constrained solution.

[0055] The construction process of the convolution operator matrix A is as follows: Specifically, under the small bias approximation model adopted in this invention, the integral is discretized using the trapezoidal rule, and the derivative is processed using the piecewise difference method.

[0056] According to formula (2) It can be represented as: (4) For any function in the formula that needs to be differentiated Both forward and backward differencing methods are used at the boundaries, while a two-point central difference scheme is used at the interior points to determine their partial derivatives. The specific form is as follows, among which : (5) In the formula, For the function that needs to be differentiated, Index of discrete points For function In the index discrete values ​​at that point It's a long walk away.

[0057] After applying the derivative using the difference method, the trapezoidal integral method is further used to obtain... Discretized expression: (6) Among them, trapezoidal integral weights The value can be: (7) This yields the matrix-form expression. Convolution operator matrix The elements in are: (8) in,

[0058] In the formula, , For the row and column indices of the matrix, For reference characteristic state density in The value at that location, For reference characteristic state density in The derivative value at that point, It is a symbolic function.

[0059] Understandably, this is because the reference differential conductance spectrum ( Inverting the effective tip-related tunneling function ( The deconvolution problem is ill-conditioned and has multiple solutions. This invention introduces a loss function with a regularization term, using prior physical knowledge of the smoothness of the response function as a constraint. The optimization process uses an iterative algorithm to find the solution that best fits the data (i.e., the data fitting term) among many possible solutions. Minimum) and satisfying the smoothness requirement (i.e.) The unique optimal solution (minimum), that is, the required solution. .

[0060] The optimization problem can be solved using the L-BFGS-B iterative algorithm, a quasi-Newton iterative algorithm for solving large-scale, nonlinear, nonconvex optimization problems that supports variable boundary constraints. Alternatively, in other embodiments, other iterative optimization algorithms such as BFGS and Newton-CG can also be used.

[0061] It is worth noting that this invention introduces Tikhonov-Phillips regularization (TPR) to impose physically reasonable constraints on smoothness, thereby achieving smoothness and physical reasonableness by minimizing the loss function. This effectively solves the ill-conditioned nature of the inversion. The resulting effective tip-correlated tunneling function ( ), which includes not only the density of states at the needle tip ( It also implicitly includes the influence of tunnel barrier shape, needle geometry, and orbit on tunneling probability.

[0062] Part Two: Restoring the Sample Density of States Step 4: Obtain the sample spectrum.

[0063] Keeping the tip structure and tunneling conditions unchanged, the tip is moved above the sample to be tested, and scanning tunneling spectrum measurement is performed on the sample to obtain the sample's differential conductivity spectrum. ).

[0064] It should be noted that the scanning tunneling spectrum measurements in steps 1 and 4 above were performed in constant height mode, meaning the tunnel junction spacing remained constant, and both satisfied the physical characteristics of the differential conductivity spectrum of the sample being measured, where the energy scale was greater than the thermal broadening scale. and AC modulation voltage conditions.

[0065] Step 5: Repeat the measurement of the reference spectrum.

[0066] Repeat step 1 to obtain a new reference differential conductance spectrum. Repeated measurements can be obtained. The measurements obtained in step 1 A comparison is made to determine if the electronic structure of the needle tip has changed. If the electronic structure of the needle tip has not changed, proceed to the next step. If the electronic structure of the needle tip has changed, the measurement needs to be repeated starting from step 2.

[0067] It is important to note that by comparing the differential conductance spectra measured twice on the same reference feature, it is possible to monitor whether the tip density of states has undergone significant changes during the measurement process due to external factors such as contamination, atomic reconstruction, or magnetic fields. This ensures the accuracy of the calibration in step 3. This step remains valid in subsequent sample measurements and is a crucial quality control step to ensure the reliability of the entire method. However, it should be noted that the electronic structure of the needle tip generally does not change; therefore, this step is not mandatory and can be set periodically or selected according to actual needs. If changes occur during repeated measurements, simply start measuring again from step 2.

[0068] Step 6: Deconvolution inversion to retrieve the density of states of the sample under test ( ).

[0069] Using the effective tip-related tunneling function obtained in step 3 ( ) and the differential conductivity spectrum of the sample to be tested measured in step 4 ( Based on the convolution integral equation (2), the density of states of the sample to be tested can also be obtained by solving an optimization problem containing a regularization term. ).

[0070] Similarly, this process can be implemented using a loss function. Likewise, to achieve numerical solutions, the continuous convolution integral equation (2) needs to be discretized. Accordingly, the loss function formula is as follows: (9) In the formula, The value of the loss function. Let be the discretized vector of the sample's differential conductance spectrum. It was determined based on step 3. And based on the convolution integral equation (2), a convolution operator matrix is ​​pre-constructed. Let be the sample density of states to be solved. For regularization parameters, for OK, The forward difference matrix of the columns.

[0071] The construction process of convolution operator matrix B is basically the same as that of convolution operator matrix A in step 3. Based on this construction process, we can obtain: (10) In the formula for this loss function, For data fitting terms, the goal is to make the data based on... and The difference between the theoretical sample differential conductivity spectrum obtained by model calculation and the measured sample differential conductivity spectrum is minimal; This is a regularization term used to constrain the smoothness and other physical plausibility of the solution. The optimization problem can also be solved iteratively using algorithms such as L-BFGS-B, BFGS, and Newton-CG to find data-fitting terms. Minimum and regularized term The smallest unique optimal solution, which is also the density of states of the sample. .

[0072] The invention will be further described below with reference to specific experimental data and accompanying drawings to demonstrate its feasibility and effectiveness.

[0073] In one specific embodiment, the surface artificial nanostructure density measurement method of the present invention was applied to the scanning tunneling spectrum measurement and deconvolution experiment of artificial quantum fences and their internal adsorbed atoms. The results showed that the present invention has stable repeatability under different needle tip structures.

[0074] In Experiment 1, the known step function state density of the Ag(111) surface states was selected as the reference feature. The broadened arctangent function was used, combined with the volume contribution, to calculate its specific value. The results are as follows: Figure 2 (c) is shown by the solid black line.

[0075] To verify whether the method in this patent can be applied to different tip structures and different sample local state densities, Experiment 1 used the following procedure to obtain all the parameters required for the deconvolution algorithm: Step 1: In an ultra-high vacuum environment, after cooling the clean Ag(111) substrate to a temperature of approximately 4.7 K, iron atoms are deposited; Step 2: Set the tunnel junction bias voltage to -0.3 V and the tunneling current to 1 nA. The scanning tunneling microscope automatically adjusts the distance between the tip and the sample to establish a stable tunnel junction and begins scanning the surface morphology of the sample. Step 3: Find such Figure 2(a) shows a broad Ag(111) platform surface. Using atomic manipulation techniques, three circular quantum fence structures with fence radii of 3.2 nm, 5.1 nm, and 9.2 nm were simultaneously constructed within the same atomically flat region, as shown in the diagram. Figure 2 As shown in (d), (g), and (j); Step 4: Collect scanning tunneling spectra in sequence at the wide Ag(111) platform surface, the center of the quantum fence with a radius of 3.2 nm, the center of the quantum fence with a radius of 5.1 nm, and the center of the quantum fence with a radius of 9.2 nm. Monitor the measurement data to ensure that there are no abnormal jumps and that the morphology of each atom in the topography does not change significantly during the measurement process. This is to ensure that the tip structure does not change during the entire measurement process. During the measurement process, the scanning tunneling microscope will automatically lock the feedback loop to enter the constant height mode and adjust the tunnel junction bias voltage from -0.3 V to measure the differential conductance value of each of the 201 sampling points uniformly distributed in the range of -0.3 to 0.3 V, thus forming scanning tunneling spectrum data. Step 5: By applying a voltage pulse with an amplitude of 10V and a duration of 100 milliseconds, the tip structure is changed, and the scanning tunneling spectrum of the broad Ag(111) plateau surface is measured and compared with the previous data to ensure that the tip structure has changed; Step 6: Collect scanning tunneling spectra again in sequence at the surface and the center of the quantum fence, and monitor that the tip structure does not change during the measurement process; Step 7: The tip structure is changed for the second time by applying a voltage pulse; the scanning tunneling spectrum at the surface and the center of the quantum fence is collected again in sequence, and the tip structure is monitored to ensure that it does not change during the measurement process.

[0076] Through the above process, that is, using three needle-tip structures to measure four different objects, such as Figure 2 As shown in (e), (h), and (k), a total of twelve scanning tunneling spectrum data were obtained, and deconvolution experiments were conducted.

[0077] The known step function state density of Ag(111) surface states and the experimentally measured scanning tunneling spectrum of the broad Ag(111) platform surface are input into the deconvolution program constructed based on the two-step deconvolution algorithm described in this patent to solve the tip-related tunneling function.

[0078] To verify the effectiveness of the tip-correlated tunneling function, the solved tip-correlated tunneling function was used to deconvolve the scanning tunneling spectrum of the same surface state measured in the previous experiment. The results are as follows: Figure 2 As shown in (c), it can be seen that the deconvolution results of the measured scanning tunneling spectrum all coincide with the black reference line, indicating that the first step of determining the tip-related tunneling function has been completed.

[0079] Subsequently, based on the tip-correlated tunneling function and the scan tunneling spectra measured at the centers of the three quantum fences, the density of states at the centers of the three quantum fences was inverted using deconvolution. For quantum fences of different radii, the density of states at their centers should exhibit a series of distinct peaks, which can be observed in the measured scan tunneling spectra. Figure 2 (e) Taking the data shown as an example, the measurement results of the three tips all show peaks at bias voltages of approximately 0.0 V and 0.25 V. However, the specific values ​​of the three curves given by the three tips at the same bias voltage are different. At 0.0 V, the differential conductance spectrum measurement values ​​corresponding to tips #1 and #2 are approximately 15 nS, but the differential conductance spectrum measurement value corresponding to tip #3 is close to 20 nS, which is significantly different. Figure 2 (f) shows the results of deconvolution of the measurements from the three tips. It can be seen that the deconvolution results of the three tips almost overlap within the measured bias voltage range. At the same time, in the range of -0.065 to 0.3 volts, the deconvolution results of the three different tips are almost consistent with the theoretical density of states (black solid line) calculated using the multiple scattering Green's function method, indicating that the deconvolution algorithm is stable and repeatable.

[0080] Corresponding to Figure 2 The different measurement results for different fence radii in (e), (h), and (k) all yielded results in the deconvolution procedure that were very consistent with the theoretical characteristics (i.e., the theoretical density of states), such as... Figure 2 As shown in (f), (i), and (l), this demonstrates that the deconvolution algorithm can indeed recover the high-fidelity sample local state density.

[0081] In Experiment 2, keeping the experimental conditions unchanged, the density of states of isolated iron adsorbed atoms was chosen as the reference characteristic. Figure 3 (b) The black solid line indicates that the density of states was obtained by fitting an analytical model established in published literature (e.g., L. Limot, E. Pehlke, J. Kröger, and R. Berndt, Surface-state localization at adatoms, Physical Review Letters 94, 036805 (2005).). On the broad Ag(111) platform surface and at the center of a quantum fence of a specific size (radius 5.1 nm) prepared in preparation, a single iron atom was precisely placed (as shown in the figure). Figure 3(a) and (c) are shown in the upper right corner). Similar to the procedure in Experiment 1, the differential conductivity spectra of the iron single atom at the center of the fence were collected using scanning tunneling microscope tips with two different tip structures on this nanostructure. The differential conductivity spectra of the central atom obtained under the two different tip structures were input together with the differential conductivity spectra measured on the reference feature (isolated impurity iron atoms adsorbed on the wide steps of the Ag(111) surface) into the deconvolution algorithm described in this invention for processing.

[0082] The results are as follows Figure 3 As shown in (b) and (d), although the original differential conductivity spectra show significant differences due to variations in the needle tip structure, the results obtained after deconvolution processing are in high agreement. Compared with the density of states characteristics of isolated iron atoms, the iron atom at the center of the 5.1 nm radius fence exhibits two density of states peaks around 0 V and 0.15 V, as well as a weak feature at approximately -0.065 V. This result demonstrates that the two-step deconvolution algorithm proposed in this invention is stable and repeatable, and can recover the weak density of states characteristics of the sample, exhibiting high fidelity.

[0083] It can be noted that Experiment 1 and Experiment 2 selected different reference features. Both deconvolution algorithms gave stable, repeatable and high-fidelity results, indicating that they have a wide range of applications and are not limited to specific material systems. The reference features can be selected in various forms such as surface states and isolated atoms, which are suitable for studying various artificial nanostructures and surface quantum phenomena.

[0084] Therefore, applying the surface artificial nanostructure density of states measurement method based on the two-step deconvolution algorithm described in this invention to experimental data of artificial quantum fences and their internal adsorbed atoms shows that it can recover high-fidelity sample local density of states under different tip structures, and possesses stability and repeatability, revealing subtle modulation of the resonance states of adsorbed atoms in quantum-confined systems. Furthermore, by extracting high-fidelity density of states, this method provides a powerful and repeatable data processing tool for studying artificial nanostructures and related quantum phenomena.

[0085] Finally, it should be noted that although the embodiments of the present invention have been described above in conjunction with the accompanying drawings, the present invention is not limited to the specific embodiments and application fields described above. The specific embodiments described above are merely illustrative and instructive, and not restrictive. Those skilled in the art, guided by this specification, can make many other forms without departing from the scope of protection of the claims of the present invention, and all of these are within the scope of protection of the present invention.

Claims

1. A method for measuring the density of states of artificial nanostructures on surfaces based on a two-step deconvolution algorithm, characterized in that, include: Select a reference feature with a known density of states within a preset energy window; Using the tip of a scanning tunneling microscope, the reference feature is scanned and tunneled in constant height mode within the preset energy window to obtain the reference differential conductivity spectrum. Using the reference density of states of the aforementioned reference feature and the measured reference differential conductance spectrum, an effective tip-related tunneling function is obtained by solving a first optimization problem that includes a regularization term. Keeping the measurement conditions unchanged, the sample to be tested is subjected to scanning tunneling spectrum measurement in the preset energy window and constant height mode to obtain the sample differential conductivity spectrum. The measurement conditions include the tip structure and tunneling conditions. Using the effective tip-correlated tunneling function and the sample differential conductivity spectrum, the density of states of the sample under test within the preset energy window is obtained by solving a second optimization problem including a regularization term. The first optimization problem and the second optimization problem are both constructed based on the same physical model describing the convolution relationship between the differential conductance spectrum and the density of states; the effective tip-related tunneling function is the product of the tip density of states and the tunneling probability containing the tip structure.

2. The method for measuring the density of artificial nanostructure states on a surface as described in claim 1, characterized in that, The reference feature is selected from any of the following: a noble metal surface with a known surface state density; a two-dimensional electron gas system with a known surface state density other than a noble metal surface; a single isolated atom adsorbed on an atomically flat platform; or a superconductor surface with a known state density near the superconducting bandgap.

3. The method for measuring the density of artificial nanostructure states on a surface as described in claim 1, characterized in that, The physical models underlying the first and second optimization problems correspond to the following convolution integral equations: In the formula, For reference characteristic R or the differential conductivity spectrum of the sample S to be tested, For elementary charge, For bias voltage, For the reference characteristic R or the density of states of the sample S, This is an effective tip-related tunneling function.

4. The method for measuring the density of artificial nanostructure states on a surface as described in claim 3, characterized in that, The first optimization problem is solved by minimizing the following first loss function value. Determine the effective tip-related tunneling function : In the formula, The discretized vector of the reference differential conductance spectrum; It is based on the reference density of states And a convolution operator matrix pre-constructed based on the convolution integral equation; This is the first regularization parameter; It is a forward difference matrix.

5. The method for measuring the density of artificial nanostructure states on a surface as described in claim 3, characterized in that, The second optimization problem is to minimize the following second loss function value. To determine the density of states of the sample to be tested. : In the formula, This is the discretized vector of the sample's differential conductance spectrum; It is based on the above And a convolution operator matrix pre-constructed based on the convolution integral equation; This is the second regularization parameter; It is a forward difference matrix.

6. The method for measuring the density of artificial nanostructure states on a surface as described in claim 1, characterized in that, The first and second optimization problems are solved using an iterative optimization algorithm, which is selected from any one of L-BFGS-B, BFGS, or Newton-CG.

7. The method for measuring the density of states of artificial nanostructures on a surface as described in any one of claims 1 to 6, characterized in that, Also includes: After obtaining the differential conductivity spectrum of the sample and before inverting the density of states of the sample to be tested, the constant height mode scanning tunneling spectrum measurement is performed again on the reference feature surface to obtain a new reference differential conductivity spectrum. This new reference differential conductivity spectrum is then compared with the initially measured reference differential conductivity spectrum to verify that the electronic structure of the tip has not changed. After verifying that the electronic structure of the tip has not changed, the inversion of the density of states of the sample to be tested is then performed. If the electronic structure of the tip has changed, the measurement is restarted from measuring the reference differential conductivity spectrum.

8. The method for measuring the density of states of artificial nanostructures on a surface as described in any one of claims 1 to 6, characterized in that, The preset energy window is centered on the Fermi level. The energy window range.

9. The method for measuring the density of states of artificial nanostructures on a surface as described in any one of claims 1 to 6, characterized in that, The tunnel junction spacing in the constant height mode is 0.1~1nm.

10. An electronic device, characterized in that, It includes a processor and a memory; the processor is used to call a computer program stored in the memory to execute the surface artificial nanostructure state density measurement method based on a two-step deconvolution algorithm as described in any one of claims 1 to 9.

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