Method for determining the spatial distribution of current density in a two-dimensional material device

Abstract

According to the present invention, the current density distribution in an electronic device is determined, which comprises a first and a second electrode and a layer of two-dimensional conductive material extending between the first and the second electrode. The total current through the electrodes is measured when the device is in operation, and then a first current measurement probe is placed at a number of positions as close as possible to the interface between the two-dimensional material and the first electrode. The probe is coupled to the same voltage as the first electrode, thereby locally shunting the current. The same is done at the interface between the channel and the second electrode. In this way boundary conditions are determined for the current, and assuming that the current density vector is orthogonal to the interface, this yields a boundary condition for the current density vector. Finally, the continuity equation is solved taking the boundary conditions into account.

Description

Invention Field

[0001] The present invention relates to semiconductor devices comprising two-dimensional conductive materials, and more particularly to a method for determining the current density in the two-dimensional material when the device is in operation. Background Technology

[0002] While the theory of carrier transport in solid-state materials in the absence of defects is well established, experimental observations often differ from predictions when non-ideal conditions exist in devices or measurement settings, especially for nanoscale materials which are more sensitive to defects and environmental conditions. This can be observed in the relatively large variations in the electrical properties of conductive 2D materials (such as graphene) or TMDC materials (transition metal dichalcogenides) (such as MoS2), which are dependent on a large number of parameters, such as grain size, impurity density, etc. Understanding how these parameters affect electronic conduction would be greatly beneficial if current density could be measured at grain boundaries or impurities. However, conventional electrical characterization only measures macroscopic properties (e.g., the total current flowing between electrical contacts to an operating device), not the magnitude and direction of current density at points of interest in 2D materials. Therefore, only average values ​​of transport properties such as carrier mobility can be determined. Summary of the Invention

[0003] The objective of this invention is to provide a solution to the aforementioned problems. This objective is achieved by the method according to the appended claims. According to the invention, a current density distribution in an electronic device is determined, the electronic device comprising first and second electrodes and a two-dimensional conductive material layer extending between the first and second electrodes. When the device is in operation, i.e., when a bias voltage is applied between the electrodes, the total current passing through the electrodes is measured, and then a first current measuring probe is placed at multiple locations as close as possible to the interface between the two-dimensional material and the first electrode. This probe is coupled to the same voltage as the first electrode, thereby locally shunting the current. The same operation is performed at the interface between the channel and the second electrode by placing a second probe, possibly identical to the first probe, coupled to the same voltage as the second electrode. According to one embodiment, the pressure applied to the probe at both interfaces is such that the sum of the currents passing through the probe and the corresponding electrodes is substantially equal to the current passing through the corresponding electrodes if the probes were not placed on the two-dimensional material. In this way, boundary conditions for the current are determined when the device is in operation, and a current density vector is assumed. Orthogonal to the interface, this generates the boundary conditions for the current density vector. Finally, the continuity equation is considered in light of these boundary conditions. Please provide a solution.

[0004] The present invention particularly relates to a method for determining a current density distribution in a microelectronic device, the microelectronic device including first and second electrodes and a two-dimensional conductive material layer extending between the first and second electrodes. The method includes the following steps, performed when the device is in operation, i.e., when a bias voltage is applied between the electrodes to cause charge carriers to flow from the first electrode to the second electrode or vice versa in the two-dimensional material, and wherein at the interface between the two-dimensional material and the electrodes, a current density vector is... Basically orthogonal to the stated boundary direction,

[0005] - Measure the total current passing through the first and second electrodes without a measuring probe being placed on the two-dimensional material.

[0006] - A first current measuring probe is placed as close as possible to a first electrode at a plurality of first locations along the interface between the first electrode and the two-dimensional material, wherein the first probe is coupled to the same voltage as the first electrode, and wherein the first probe at each of the first locations is configured to locally shunt the current passing through the first electrode.

[0007] - Determine the current passing through the first probe at the first position, and derive a first set of current density values ​​from it, which represent the boundary conditions of the current density vector in the two-dimensional material at the interface between the first electrode and the two-dimensional material.

[0008] - A second current measuring probe, which may be the same as or different from the first probe, is placed as close as possible to the second electrode at multiple second locations along the interface between the second electrode and the two-dimensional material, wherein the second probe is coupled to the same voltage as the second electrode, and wherein the second probe at each of the second locations is configured to locally shunt the current passing through the second electrode.

[0009] - Determine the current passing through the second probe at the second position, and derive a second set of current density values ​​from it, which represent the current density in the two-dimensional material (3) at the boundary condition of the interface between the second electrode and the two-dimensional material.

[0010] -The equations taking the aforementioned boundary conditions into account Solve to obtain the current density distribution in the two-dimensional material.

[0011] According to one embodiment, the method includes the following features:

[0012] - When the first probe is placed in the first position, pressure is applied to the first probe such that the sum of the currents passing through the first electrode and the first probe is substantially equal to the total current passing through the first electrode when no probe is present on the two-dimensional material.

[0013] - When the second probe is placed in the second position, pressure is applied to the second probe such that the sum of the currents passing through the second electrode and the second probe is substantially equal to the total current passing through the second electrode when no probe is present on the two-dimensional material.

[0014] According to one embodiment, a first predefined pressure is applied to a first probe at each of the first positions and a second predefined pressure is applied to a second probe at each of the second positions.

[0015] According to one embodiment, the first probe is formed of the same material as the first electrode and / or the second probe is formed of the same material as the second electrode.

[0016] According to one embodiment, a single probe is used as both the first and second probes.

[0017] According to one embodiment, the interface between the electrode and the two-dimensional material forms a closed boundary.

[0018] According to one embodiment, the electrodes are rectangular, oriented parallel to each other, and spaced apart by a distance L. The length of the interface corresponds to the width W of the device, and L is much smaller than W.

[0019] According to one embodiment, the equation determines the vector The initial guess and by updating the initial guess until The solution is obtained by fitting the previously determined boundary conditions to a numerical solution.

[0020] According to one embodiment, the method further includes using coordinates (x, y) in a two-dimensional material layer. m y n Nanopotential measurements were performed at multiple points defined by the equation. Written as:

[0021] in:

[0022] -σ(x m y n ) is the effective value of the position-dependent conductivity in this channel, and E F It is the position-dependent quasi-Fermi level in this channel.

[0023] - and It was obtained from nanopotential measurements.

[0024] -σ(x m y n ) is used as a fitting parameter to be used based on the relation, taking into account previously determined boundary conditions. To determine the current density. Attached Figure Description

[0025] Figure 1a and 1b A measuring apparatus for measuring a first boundary current near a first electrode is shown according to an embodiment of the present invention.

[0026] Figure 1c , 1d Figures 1 and 1e show the current in the region of interest when the probe is placed near the first electrode.

[0027] Figure 2a and 2b A measuring apparatus for measuring a second boundary current near a second electrode is shown according to an embodiment of the present invention.

[0028] Figure 2c , 2d Figures 2e and 2e show the current in the region of interest when the probe is placed near the second electrode.

[0029] Figure 3 A flowchart of a method according to a first embodiment for obtaining the current density in a two-dimensional material by numerically solving a continuity equation.

[0030] Figure 4 This demonstrates how nanopotential metrology can be applied to determine quasi-Fermi levels in two-dimensional materials according to embodiments of the invention.

[0031] Figure 5 A flowchart is shown of a method for obtaining the current density in a two-dimensional material by numerically solving a continuity equation, according to one embodiment, wherein the QFL was previously measured by nanopotentialometry.

[0032] Figure 6a This illustrates how the method of the present invention can be applied to a two-dimensional material extending between two parallel electrodes.

[0033] Figure 6b and 6c It shows Figure 6a Method steps for measuring boundary current in a measuring device. Detailed Implementation

[0034] Figures 1a-1eFigures 2a-2e illustrate a measuring apparatus for performing a method according to an embodiment of the invention, the method being applied to a device including a first electrode 1 and a second electrode 2. The first electrode 1 is a circular electrode placed at the center. The second electrode 2 is annular and arranged around the first electrode 1. The invention is not limited to this particular device architecture, which is used as an example to explain the characteristic steps of the method.

[0035] Electrodes 1 and 2 are placed on a two-dimensional conductive material layer 3. This can be, for example, a two-dimensional semiconductor material, such as MoS2. Layer 3 is presented in cross-section as having a distinguishable thickness, which is merely one way of visualizing the layer. In reality, the two-dimensional layer 3 is very thin. The thickness of layer 3 can be based on known prior art devices. The device can be a transistor, where electrodes 1 and 2 are the corresponding drain and source electrodes of the transistor, and the portion of the two-dimensional layer 3 extending between the electrodes is the channel of the transistor. The transistor gate is not shown for the purpose of not complicating the figures. However, the invention is not limited to transistors. The term 'channel' is used in the specific embodiments but should not be construed as limiting the scope of the invention. The two-dimensional layer 3 is present on a dielectric layer 4, which in turn covers the substrate 5. The dielectric layer can be a SiO2 layer 4 deposited on a silicon substrate 5.

[0036] When the device is in operation, a DC bias voltage V is applied between electrodes 1 and 2. DS In the illustrated embodiment, the first electrode 1 is coupled to a DC source configured to deliver the bias voltage, while the second electrode 2 is coupled to ground. However, the bias voltage can be applied in any manner known in the art and relative to any reference voltage. The current direction in the channel is considered to be from the first (center) electrode 1 outwards to the second ring electrode 2.

[0037] The measuring apparatus further includes a conductivity current measuring probe 10, which may be a probe known from scanning probe microscopy techniques such as conductivity AFM (atomic force microscopy) or EFM (electric power microscopy). The probe has a conductive probe tip mounted on a cantilever. A circuit system is integrated into the cantilever for conducting current through the tip when the tip is positioned in physical contact with a conductive surface.

[0038] The following measurements are performed to determine the boundary currents at the perimeters of electrodes 1 and 2. The measuring apparatus for electrode 1 is... Figure 1a and 1b As shown in the diagram, a current measuring device (hereinafter referred to as a meter) A1 is coupled to a voltage source V. DS Between the first electrode 1 and the second meter A2, which is coupled to probe 10 and configured to measure the current passing through probe 10, the second meter A2 is coupled to bias voltage V. DSThat is, the probe is biased to the same voltage as the first electrode 1.

[0039] When probe 10 is not in contact with the channel, such as in Figure 1a Under the conditions shown, meter A1 measures the current. That is, the total current passing through the first electrode 1 when the probe 10 is not placed on the two-dimensional material. The value is recorded.

[0040] Then, the probe 10 is placed on the channel as close as possible to the interface between the first electrode 1 and the channel, such as... Figure 1b As shown, the interface is defined by the perimeter of the first electrode 1. Placing the probe at the same bias voltage as the first electrode 1 and positioning the probe as close as possible to the electrode 1 on the two-dimensional material has the effect of locally shunting the current path, such as... Figure 1c and 1d As shown. Consider point P1 (index 1 refers to the first electrode 1) along the perimeter of the electrode. The probe tip 12 will be placed on this channel as close as possible to point P1. When the probe is positioned in this manner, the probe tip 12 affects the current passing through a length d extending along the perimeter. The value of d is primarily determined by the tip size, which can be defined or approximated as the diameter of the tip when the tip is equal to or approximately hemispherical.

[0041] To enable localized current shunting, the probe tip size is larger than the transmission length of the contact resistance between the probe tip 12 and the two-dimensional material. This ensures that when the probe is placed on the two-dimensional material, a large portion of the current flows through the probe and not through the two-dimensional material in contact with it. This transmission length is defined as the ratio between the contact resistivity (in Ωm) and the sheet resistivity of the two-dimensional material (in Ω). The material of the probe 10 (i.e., the probe tip in contact with the two-dimensional material) is preferably the same as the material of the first electrode 1. This minimizes the difference between the contact resistivity between the probe and the two-dimensional material and the contact resistivity between the two-dimensional material and the electrode 1. Typically, the materials of the first and second electrodes are the same so that a single probe 10 can be used. If the second electrode 2 has a different material than the first electrode 1, a different probe (preferably with the same material as the second electrode) can be used to measure the boundary current at the second electrode (see also).

[0042] When the probe tip is not in contact with the channel (e.g.) Figure 1a and 1c As shown), current I P1 The current flows through a length d. Placing the probe as close as possible to point P1 in contact with this channel shunts the current through the d-channel, resulting in a current I′. P1 After passing through the probe (see Figure 1dThe current is measured by meter A2, while meter A1 measures the total current passing through the first electrode 1. (That is, the total current flowing through electrode 1 when the probe is positioned close to point P1). All of these currents flow into the two-dimensional material 3. Current I′ P1 This can be influenced by adjusting the pressure on the probe. The pressure is now adjusted until… When this is the case, the current through probe 10 is a good approximation of the current through length d when the probe is not present, i.e., I′ P1 Basically equal to I P1 I' P1 This value is taken as I P1 The measured value represents the boundary condition of the current passing through the channel at point P1 when the device is in operation.

[0043] I in the above manner P1 The measurement was then performed at multiple points P along the perimeter of the first electrode 1. 1i Execution at the location, such as Figure 1e As shown. At each point, the probe is placed as close as possible to the interface between the first electrode 1 and the channel, so that the same length d is affected by the probe at each point, and the pressure is adjusted until... In this way, at a distance Δ1 1i N points P (i = 1, ..., N) distributed along the perimeter of the first electrode 1 1i N currents I are obtained at (i = 1, ..., N). P1i (i = 1, ..., N). Preferably, these points are regularly distributed such that the distance Δ1 1i They are basically equal.

[0044] Then, the boundary current at the interface with the second motor 2 is used Figure 2a and 2b The device is used for measurement. Meter A1 is now coupled between the second motor 2 and the ground reference. Probe 10 (in this case, the same probe as in the first measuring device) is now also coupled to the ground reference, and meter A2 remains coupled to this probe for measuring the current passing through it. When the probe is not in contact with the channel ( Figure 2a Meter A1 measures the total current. In theory, equal However, slight differences may exist due to factors such as current leakage. According to some embodiments, for example, if the leakage current is known to be very small, then a probe can be used without it. Figure 1a or Figure 2a The device measures the total current passing through electrodes 1 and 2 only once.

[0045] The probe 10 is then positioned close to the interface between the channel and the second electrode 2, as in Figure 2b Shown in and in Figure 2c and 2d This is shown in more detail below. The probe is positioned as close as possible to point P2 on the perimeter of the second electrode 2, thereby affecting the current through a length d of said perimeter. In the case where the probe does not contact the channel at that point ( Figure 2c ), current I P2 The current flows through a length d. When the probe is placed on the channel, the current also flows because the probe 10 contacts the channel at that position, causing the current I′ to flow. P2 After the probe ( Figure 2d The current I′ is diverted. P2 Measured by meter A2, while measured by meter A1. That is, the total current passing through the second electrode 2 when the probe approaches P2 and contacts the channel.

[0046] Similar to the first electrode 1, the pressure on this probe is adjusted until... That is I′ P2 Basically equal to I P2 This is how to determine I P2 , the I P2 This represents the boundary condition at point P2 for the current flowing through the channel when the device is in operation. The probe is then placed at multiple (M) points P along the perimeter of the second electrode. 2i These points are located at (I = 1...M). They are spaced apart by a distance Δ1. 2i (i = 1...M), preferably all of these distances are equal to the same value. The repetition of the above pressure adjustment thus generates the measured boundary current I. P2i (I = 1...M).

[0047] In the next step, the current density in the channel is determined. The current density is a two-dimensional vector within the channel. J x and J yThe coordinates x and y in the orthogonal axis system shown in the accompanying figures are used to determine the bias. Under low field and diffusion, low current density conditions, i.e., under low to moderate bias between the first and second electrodes 1 and 2, the quasi-Fermi level (QFL) on the electrodes can be assumed to be constant such that the current density proportional to the gradient of the QFL is substantially orthogonal to the perimeter of the electrode. This assumption that the current density vector is orthogonal to the perimeter is necessary for carrying out the method of the present invention. As described, achieving this orthogonal orientation requires the application of a low to moderate bias. The value of such bias can depend on the material used, the device size, and the device architecture. Generally, and as is known in the art, a low to moderate bias is a bias that results in only a negligible potential drop within the electrode (i.e., a large portion of the voltage drop across the two-dimensional material). This is typically achieved when the electric field within the two-dimensional material is too small to induce ballistic transport effects such as the Fowler-Nordheimer tunneling effect or impact ionization. When the current density is orthogonal to the perimeter of the first electrode 1, the magnitude of the current density at point P1 on the boundary between the first electrode and the two-dimensional channel is determined by... Given. Due to the total current in the absence of probes. Therefore, we can conclude that:

[0048] For i = 1...N (1)

[0049] In the measured quantity I P1i and Δl 1i In this respect, this provides a reliable estimate of the magnitude of the current density at the perimeter of the first electrode 1. Assuming the current density vector is orthogonal to the perimeter of the first electrode 1 (i.e., the orientation of this vector is known), the vector... Therefore, at discrete point P along this perimeter 1i The location is known.

[0050] Similarly, the current density at the perimeter of the second electrode 2 is given by the following equation:

[0051] For i = 1...M (2)

[0052] Similarly assuming the current density vector is orthogonal to the perimeter of the second electrode 2, equation (2) provides the current density vector along said perimeter.

[0053] These boundary conditions for current density allow for the solution of the continuity equation, which expresses the law of continuity for current in any system under steady-state conditions:

[0054]

[0055] Equation (3) indicates that if the total charge remains the same, then the current density vector... The divergence is zero at all points. If the current density vector at the boundary of a given region is known, the current density vector in that region can be found by numerically solving this differential equation, a similar result to the uniqueness theorem in electrostatics. Therefore, the current density at all points of a two-dimensional channel can be numerically calculated if the current density at the boundary of the channel is known.

[0056] As described above and referring to Figures 1a-1e The boundary conditions (1) and (2) determined by 2a-2e allow for the numerical solution of equation (3) using finite element methods known in the art. Essentially, the continuity equation... The discrete point (x) is defined by the following operation. m y n The solution involves a network that divides the channel region into subspaces (e.g., triangles) and iteratively adjusts the size of these subspaces based on the geometry of the channel and its boundaries, as well as the complexity of the differential equations and boundary conditions. At each node of this network, an initial guess about the current density distribution is assumed. This is typically achieved through interpolation of boundary conditions at each node. The initial guess is iteratively modified until it reaches a point P on the boundary. 1i and P 2i The values ​​on the probe match those measured by probe 10. Figure 3 is an overview of the current density vector (J) used to obtain the current density vector in the channel. x J y The flowchart of the numerical method is shown below.

[0057] Depending on the level of detail in the approximately continuous channel region of the discrete-point network, the solution found for the density vector allows for determining the current density near a local defect of a given size. The achievable level of detail depends on the resolution of the boundary conditions, i.e., the point P measured along the electrode perimeter. 1i and P 2i The more points there are, the more points in the channel where the current density can be determined.

[0058] The above embodiments involve adjusting the pressure applied to the probe and measuring two currents passing through meters A1 and A2 at each probe location to determine the best possible approximation of the boundary current. Although this is a preferred embodiment, the scope of the invention is not limited thereto. According to another embodiment, a predefined pressure is applied to the probe, and the same pressure is applied at every point along the electrode perimeter (preferably different predefined pressures are applied to the two electrodes). For example, if it is known in advance that the channel includes several defects, this approximation will allow for a reliable estimate of the relative current density distribution (i.e., not the correct absolute value of the density, but the correct value relative to each other at different locations in the channel). In this case, the value J in equation (1) P1i It needs to be multiplied by a constant so that point P 1i The sum of the currents on the probe is equal to the currents on a probe that do not exist in a two-dimensional material (e.g., Figure 1a The total current measured by meter A1 coupled to the first electrode 1 in the case of (medium) Similarly, the value J in equation (2) P12 A constant needs to be multiplied so that the sum of the currents at point P2i equals the sum of the currents at points P2i when the probe does not exist in the two-dimensional material (e.g., ...). Figure 2a The total current measured by meter A2 coupled to the second electrode 2 in the case of (medium)

[0059] According to an alternative embodiment, an additional measurement is performed before solving the continuity equation (3): the direction of the current density on the channel is measured directly using nanopotentialometry. Under steady-state conditions for unipolar charge carriers (i.e., electrons or holes), the current density at any point is:

[0060]

[0061] Where σ is the effective value of the position-dependent conductivity, and E F This is the position-dependent QFL. The effective value of the position-dependent conductivity is a scalar function of x and y, thus, at each position in the channel, equation (4) indicates that the current density is proportional to the gradient of the QFL. The QFL in two-dimensional material devices can be measured by nanopotentialometry, such as... Figure 4As shown. Nanopotentiometry is well-known and is described, for example, in the paper “New aspects of nanopotentiometry for complementary metal-oxide-semiconductor transistors” published by Trenkler, Thomas et al. in Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures Processing, Measurement, and Phenomena 18.1 (2000): 586-594. Figure 4 As shown, probe 10 is scanned across channels while being connected to variable voltage source 15 and meter A2 to allow zero current through the probe, i.e., the probe is used as a zero current source or voltmeter: the voltage required to achieve zero current is the measured voltage V. 探针 V 探针 The measurements are taken at multiple discrete points across the channel, preferably by scanning the probe along multiple radial lines from electrode 1 to electrode 2. Under steady-state conditions, for the applied bias voltage V... DS At the discrete point (x) m y n The voltage measured at point ) can be written as:

[0062] V 探针 (V DS x m y n )=ΔΦ0+E F (V DS x m y n (5)

[0063] Where ΔΦ0 is the work function of the probe and the work function under equilibrium conditions (i.e., V). DS =0) The difference between the Fermi levels of the entire system, and E F The QFL is related to the device position and device bias. Since Δφ0 depends on the intrinsic properties of the probe and the two-dimensional material and is independent of the device bias, the gradient of equation (5) is generated: So that the current density can be expressed as follows:

[0064]

[0065] This enables an alternative approach for determining the current density vector throughout the device: replacing the continuity equation. Relationships in produce:

[0066]

[0067] Now we can apply the boundary conditions obtained as described above. and Taking this into account, σ(x) is used as the fitting parameter. m y n ), using data obtained from nanopotential measurements We can solve this equation numerically. Equation (7) is also solved numerically. Figure 5 The finite element method is used to solve the problem as described in the flowchart. Due to the fitting parameter σ(x)... m y n ) is a scalar function of x and y, therefore the number of quantities to be fitted is Figure 3 The method shown requires half the number of solutions. Therefore, using the same mathematical resources, this embodiment allows for a faster solution arrival and / or allows for a higher resolution.

[0068] The method described so far can be applied when the outer electrode 2 completely surrounds the inner electrode 1. In this case, both electrodes have closed boundaries, allowing boundary currents to be measured at any number of points along a well-defined perimeter. However, the method of the invention can be applied to architectures without closed boundaries, provided the effects of the non-closed boundaries are negligible or confined to a finite region. Examples of devices to which this approximate version of the method of the invention is applicable are in Figure 6a As shown in the diagram, the first and second electrodes 1 and 2 are parallel rectangular electrodes covered with a conductive two-dimensional material 3, thereby defining a channel region between the electrodes when a bias voltage is applied between them. Similarly, the electrodes and channels can be the source, drain, and channel of a transistor, the gate of which is not shown.

[0069] As seen in the attached diagram, the width W of the device is much larger than the length L of the channel. This means that the two sides 20 and 21, whose boundaries are not closed, are small compared to the total device perimeter, i.e., 2L << 2(L + W). Multiple points P along the boundaries 22 and 23 between the electrodes and the channel... 1i and P 2i The current measurement is completed at this point. The connectivity between probe 10 and meters A1 and A2 is... Figure 6b and 6cThe measurements shown are similar to those shown in Figures 1 and 2, respectively. Such boundary current measurements are impossible at the narrow sides 20 and 21 of the channel because the direction of the current density cannot be assumed in these regions. However, since these regions are small compared to the total channel perimeter, they can be neglected, and the continuity equation (3) can be solved by taking only the boundary currents measured along sides 22 and 23 into consideration. This solution will approximate the correct solution across the central portion of the width W and will aim to deviate from this correct solution at the sides. The current density in most of the channel region can thus be correctly determined.

[0070] According to embodiments of the present invention, additional information regarding the electron transport parameters in a two-dimensional device is obtained by supplementing the above procedures with existing measurement techniques, as follows:

[0071] -Position-dependent carrier mobility: Once the effective value of the position-dependent conductivity σ(x) is obtained... m y n Using the above procedure, the position-dependent electrostatic potential Φ(x) in a two-dimensional material can be found using a Kelvin probe force microscope (KPFM). m y n From this, the carrier density can be found. This, in turn, allows the calculation of the location-dependent mobility μ(x) m y n )=σ(x m y n ) / qρ(x m y n ), and ∈ and q are the dielectric constant and electron charge, respectively.

[0072] -Power density: KPFM also allows for the calculation of location-dependent electric fields. It can be used together with the calculated current density to find the power density.

[0073] Although the invention has been illustrated and described in detail in the accompanying drawings and the foregoing description, such illustrations and descriptions should be considered illustrative or exemplary rather than restrictive. Other variations of the disclosed embodiments may be understood and implemented by those skilled in the art from a study of the drawings, this disclosure, and the appended claims when implementing the claimed invention. In the claims, the word "comprising" does not exclude other elements or steps, and the indefinite articles "a" or "an" do not exclude a plural. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that combinations of these measures cannot be advantageously used. Any reference numerals in the claims should not be construed as limiting the scope.

Claims

1. A method for determining a current density distribution in a microelectronic device, the microelectronic device comprising a first electrode and a second electrode and a two-dimensional conductive material layer extending between the first electrode and the second electrode, the method comprising the steps thereof being performed when the device is in operation, i.e., when a bias voltage is applied between the electrodes to cause charge carriers to flow from the first electrode to the second electrode or vice versa in the two-dimensional material, and wherein at the interface between the two-dimensional material and the electrodes, a current density vector is... Basically orthogonal to the interface, The total current passing through the first and second electrodes is measured without a measuring probe being placed on the two-dimensional material. On the two-dimensional material, a first current measurement probe is placed as close as possible to the first electrode at multiple first locations (P) along the interface between the first electrode and the two-dimensional material. 1i In the first position, the first current measuring probe is coupled to the same voltage as the first electrode, and the first current measuring probe is configured in each of the first positions to locally shunt the current passing through the first electrode. The current passing through the first current measurement probe at the first position is determined, and a first set of current density values ​​is derived therefrom, the first set of current density values ​​representing the boundary conditions of the current density vector in the two-dimensional material at the interface between the first electrode and the two-dimensional material. On the two-dimensional material, a second current measuring probe, which may be the same as or different from the first current measuring probe, is placed as close as possible to the second electrode at multiple second locations along the interface between the second electrode and the two-dimensional material. The second current measuring probe is coupled to the same voltage as the second electrode, and in each of the second locations, the second current measuring probe is configured to locally shunt the current passing through the second electrode. The current passing through the second current measurement probe at the second position is determined, and a second set of current density values ​​is derived therefrom. This second set of current density values ​​represents the boundary condition of the current density in the two-dimensional material at the interface between the second electrode and the two-dimensional material. Taking the aforementioned boundary conditions into account, the equations are... Solve to obtain the current density distribution in the two-dimensional material.

2. The method as described in claim 1, characterized in that: When the first current measuring probe is placed in the first position, pressure is applied to the first current measuring probe such that the sum of the currents passing through the first electrode and the first current measuring probe is substantially equal to the total current passing through the first electrode when no probe is present on the two-dimensional material. When the second current measuring probe is placed in the second position, pressure is applied to the second current measuring probe such that the sum of the currents passing through the second electrode and the second current measuring probe is substantially equal to the total current passing through the second electrode when no probe is present on the two-dimensional material.

3. The method of claim 1, wherein a first predefined pressure is applied to the first current measuring probe at each of the first positions and a second predefined pressure is applied to the second current measuring probe at each of the second positions.

4. The method of claim 1, wherein the first current measuring probe is formed of the same material as the first electrode and / or wherein the second current measuring probe is formed of the same material as the second electrode.

5. The method of claim 1, wherein a single probe is used as the first current measuring probe and the second current measuring probe.

6. The method of claim 1, wherein the interface between the electrode and the two-dimensional material forms a closed boundary.

7. The method of claim 1, wherein the electrodes are rectangular, oriented parallel to each other, and spaced apart by a distance L, the length of the interface corresponds to the width W of the device, and wherein L is much smaller than W.

8. The method of claim 1, wherein the equation is determined by applying the vector The initial guess and by updating the initial guess until The boundary conditions are fitted to the previously determined boundary conditions for numerical solution.

9. The method of claim 1, further comprising, in the two-dimensional conductive material layer, a coordinate system... Nanopotential measurements were performed at multiple defined points, and the equations were... Written as: ,in: It is the effective value of the position-dependent conductivity in the channel, and E F It is the position-dependent quasi-Fermi level in the channel. and It is obtained from the nanopotential measurement. These are used as fitting parameters to, based on the relational formula, take into account the previously determined boundary conditions. To determine the current density.

Images