Innovative Design of Bismuth-Telluride-Based Thermoelectric Transistors

Conventional thermoelectric generators, predominantly based on the π-type structure, are severely limited in their applications due to the relatively low conversion efficiency. In response to the challenge, in this work, a Bi2Te3-based thermoelectric transistor driven by laser illumination is demonstrated. Under laser illumination, a temperature difference of 46.7 °C is produced between the two ends of the transistor structure. Further, the hole concentrations in each region redistribute and the built-in voltages decrease due to the temperature difference, leading to the formation of the transistor circuit. Additionally, the operation condition of the thermoelectric transistor is presented. The calculation results demonstrate that the maximum output power of such a designed thermoelectric transistor is 0.7093 μW.


Introduction
Fossil fuels, being the primary and nonrenewable energy source that has driven human society and industrial development since the Industrial Revolution, are facing an inevitable depletion due to the ever-growing demand.The resulting concerns over energy security have also contributed to escalated global conflicts [1].Therefore, it is imperative to focus on the development of clean, low-carbon, secure, and efficient renewable energy sources.
Thermoelectric generators, capable of converting heat into electricity through the Seebeck effect, have emerged as a promising energy conversion technology [2].Conventional thermoelectric generators typically consist of multiple thermoelectric modules, offering numerous advantages, such as high safety, extended service life, zero waste generation, no noise, and simple structure [3].However, the applications of conventional thermoelectric generators are limited due to their relatively poor output performance.Therefore, thermoelectric generators can only be used in a few specific scenarios, such as medicine, aerospace, and military sectors.
In general, the output performance of thermoelectric generators was assessed based on the zT value of materials, output power, and conversion efficiency.To enhance the output performance, the combination of thermoelectric generators with transistor technology has been proposed [4,5].Bejenari et al. studied the thermoelectric performance of Bi 2 Te 3 nanowires in the transistor structure under the gate voltage.Theoretical results indicated that the zT value could reach 3.4 for Bi 2 Te 3 nanowires [6].Subsequently, Qin et al. found that the zT values of N-type and P-type Bi 2 Te 3 films could reach 1.22 and 1.02 in the transistor structure through experimental measurement [7].Furthermore, Nan et al. studied the output performance of a thermoelectric transistor driven by the Seebeck effect, in which the built-in electric field is perpendicular to the gradient temperature field.Results showed that the output power was 17.8 mW and conversion efficiency could reach 8.69% at a temperature difference of 50 • C [8,9].
These research results demonstrated that the output performance of thermoelectric generators could be significantly improved through the transistor effect.In this work, a Bi 2 Te 3 -based thermoelectric transistor driven by the laser illumination is presented.On the one hand, the short pulse width of a laser allows for rapid creation of a temperature difference between the two ends of the device, reaching nanosecond levels [10].On the other hand, Bi 2 Te 3 -based materials exhibit excellent thermoelectric properties at room temperature [11].Moreover, utilizing Bi 2 Te 3 -based materials as research subjects can improve the accuracy of theoretical calculations due to their comprehensive performance parameters [12,13].As a result, in this work, theoretical results indicated that the maximum output power of a single Bi 2 Te 3 -based thermoelectric transistor could reach 0.7093 µW with a temperature difference of 46.7 • C. Conventional thermoelectric generators would require multiple thermoelectric modules connected in series to achieve such output performance [14][15][16].

Theoretical Foundations
The structure of the designed structure is depicted in the following Figure 1.
effect, in which the built-in electric field is perpendicular to the gradient temperature field.Results showed that the output power was 17.8 mW and conversion efficiency could reach 8.69% at a temperature difference of 50 °C [8,9].
These research results demonstrated that the output performance of thermoelectric generators could be significantly improved through the transistor effect.In this work, a Bi2Te3-based thermoelectric transistor driven by the laser illumination is presented.On the one hand, the short pulse width of a laser allows for rapid creation of a temperature difference between the two ends of the device, reaching nanosecond levels [10].On the other hand, Bi2Te3-based materials exhibit excellent thermoelectric properties at room temperature [11].Moreover, utilizing Bi2Te3-based materials as research subjects can improve the accuracy of theoretical calculations due to their comprehensive performance parameters [12,13].As a result, in this work, theoretical results indicated that the maximum output power of a single Bi2Te3-based thermoelectric transistor could reach 0.7093 μW with a temperature difference of 46.7 °C.Conventional thermoelectric generators would require multiple thermoelectric modules connected in series to achieve such output performance [14][15][16].

Theoretical Foundations
The structure of the designed structure is depicted in the following Figure 1.Considering the one-dimensional model, the length (x1) of P1 region (P1-Bi2Te3) is 1 μm, while the length of (x2) P2 region (P2-Si) is 4 μm.The scale of N region (N-Si) is the nanometer range, which can be neglected for calculation purposes.
Using a laser as the heat source, serving as the triggering condition, the P1-Bi2Te3 is irradiated by the laser at one end, causing its temperature to rise.Subsequently, the heat is transmitted throughout the entire structure via thermal conduction, forming a temperature field along the X direction.The temperature field induces the redistribution of majority charge carriers (holes) within the structure.The holes in P1 region and P2 region diffuse from the hot end to the cold end, accumulating at the cold end.Simultaneously, the built-in voltage across P1-N junction decreases from the cold end to the far end.Consequently, the balance between the diffusion potential and drift potential on both sides of P1-N junction is disrupted, leading to the injection of holes from P1 region into N-type region at the cold end.Additionally, since the length of N-type region is on the nanoscale, the holes further migrate towards P2 region.Subsequently, the accumulated holes in P2 region flow back to the N-type region along the wires.The migration of holes is consistent with the common base circuit of a bipolar transistor.Thus, it is inferred that the transistor circuit can be formed in the PNP structure under laser irradiation.In other words, the cold side of the P1 region (x1) functions as the emitter of the thermoelectric transistor, while the hot side of the P2 region (x2) serves as the collector, with the N-type region acting as the base [17].Considering the one-dimensional model, the length (x 1 ) of P 1 region (P 1 -Bi 2 Te 3 ) is 1 µm, while the length of (x 2 ) P 2 region (P 2 -Si) is 4 µm.The scale of N region (N-Si) is the nanometer range, which can be neglected for calculation purposes.
Using a laser as the heat source, serving as the triggering condition, the P 1 -Bi 2 Te 3 is irradiated by the laser at one end, causing its temperature to rise.Subsequently, the heat is transmitted throughout the entire structure via thermal conduction, forming a temperature field along the X direction.The temperature field induces the redistribution of majority charge carriers (holes) within the structure.The holes in P 1 region and P 2 region diffuse from the hot end to the cold end, accumulating at the cold end.Simultaneously, the built-in voltage across P 1 -N junction decreases from the cold end to the far end.Consequently, the balance between the diffusion potential and drift potential on both sides of P 1 -N junction is disrupted, leading to the injection of holes from P 1 region into N-type region at the cold end.Additionally, since the length of N-type region is on the nanoscale, the holes further migrate towards P 2 region.Subsequently, the accumulated holes in P 2 region flow back to the N-type region along the wires.The migration of holes is consistent with the common base circuit of a bipolar transistor.Thus, it is inferred that the transistor circuit can be formed in the PNP structure under laser irradiation.In other words, the cold side of the P 1 region (x 1 ) functions as the emitter of the thermoelectric transistor, while the hot side of the P 2 region (x 2 ) serves as the collector, with the N-type region acting as the base [17].When a laser is incident upon Bi 2 Te 3 , according to Lambert's law, it is known that the amount of light absorbed is closely related to the position in the one-dimensional model.The closer the position is to the source of light, the greater the light absorption.Assuming material thermal capacitance is uniform, a temperature gradient will form parallel to the direction of illumination.According to Fourier's law, heat conduction will also occur along the direction of the incident light.Taking all these considerations into account under the one-dimensional condition, the following equation can be derived [18]: where ρ represents material density, c represents material-specific heat capacity, T 1 is the temperature, k represents thermal conductivity, R represents reflectivity, I 0 represents light intensity, t represents time, x is the material length along the direction of illumination, and α represents absorption coefficient.Due to the extremely short duration of the pulse laser, it can be assumed that there is no heat transfer occurring at the ends of the sample.Therefore, the following conditions hold: In addition, the initial temperature distribution of the sample is assumed to be uniform, with no heat transfer occurring between different parts of the sample.In other words, the temperature at each position is a constant, given by: By solving these equations in MATLAB R2020a, the temperature distribution T 1 (x 1 ) in P 1 -Bi 2 Te 3 can be obtained.

Calculation of Temperature Distribution in P 2 -Si
Unlike the calculation of temperature distribution in Bi 2 Te 3 , the differential equation for the temperature distribution in P 2 -Si does not include the influence of illumination; thus, it becomes a one-dimensional heat conduction problem without a heat source: Similarly, the boundary conditions can be set as follows: The initial condition for the P 2 -Si sample is: By solving these equations in MATLAB R2020a, the temperature distribution T 2 (x 2 ) in P 2 -Si can be obtained.When P 1 -Bi 2 Te 3 is subjected to a stable temperature field at 20 • C, the hole concentration is uniformly distributed, which is equal to the acceptor concentration P a .Subsequently, when P 1 -Bi 2 Te 3 is influenced by a temperature gradient (dx/dt = 0), the holes redistribute, while the acceptor ions remain fixed.The redistribution of holes can be obtained through theoretical analysis, as shown below.

Hole Concentration Distribution in
DC transport equation for current [19]: where J p (x) represents the current density, ∇T(x) is the temperature gradient, E(x) denotes the electric field, ϕ(x) represents the electric potential, σ p (x) is the electrical conductivity of P 1 -Bi 2 Te 3 , and S p (x) is the Seebeck coefficient p-type Bi 2 Te 3 .
One-dimensional Poisson equation [20]: where e is the electronic charge, ε r is the relative permittivity, ε 0 is the vacuum permittivity, p 1 (x 1 ) represents the hole concentration distribution in P 1 -Bi 2 Te 3 , and P a is the constant concentration of acceptor holes assuming complete ionization of Bi 2 Te 3 .
Continuity equation [21]: where G p is the hole carrier generation rate in Bi 2 Te 3 , and R p is the hole carrier recombination rate.
Since P 1 -Bi 2 Te 3 discussed in this work is a degenerate semiconductor, σ p (x) and S p (x) can be expressed as [11,22]: where µ p represents the hole mobility of P 1 -Bi 2 Te 3 , r is the scattering factor (= −1/2), h is the Planck constant, m * p denotes the effective mass, and T 1 (x 1 ) is temperature distribution of P 1 -Bi 2 Te 3 along the x direction.
Since the simulated temperature distribution is significantly lower than the intrinsic excitation temperature of Bi 2 Te 3 , it can be assumed that G p = R p and the total hole concentration in Bi 2 Te 3 remains constant after applying the temperature gradient: Substituting the above equation into Equation ( 10) can obtain: Substituting Equation ( 16) into Equation (9) yields: Furthermore, assuming that the entire P 1 -Bi 2 Te 3 reaches a stable state after the temperature distribution is established, Equation ( 12) can be written as: By solving these equations in MATLAB, the hole concentration distribution of P 1 -Bi 2 Te 3 can be obtained under a temperature gradient.

Hole Concentration Distribution in P 2 -Si
Since the carrier concentration of P 2 -Si studied in this work is less than 10 18 cm −3 , it is considered a nondegenerate semiconductor.Therefore, the Seebeck coefficient can be expressed as [23]: Then, based on the temperature distribution T 2 (x 2 ) in P 2 -Si and Equations ( 13), ( 15), (17), and (18), the hole concentration distribution of P 2 -Si can be derived.

Operation Conditions of Thermoelectric Transistor
In order for the thermoelectric transistor to operate properly, it must be in a forwardactive mode [17].Since the formation for the circuit of the thermoelectric transistor is only based on the Seebeck effect, the potential within it can be represented by the concentration distribution in each region.Therefore, it is crucial to determine appropriate concentration ranges within each region to ensure the normal operation of the thermoelectric transistor.
Firstly, the thermoelectric transistor is composed of P 1 -Bi 2 Te 3 , N-Si, and P 2 -Si materials.Before applying a temperature gradient (the entire transistor is at T = 20 • C), the respective donor or acceptor doping ion concentrations must be greater than their intrinsic carrier concentrations.Mathematically, it can be expressed as follows: where P a1 , N d , and P a2 represent the doping concentrations in P 1 region (P 1 -Bi 2 Te 3 , emitter), N region (N-Si, base), and P 2 region (P 2 -Si, collector), respectively.p i1 , n i , and p i2 represent the intrinsic carrier concentrations in the respective regions at T = 20 • C.

Implementation of Forward Bias and Forward Conduction at the Emitter-Base
After applying a temperature gradient, the emitter (P 1 region) and the base (N region) should simultaneously satisfy the forward bias and forward conduction.The temperature distribution T 1 (x 1 ) between P 1 region and N region can be obtained from the previous section.According to the relevant literature [24], the built-in voltage at the emitter-base junction varies with temperature: where k B is the Boltzmann constant, p i1 (x 1 ) is the intrinsic carrier concentration affected by the temperature distribution T 1 (x 1 ), while P a1 and N d remain constant regardless of the applied temperature gradient.The variations in p i1 (x 1 ) and n i (x 1 ) with the temperature distribution can be described by the following [25,26]: where m * p1 is the effective mass of the emitter, m e is the electron mass, and E gp1 is the bandgap of the emitter.
To achieve forward bias and forward conduction at the emitter-base junction, the built-in voltage V bi1 (x 1 ) should satisfy: Additionally, after applying the temperature gradient, the potential of the emitter should be higher than that of the base: Since the potential of each region in this work is mainly determined by its carrier concentration, Equation ( 27) can be written as: where p 1 (x 1 ) is the hole concentration distribution of the emitter after applying the temperature gradient.However, the concentration of the base remains constant (N d ) because it is completely depleted.Finally, in order to ensure the forward conduction in the thermoelectric transistor, the generated voltage due to the Seebeck effect (the Seebeck voltage V s1 (x 1 )) should be greater than the built-in voltage V bi1 (x 1 ).In other words, the emitter voltage V e (x 1 ) should be greater than 0: (31 where S 1 represents the Seebeck coefficient of the emitter and ∆T(x 1 ) is the temperature distribution between the emitter and the base.By combining these equations, the forward bias and forward conduction at the emitter-base can be achieved.

Realization of Reverse Bias at the Base-Collector
Similarly, the intrinsic carrier concentration distribution of P 2 -type region (P 2 -Si) under the applied temperature gradient can be expressed [25]: In order ensure the normal operation of the thermoelectric transistor, the base-collector junction should be the reverse bias.Under the influence of the temperature distribution T 2 (x 2 ), the built-in voltage at the base-collector junction can be expressed as: To ensure proper operation, V bi2 (x 2 ) must be positive: Similar to the emitter-base junction, the potential of the base should be higher than that of the collector: Furthermore, the Seebeck coefficient and the Seebeck voltage for P 2 -Si are expressed as follows: (37) where p 2 (x 2 ) is the hole concentration distribution in the collector region after applying the temperature gradient.By solving these equations, the operation conditions of the base-collector junction can be determined.

The Output Performance of Thermoelectric Transistors
For bipolar thermoelectric transistors, the base-collector circuit can be considered as the output circuit.In this work, the early voltage effects of the emitter, base, and collector, as well as the contact resistance between the transistor and the electrodes, can be neglected.The corresponding equivalent DC circuit diagram is shown in Figure 2. The left side of the circuit diagram represents the equivalent circuit of the emitter-base junction.V s1 (x 1 ) denotes the Seebeck voltage generated in the P 1 region, serving as the external voltage at the emitter.R e (x 1 ) represents the resistance of the P 1 region of the emitter.V bi1 (x 1 ) is assumed to be the forward voltage.On the right side of the circuit, the equivalent circuit of the base-collector junction is shown.V s2 (x 2 ) signifies the Seebeck voltage generated in the P 2 region, acting as the external voltage at the collector.R c (x 2 ) corresponds to the resistance of the P 2 region of the collector.V bi2 (x 2 ) is the voltage influenced by the emitter current I e (x 1 ).Furthermore, considering that the thickness of the base can be neglected compared to the emitter and collector and that it is fully depleted, the collector current I c (x 2 ) can be assumed to be equal to the emitter current I e (x 1 ).
To ensure proper operation, Vbi2(x2) must be positive: Similar to the emitter-base junction, the potential of the base should be higher than that of the collector: Furthermore, the Seebeck coefficient and the Seebeck voltage for P2-Si are expressed as follows: ( ) = ∆( ) ×  (37) where p2(x2) is the hole concentration distribution in the collector region after applying the temperature gradient.By solving these equations, the operation conditions of the base-collector junction can be determined.

The Output Performance of Thermoelectric Transistors
For bipolar thermoelectric transistors, the base-collector circuit can be considered as the output circuit.In this work, the early voltage effects of the emitter, base, and collector, as well as the contact resistance between the transistor and the electrodes, can be neglected.The corresponding equivalent DC circuit diagram is shown in   According to the Kirchhoff's law, the relationships among these electrical parameters can be expressed: where V e (x 1 ) is the emitter voltage, V c (x 2 ) is the collector voltage, and µ p1 and µ p2 are the hole mobility of the P 1 -type material (P 1 -Bi 2 Te 3 ) and P 2 -type material (P 2 -Si), respectively.lx 1 and lx 2 represent the lengths of the P 1 -type and P 2 -type materials along the x direction, while w y and h z denote the lengths of the device in y and z directions, respectively, with values set to 1 µm.S p2 is the Seebeck coefficient of the collector and m* p2 is the effective mass of the collector.
The output power of the transistor can be expressed as the product of the open-circuit voltage and the short-circuit current: (44)

The Material Parameters of Thermoelectric Transistor
In this work, P 1 -Bi 2 Te 3 , N-Si, and P 2 -Si are selected as the emitter material, base material, and collector material, respectively.The material parameters of the thermoelectric transistor studied are listed in Table 1.As depicted in Figure 3, it is apparent that, with an increase in laser irradiation time, the sample initiates heating from the irradiated surface at x = 0 μm.Subsequently, the heat gradually propagates throughout the entire device, and the lowest temperature is observed at x = 1 μm within P1 region.Figure 4 illustrates the temperature distribution in P1 region at T = 100 ns.Notably, after 100 ns of illumination, the temperature distribution in P1 region (Bi2Te3) exhibits a pattern of decreasing temperature as the distance from the laser increases.Specifically, at x = 0 μm, the temperature at 100 ns is Tb1 = 66.7 As depicted in Figure 3, it is apparent that, with an increase in laser irradiation time, the sample initiates heating from the irradiated surface at x = 0 µm.Subsequently, the heat gradually propagates throughout the entire device, and the lowest temperature is observed at x = 1 µm within P 1 region.Figure 4 illustrates the temperature distribution in P 1 region at T = 100 ns.Notably, after 100 ns of illumination, the temperature distribution in P 1 region (Bi 2 Te 3 ) exhibits a pattern of decreasing temperature as the distance from the laser increases.Specifically, at x = 0 µm, the temperature at 100 ns is T b1 = 66.7 • C, whereas, at x = 1 µm, the temperature reaches its minimum value of T b2 = 37 • C. By employing a quadratic function, the temperature variation T 1 (x 1 ) in P 1 region as a function of x can be obtained: As depicted in Figure 3, it is apparent that, with an increase in laser irradiation time, the sample initiates heating from the irradiated surface at x = 0 μm.Subsequently, the heat gradually propagates throughout the entire device, and the lowest temperature is observed at x = 1 μm within P1 region.Figure 4 illustrates the temperature distribution in P1 region at T = 100 ns.Notably, after 100 ns of illumination, the temperature distribution in P1 region (Bi2Te3) exhibits a pattern of decreasing temperature as the distance from the laser increases.Specifically, at x = 0 μm, the temperature at 100 ns is Tb1 = 66.7 °C, whereas, at x = 1 μm, the temperature reaches its minimum value of Tb2 = 37 °C.By employing a quadratic function, the temperature variation T1(x1) in P1 region as a function of x can be obtained:  ( ) = 14.35 − 43.95 + 66.7

Temperature Distribution in p-Si
By solving the equations in MATLAB, the temperature distribution in P 2 region (P 2 -Si) along the x direction can be obtained.After 100 ns, the temperature distribution is shown in Figure 5. Based on Figure 5, it can be observed that the temperature distribution in P2-type region gradually decreases with increasing x.It is important to note that n-type region in this work is extremely thin.Therefore, the length of the n-type region along the x direction is assumed to be 0, implying that x1 = 1 μm coincides with x2 = 0 μm.At x2 = 0 μm, P2-Si exhibits the highest temperature, Ts1 = 37 °C, while, at x2 = 4 μm, p-Si reaches its lowest temperature, Ts2 = 20 °C.By fitting a curve to the data in the figure, a fitting function Based on Figure 5, it can be observed that the temperature distribution in P 2 -type region gradually decreases with increasing x.It is important to note that n-type region in this work is extremely thin.Therefore, the length of the n-type region along the x direction is assumed to be 0, implying that x 1 = 1 µm coincides with x 2 = 0 µm.At x 2 = 0 µm, P 2 -Si exhibits the highest temperature, T s1 = 37 • C, while, at x 2 = 4 µm, p-Si reaches its lowest temperature, T s2 = 20 • C. By fitting a curve to the data in the figure, a fitting function T 2 (x 2 ) for the temperature distribution in P 2 -Si can be obtained:

Hole Concentration Distribution within Thermoelectric Transistor
The thermoelectric transistor consists of P 1 -type region, N-type region, and P 2 -type region.As mentioned in the previous section, the temperature distribution along the x direction of the transistor is represented by Equations ( 45) and ( 46).
Considering that N-type region is assumed to be completely depleted, it contains no free mobile carriers.However, for both P 1 -type region and P 2 -type region, under the influence of temperature distribution, the holes migrate from the hot end to the cold end, leading to their accumulation at the cold end.By utilizing the calculation formula from the previous section, the concentration distribution of holes in P 1 -type region can be obtained: From Figure 6, it is evident that p 1 (x 1 ) increases with increasing x 1 .This indicates that the hole concentration in P 1 -type region gradually increases from the hot end to the cold end.Additionally, as P a1 increases, p 1 (x 1 ) also increases.Of note, p 1 (x 1 ) = P a1 is near the midpoint of the P 1 -type region (x 10 = 0.501 µm), as shown in the dashed line in Figure 6.This is because the hole carriers in the P 1 -type region (Bi 2 Te 3 ) redistribute only along the x direction, and the temperature difference is fixed.Therefore, the position x 10 remains constant.As a result, it can be inferred that, for x 1 < x 10 , p 1 (x 1 ) < P a1 , and, for x 10 ≤ x 1 ≤ 1 µm, p 1 (x 1 ) ≥ P a1 , with equality only at x 1 = x 10 .Figure 7 illustrates the variation in p 2 (x 2 ) with respect to x 2 for different P a2 .It can be observed that p 2 (x 2 ) also increases with the increase in x 2 .Additionally, as P a2 increases, p 2 (x 2 ) increases.Similarly, at x 20 = 2.009 µm, p 2 (x 2 ) equals P a2 , as indicated by the dashed line in Figure 7.In other words, for 0 ≤x 2 < x 20 = 2.009 µm, P a2 > p 2 (x 2 ), and, for x 20 ≤ x 2 ≤ 4 µm, p 2 (x 2 ) ≥ P a2 , with equality only at x 2 = x 20 .Thus, it can be inferred that the Seebeck effect causes the redistribution of hole concentrations in both P1-type and P2-type regions, with holes diffusing from the hot end to the cold end.Consequently, the built-in voltage at both the P1-N junction and the N-P2 junction decreases from the cold end to the hot end.This unique characteristic enables such PNP heterojunctions to function as thermoelectric transistors.The emitter corresponds to the region with x10 ≤ x1 ≤ 1 μm in the P1-type region, the base corresponds to the entire N-type region, and the collector corresponds to the region with 0 ≤ x2 < x20 = 2.009 μm in the P2-type region.Thus, it can be inferred that the Seebeck effect causes the redistribution of hole concentrations in both P 1 -type and P 2 -type regions, with holes diffusing from the hot end to the cold end.Consequently, the built-in voltage at both the P 1 -N junction and the N-P 2 junction decreases from the cold end to the hot end.This unique characteristic enables such PNP heterojunctions to function as thermoelectric transistors.The emitter corresponds to the region with x 10 ≤ x 1 ≤ 1 µm in the P 1 -type region, the base corresponds to the entire N-type region, and the collector corresponds to the region with 0 ≤ x 2 < x 20 = 2.009 µm in the P 2 -type region.
The temperature distribution T 1 (x 1 ) and the temperature difference ∆T 1 (x 1 ) for the emitter-base junction can be written: Under this temperature distribution, the calculated values for the Seebeck coefficient S 1 and the Seebeck voltage V s1 in P 1 region are shown in Figure 8.
Under this temperature distribution, the calculated values for the Seebeck coefficient S1 and the Seebeck voltage Vs1 in P1 region are shown in Figure 8.As shown in Figure 8, the Seebeck coefficient S1 of P1 region increases with the temperature T1(x1).Moreover, for different doping concentrations Pa1, the smaller doping concentration leads to a larger Seebeck coefficient.As for the Seebeck voltage Vs1, it exhibits a trend where it increases as the distance from the hot end (x1 = 0) increases due to the larger temperature difference.
Additionally, the built-in voltage Vbi1 within P1-N region can be obtained, as shown in Figure 9.As shown in Figure 8, the Seebeck coefficient S 1 of P 1 region increases with the temperature T 1 (x 1 ).Moreover, for different doping concentrations P a1 , the smaller doping concentration leads to a larger Seebeck coefficient.As for the Seebeck voltage V s1 , it exhibits a trend where it increases as the distance from the hot end (x 1 = 0) increases due to the larger temperature difference.
Additionally, the built-in voltage V bi1 within P 1 -N region can be obtained, as shown in Figure 9.Under different values of P a1 and N d , the built-in voltage increases with the increase in x 1 .This means that, as the temperature decreases, the built-in voltage increases.The increase is primarily attributed to the decrease in intrinsic carrier concentration with decreasing temperature.
Based on Section 2.3.1, it can be concluded that achieving the forward bias and forward conduction of the emitter-base junction depends on three conditions: V bi1 (x 1 ) > 0, V e (x 1 ) = V s1 (x 1 )-V bi1 (x 1 ) > 0, and p 1 (x 1 ) > N d .These conditions are all influenced by P a1 , x 1 , and N d .The appropriate concentration ranges for P a1 and N d are shown in Figure 10.
It can be observed that the suitable range for N d is 3.2 × 10 10 cm −3 ~15.8 × 10 10 cm −3 .As the range of N d increases, the range of P a1 decreases.When N d is 3.2 × 10 10 cm −3 , the range of P a1 is 1.61 × 10 18 cm −3 ~5.6 × 10 18 cm −3 , as indicated by the black line.When N d increases to 15.8 × 10 10 cm −3 , the range of P a1 is only 1.61 × 10 18 cm −3 ~2.0 × 10 18 cm −3 , as shown by the red line.This is because the built-in voltage V bi1 (x 1 ) is positively correlated with N d and P a1 .As N d and P a1 increase, V bi1 (x 1 ) also increases, resulting in a smaller range of P a1 for achieving V s1 (x 1 )-V bi1 (x 1 ) > 0.
in x1.This means that, as the temperature decreases, the built-in voltage increases.The increase is primarily attributed to the decrease in intrinsic carrier concentration with decreasing temperature.
Based on Section 2.   Additionally, it can be found that x 1 also changes with N d and P a1 .As N d and P a1 increase, the range of x 1 gradually approaches 0.501 µm.This is because, at x 1 = 1 µm, both V bi1 (x 1 ) and V s1 (x 1 ) reach their maximum values.However, in comparison to V s1 (x 1 ), V bi1 (x 1 ) increases at a faster rate as x 1 increases.Therefore, it is relatively easier to achieve the operation conditions of the emitter-base junction at x 1 = 0.501 µm.Thus, when N d = 3.2 × 10 10 cm −3 , the maximum range of x 1 is 0.608 µm to 1 µm.While N d = 15.8 × 10 10 cm −3 , the maximum range of x 1 is 0.501 µm to 0.537 µm.Therefore, the forward bias and forward conduction of the emitter-base junction can be achieved by adjusting the concentrations of P a1 and N d .Such a deigned doping concentration for P 1 -Bi 2 Te 3 can be obtained directly by doping process, e.g., phosphorus as acceptor.

Reverse Bias of Base-Collector Junction
When a temperature gradient is applied in the base-collector junction, the temperature distribution equation T 2 (x 2 ) and temperature difference ∆T 2 (x 2 ) can be expressed as follows: It should be noted that ∆T(x 1 ) represents the temperature difference between the emitter and base, with a positive value.However, ∆T 2 (x 2 ) represents the temperature difference between the collector and base, with a negative value.Therefore, V s1 (x 1 ) has a positive value, while V s2 (x 2 ) has a negative value.
For P 2 region, the values of the Seebeck coefficient S 2 and the Seebeck voltage V s2 are shown in Figure 11.
It can be observed that the Seebeck coefficient S 2 of P 2 region increases with temperature T 2 (x 2 ).For different doping concentrations P a2 , a smaller doping concentration leads to a larger Seebeck coefficient.Regarding the Seebeck voltage V s2 , it exhibits a trend as the position approaches x 2 = 0 (closer to the hot end) due to the increasing temperature difference.
The built-in voltage V bi2 in the N-P 2 region is depicted in Figure 12.It can be found that the built-in voltage V bi2 increases with increasing x 2 at different values of P a2 and N d .In other words, as the temperature decreases, the built-in voltage increases.This increase primarily stems from the decrease in intrinsic charge carriers with decreasing temperature. ( ) = −0.1982+ 2.798 − 12.48 + 37 (55) It should be noted that ∆T(x1) represents the temperature difference between the emitter and base, with a positive value.However, ∆T2(x2) represents the temperature difference between the collector and base, with a negative value.Therefore, Vs1(x1) has a positive value, while Vs2(x2) has a negative value.
For P2 region, the values of the Seebeck coefficient S2 and the Seebeck voltage Vs2 are shown in Figure 11.It can be observed that the Seebeck coefficient S2 of P2 region increases with temperature T2(x2).For different doping concentrations Pa2, a smaller doping concentration leads to a larger Seebeck coefficient.Regarding the Seebeck voltage Vs2, it exhibits a trend as the position approaches x2 = 0 (closer to the hot end) due to the increasing temperature difference.
The built-in voltage Vbi2 in the N-P2 region is depicted in Figure 12.It can be found that the built-in voltage Vbi2 increases with increasing x2 at different values of Pa2 and Nd.In other words, as the temperature decreases, the built-in voltage increases.This increase primarily stems from the decrease in intrinsic charge carriers with decreasing temperature.Based on Section 2.3.2, it can be concluded that achieving the forward bias and forward conduction of the emitter-base junction depends on three conditions: V bi2 (x 2 ) > 0 and p 2 (x 2 ) < N d .These conditions are all influenced by P a2 , x 2 , and N d .Therefore, the reverse bias in the base-collector junction can be achieved by adjusting the concentrations of P a2 and N d .The appropriate concentration ranges for P a2 and N d are shown in Figure 13.
Unlike the previous section, the range of N d is from 5.2 × 10 10 cm −3 to 15.8 × 10 10 cm −3 .It can be observed that, as N d increases, the corresponding range of P a2 also gradually increases.This is because higher values of N d and P a2 both lead to larger values of V bi2 (x 2 ), making it easier to satisfy the reverse biasing condition V bi2 (x 2 ) > 0.
To summarize, when N d is within the range of 5.2 × 10 10 cm −3 to 15.8 × 10 10 cm −3 , the thermoelectric transistor can be activated in the forward-active mode.The corresponding ranges of P a1 , P a2 , x 1 , and x 2 are determined by the different values of N d .By adjusting the concentrations of P a1 , N d , and P a2 , the current can be generated in the thermoelectric transistor directly driven by the Seebeck effect.

Output Performance of Thermoelectric Transistor
It is evident that the output power of the transistor is primarily influenced by P a1 , N d , P a2 , x 1 , and x 2 based on Equation (44).Therefore, the impact of these parameters on the output power of the thermoelectric transistor is obtained in the following.
verse bias in the base-collector junction can be achieved by adjusting the concentrations of Pa2 and Nd.The appropriate concentration ranges for Pa2 and Nd are shown in Figure 13.
Unlike the previous section, the range of Nd is from 5.2 × 10 10 cm −3 to 15.8 × 10 10 cm −3 .It can be observed that, as Nd increases, the corresponding range of Pa2 also gradually increases.This is because higher values of Nd and Pa2 both lead to larger values of Vbi2(x2), making it easier to satisfy the reverse biasing condition Vbi2(x2) > 0.
To summarize, when Nd is within the range of 5.2 × 10 10 cm −3 to 15.8 × 10 10 cm −3 , the thermoelectric transistor can be activated in the forward-active mode.The corresponding ranges of Pa1, Pa2, x1, and x2 are determined by the different values of Nd.By adjusting the concentrations of Pa1, Nd, and Pa2, the current can be generated in the thermoelectric transistor directly driven by the Seebeck effect.It is evident that the output power of the transistor is primarily influenced by Pa1, Nd, Pa2, x1, and x2 based on Equation (44).Therefore, the impact of these parameters on the output power of the thermoelectric transistor is obtained in the following.

Impact of Nd on Output Power of Thermoelectric Transistor
Since the value of Nd directly affects the other four physical quantities, it is essential to initially examine the influence of Nd on the output power.Figure 14     It is evident that, as N d gradually increases, P outmax exhibits a trend of initially increasing and then decreasing.At N d = 8.0 × 10 10 cm −3 , P outmax reaches its maximum value of 0.7093 µW.It is important to note that different values of N d correspond to various combinations of P a1 , P a2 , x 1 , and x 2 .Considering the multitude of possible parameter combinations, it is challenging to directly explain how N d influences the output power.Therefore, a comprehensive enumeration of the parameters is necessary to identify the optimal solution, which, in this case, is N d = 8.0 × 10 10 cm −3 and P outmax = 0.7093 µW.

Impact of P a1 and x 1 on Output Power of Thermoelectric Transistor
For N d = 8.0 × 10 10 cm −3 , the range of P a1 is from 1.61 × 10 18 cm −3 to 2.6 × 10 18 cm −3 and the range of P a2 is from 1.0 × 10 10 cm −3 to 8.2 × 10 10 cm −3 .Moreover, different values of P a1 and P a2 correspond to specific ranges of x 1 and x 2 .Considering a fixed P a2 value of 7.8 × 10 10 cm −3 and x 2 value of 0.4 µm, the relationship between output power (P out ) and the variations in P a1 and x 1 is illustrated in Figure 15.It can be observed that the output power (P out ) increases as the concentration of P a1 and x 1 decrease.These results can be derived from Equation (58): When N d , P a2 , and x 2 remain constant, the value of P out is primarily influenced by I e (x 1 ): At x 1 = 1 µm, both V bi1 (x 1 ) and V s1 (x 1 ) reach their maximum values.However, compared to V s1 (x 1 ), V bi1 (x 1 ) increases more rapidly as x 1 increases.Therefore, at x 1 = 0.501 µm, V s1 (x 1 )-V bi1 (x 1 ) reaches its maximum value.Thus, as x 1 decreases within the range of x 1 ∈ (0.501 µm, 1 µm), P out increases.On the other hand, when other conditions remain unchanged, a larger P a1 leads to a higher built-in voltage in the P 1 region (V s1 (x 1 )-V bi1 (x 1 )).As a result, with an increase in P a1 , P out decreases.

Impact of P a2 and x 2 on Output Power of Thermoelectric Transistor
The impact of P a2 and x 2 on the output power of the thermoelectric transistor is examined in the following when P a1 = 1.61 × 10 18 cm −3 and x 1 = 0.508 µm.As shown in Figure 16, it can be observed that P out increases as P a2 increases and x 2 decreases.Again, referring to the expression for P out : When N d , P a1 , and x 1 are constant, the value of P out is mainly influenced by V bi2 (x 2 )-V s2 (x 2 ).An increase in P a2 leads to a higher value of V bi2 (x 2 ), resulting in a larger difference between V bi2 (x 2 ) and V s2 (x 2 ).Consequently, as P a2 increases, P out also increases.On the other hand, when x 2 increases, ∆T 2 (x 2 ) decreases, resulting in a lower V s2 (x 2 ) and higher V bi2 (x 2 ).In this case, V s2 (x 2 ) dominates the relationship and the value of V bi2 (x 2 )-V s2 (x 2 ) decreases, leading to a decrease in P out .As deduced from the above analysis, when N d = 8.0 × 10 10 cm −3 , P a1 = 1.61 × 10 18 cm −3 , P a2 = 8.2 × 10 10 cm −3 , x 1 = 0.508 µm, and x 2 = 0 µm, the maximum output power of this designed thermoelectric transistor can reach 0.7093 µW.
If only Bi 2 Te 3 is used without the thermoelectric transistor structure, the output power, can be calculated as 0.1442 µW.It can be observed that the designed thermoelectric transistor structure in this work results in an increase in the output power compared to a single thermoelectric material, with an improvement of 391.9%.

Conclusions
In this work, the temperature distribution and hole concentration distribution within the thermoelectric transistor is obtained through theoretical calculations.By considering the variation in the built-in voltage with temperature, the formation principle of the thermoelectric transistor is demonstrated.Subsequently, based on the operation conditions of the thermoelectric transistor, suitable doping concentrations of the emitter, base, and collector can be determined.Furthermore, the maximum output power of the thermoelectric transistor is obtained.The main conclusions drawn from this work are as follows: (i) The thermoelectric transistor, composed of P 1 -Bi 2 Te 3 , N-Si, and P 2 -Si, is directly irradiated by the laser.Under laser illumination and heat conduction, the temperature decreases from 66.7 • C to 20 • C, creating a temperature difference of 46.7 • C at the two ends of the thermoelectric transistor.As a result of the temperature difference, holes inside P 1 and P 2 regions migrate from the hot end to the far end, leading to increased hole concentration at the cold end.(ii) The operation conditions of the thermoelectric transistor under laser irradiation are investigated.Based on the corresponding conditions, suitable doping concentrations of the emitter, base, and collector can be determined.By adjusting these concentrations, the current can be produced in the thermoelectric transistor only driven by the Seebeck effect.(iii) The influence of these parameters on the output power of the thermoelectric transistor is also investigated.The maximum output power of the thermoelectric transistor is 0.7093 µW under a temperature difference of 46.7 • C, which is nearly quadrupling the performance compared to the single thermoelectric material structure.

Figure 1 .
Figure 1.Diagram of thermoelectric transistor in X-Y plane.

Figure 1 .
Figure 1.Diagram of thermoelectric transistor in X-Y plane.

Figure 2 .
The left side of the circuit diagram represents the equivalent circuit of the emitter-base junction.Vs1(x1) denotes the Seebeck voltage generated in the P1 region, serving as the external voltage at the emitter.Re(x1) represents the resistance of the P1 region of the emitter.Vbi1(x1) is assumed to be the forward voltage.On the right side of the circuit, the equivalent circuit of the base-collector junction is shown.Vs2(x2) signifies the Seebeck voltage generated in the P2 region, acting as the external voltage at the collector.Rc(x2) corresponds to the resistance of the P2 region of the collector.Vbi2(x2) is the voltage influenced by the emitter current Ie(x1).Furthermore, considering that the thickness of the base can be neglected compared to the emitter and collector and that it is fully depleted, the collector current Ic(x2) can be assumed to be equal to the emitter current Ie(x1).

Figure 2 .
Figure 2. Schematic diagram of equivalent circuit in thermoelectric transistor.Figure 2. Schematic diagram of equivalent circuit in thermoelectric transistor.

Figure 2 .
Figure 2. Schematic diagram of equivalent circuit in thermoelectric transistor.Figure 2. Schematic diagram of equivalent circuit in thermoelectric transistor.

3 . 20 Figure 3 .
Figure 3. Temperature distribution of Bi2Te3 after laser irradiation with time t and x.

Figure 3 .
Figure 3. Temperature distribution of Bi 2 Te 3 after laser irradiation with time t and x.

35x 2 1 −Figure 3 .
Figure 3. Temperature distribution of Bi2Te3 after laser irradiation with time t and x.

Figure 4 .Figure 4 .
Figure 4. Temperature distribution in P1 region at t = 100 ns.3.1.2.Temperature Distribution in p-SiBy solving the equations in MATLAB, the temperature distribution in P2 region (P2-Si) along the x direction can be obtained.After 100 ns, the temperature distribution is shown in Figure5.

Figure 5 .
Figure 5. Temperature distribution in P 2 region at t = 100 ns.

Figure 7 .
Figure 7. Relationship between p 2 (x 2 ) and x 2 at different hole concentrations.

Figure 8 .
Figure 8.In the P1 region, under different doping concentrations Pa1: (a) the relationship between Seebeck coefficient S1 and temperature; (b) variation in Seebeck voltage Vs1 at different x1 positions.

Figure 8 .
Figure 8.In the P 1 region, under different doping concentrations P a1 : (a) the relationship between Seebeck coefficient S 1 and temperature; (b) variation in Seebeck voltage V s1 at different x 1 positions.

Materials 2023 , 20 Figure 9 . 3 Figure 9 .
Figure 9. Different values of built-in voltage in P1-N region on different x1.Under different values of Pa1 and Nd, the built-in voltage increases with the increase in x1.This means that, as the temperature decreases, the built-in voltage increases.The increase is primarily attributed to the decrease in intrinsic carrier concentration with decreasing temperature.Based on Section 2.3.1, it can be concluded that achieving the forward bias and forward conduction of the emitter-base junction depends on three conditions: Vbi1(x1) > 0, Ve(x1) = Vs1(x1)-Vbi1(x1) > 0, and p1(x1) > Nd.These conditions are all influenced by Pa1, x1, and Nd.The appropriate concentration ranges for Pa1 and Nd are shown in Figure 10.

Figure 10 .
Figure 10.Suitable range and corresponding x 1 value of P a1 that satisfies forward bias and forward conduction of emitter-base.

Figure 11 .
Figure 11.Under different doping concentrations Pa2: (a) the relationship between Seebeck coefficient S2 and temperature; (b) variation in Seebeck voltage Vs2 at different x2 positions.

3 Figure 11 . 20 Figure 12 .
Figure 11.Under different doping concentrations P a2 : (a) the relationship between Seebeck coefficient S 2 and temperature; (b) variation in Seebeck voltage V s2 at different x 2 positions.Materials 2023, 16, x FOR PEER REVIEW 15 of 20

Figure 12 .
Figure 12.Different values of built-in voltage in P 2 -N region on different x 2 .

Figure 13 .
Figure 13.Suitable range and corresponding x2 value of Pa2 that satisfies forward bias and forward conduction of emitter-base.

Figure 13 .
Figure 13.Suitable range and corresponding x 2 value of P a2 that satisfies forward bias and forward conduction of emitter-base.

3. 4 . 1 .
Impact of N d on Output Power of Thermoelectric TransistorSince the value of N d directly affects the other four physical quantities, it is essential to initially examine the influence of N d on the output power.Figure14illustrates the relationship between N d and the maximum output power of the thermoelectric transistor.Materials 2023, 16, x FOR PEER REVIEW 16 of 20 illustrates the relationship between Nd and the maximum output power of the thermoelectric transistor.It is evident that, as Nd gradually increases, Poutmax exhibits a trend of initially increasing and then decreasing.At Nd = 8.0 × 10 10 cm −3 , Poutmax reaches its maximum value of 0.7093 μW.It is important to note that different values of Nd correspond to various combinations of Pa1, Pa2, x1, and x2.Considering the multitude of possible parameter combinations, it is challenging to directly explain how Nd influences the output power.Therefore, a comprehensive enumeration of the parameters is necessary to identify the optimal solution, which, in this case, is Nd = 8.0 × 10 10 cm −3 and Poutmax = 0.7093 μW.

Figure 14 .
Figure 14.Variation in the maximum output power Pmax of the transistor with the value of Nd.

3. 4 . 2 .
Impact of Pa1 and x1 on Output Power of Thermoelectric Transistor For Nd = 8.0 × 10 10 cm −3 , the range of Pa1 is from 1.61 × 10 18 cm −3 to 2.6 × 10 18 cm −3 and the range of Pa2 is from 1.0 × 10 10 cm −3 to 8.2 × 10 10 cm −3 .Moreover, different values of Pa1 and Pa2 correspond to specific ranges of x1 and x2.Considering a fixed Pa2 value of 7.8 × 10 10 cm −3 and x2 value of 0.4 μm, the relationship between output power (Pout) and the variations in Pa1 and x1 is illustrated in Figure 15.It can be observed that the output power (Pout) increases as the concentration of Pa1 and x1 decrease.These results can be derived from Equation (58):  =  ( ) × [ ( ) −  ( )] (58) When Nd, Pa2, and x2 remain constant, the value of Pout is primarily influenced by Ie(x1):

Figure 14 .
Figure 14.Variation in the maximum output power P max of the transistor with the value of N d .

Materials 2023 ,
16, x FOR PEER REVIEW 17 of 20 changed, a larger Pa1 leads to a higher built-in voltage in the P1 region (Vs1(x1)-Vbi1(x1)).As a result, with an increase in Pa1, Pout decreases.

Figure 15 .
Figure 15.Variation in the maximum output power of transistor Pout with the values of Pa1 and x1.

Figure 16 .Figure 15 .
Figure 16.Variation in the maximum output power of transistor Pout with the values of Pa2 and x2.

Figure 16 . 3 Figure 16 .
Figure 16.Variation in the maximum output power of transistor Pout with the values of Pa2 and x2.

Table 1 .
Material parameters of thermoelectric transistor.