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ADVANCED THEORY OF THE HEMT

To obtain a value for V0g in equation 8.10 without resorting to numerical computations we had to omit the term Ey/e in equation 8.9. If we wanted to calculate Ep we would need to solve Schrodinger’s equation along the depth z of the device, obtain the eigenvalues and then obtain ns. In fact our analysis of the HEMT so far was essentially a 1-dimensional one: we solved the Poisson equation in 1 dimension along the depth z of the device and used a simple drift equation along the x direction. From our discussion of the MOSFET in chapter 7 we saw how inefficient these 1-dimensional models are. The presence of both Vc and Vp imply a vertical (to the gate) field and a parallel field, respectively, making any transistor problem essentially a 2-dimensional problem, especially at short gate lengths. The machinery we have developed for the MOSFET can be applied equally well to the HEMT. In fact it becomes even simpler in the case of the HEMT because GaAs and several of its alloys are direct band-gap materials having the respective conduction band-edge £c at к = 0, so there is no valley dependency.

We summarize below the set of differential equations, specifically for the HEMT, that are necessary to produce a more accurate (though numerical) simulation than the simple algebraic model that we have given so far.

Poisson’s Equation

In equation 8.27 the dielectric constant is put between the 3-dimensional derivatives because HEMT structures include many layers of III—V materials with different dielectric constants.

The 2-Dimensional Schroedinger’s Equation (Effective mass form).

In equation 8.28 a unique effective mass m* is used on the one hand but on the other hand the form of the equation valid for interfaces was used (see equation 5.40) because again many layers of different effective masses are necessary for the analysis of the HEMT. The Schroedinger equation, contrary to the Poisson equation, is not solved in the entire domain of the device but only in a subdomain, called quantum box, which includes the channel layer extending well into the source and drain regions, just as in the MOSFET.

The Continuity Equation

In equation 8.29 the recombination term R„ has been omitted since it is small in unipolar devices and especially in HEMTs. Equation 8.29 can give us an equation for the Fermi level

Ep as follows: the expression for the drift-diffusion current for electrons, equation 7.24, can be written as (see problem 3.2 at the end of chapter 3)

If we substitute 8.30 in 8.29 we get

Given the eigenfunctions and the Fermi level EF, the quantum charge density

nQ (x,z) can be calculated in analogy with equation 7.30 for the MOSFET.

where /о is the Fermi- Dirac occupation function.A notable difference, however, with the MOSFET is the following. The depth of the well in the oxide/Si interface is determined by the band-edge offset AEc of Si with respect to the oxide which is AEc = 3.9eV. The corresponding quantity in HEMTs is the band-edge offset A Ec of very similar III—V alloys (e.g. InGaAs and AlGaAs), resulting in a AEc of a few tenths of an eV, i.e. much smaller than the Si/Si02 interface. This low depth of the well holding the 2-dimensional electron gas results in some eigenstates being above the top of the barrier Elop, but below £f, i.e. some occupied eigenstates are not true 2-dimensional states, see figure 8.4. The situation is erected by summing in 8.32 only eigenstates with energies E, < Et0p. For the remaining eigenstates we simply use the corresponding 3-dimensional formula, treating them as a classical semiconductor electron density, i.e.

Definition of the quantity E in a quantum well above which the energy eigenvalues Ej, E etc may be considered as 3-dimensional

FIGURE 8.4 Definition of the quantity Etop in a quantum well above which the energy eigenvalues Ej, E2 etc may be considered as 3-dimensional.

Then, the total electron density n is given by A flow chart that solves the HEMT equations is given below.

FLOWCHART 3

In the above flow chart, EPS is the accuracy with which we want to calculate n, the electron density. Note that in the return box to the Poisson equation the electron density n is updated not by the new value, but by the old value n0 plus a fraction a of the change An in n. To ensure convergence that fraction must be as low as 1%. The workload can be significantly reduced for long-channel HEMTs (L>100nm) if the Schroedinger equation is left out. Then the updated charge density is nx = nd with Elop = Ec and nQ = 0. Such a calculation is called classical whereas the one including the Schroedinger equation is frequently called quantum, although we have reserved the term quantum in this book for a calculation that uses a quantum-mechanical expression for the current.

Some typical results for the HEMT structure are given below. Figure 8.5 gives the particular HEMT structure for which the author has performed calculations for several values of the gate length [2]. In this particular structure, an InGaAs channel is sandwiched between two InAlAs barriers. The channel density for L = 100nm and Vcs = Vos = 0 is given in Figure 8.6. Note 1) the high electron density below the source and drain regions which

Layout and parameters for an InAlAs/InGaAs HEMT for which calculations have been performed by the author [2]

FIGURE 8.5 Layout and parameters for an InAlAs/InGaAs HEMT for which calculations have been performed by the author [2].

A 2-dimensional view of the electron density across the entire structure of the device of figur

FIGURE 8.6 A 2-dimensional view of the electron density across the entire structure of the device of figure 8.5. The gate and drain voltages are Vqs = Vfis = 0. Note the channel electron density, having just been formed, the high electron density below the source and drain regions, which are overdoped, and the depleted InAlAs region below the gate.

A 1-dimensional view of the electron density of a nanometric and a long-channel HEMT

FIGURE 8.7 A 1-dimensional view of the electron density of a nanometric and a long-channel HEMT: whereas the long HEMT is below its threshold the nanometric one is above its threshold.

are overdoped, 2) the depleted region below the gate in the InAlAs layer, and 3) the channel region having just been formed in the InGaAs layer. Note also that in order to achieve high accuracy a large depth of the buffer onto which the channel rests had to be included in the calculations.

A 1-dimensional cut along the channel length at the middle point of the channel depth is shown in figure 8.7 for two values of the channel length, L = 200nm and L = 30nm at Vcs = VDS = 0. It can be easily seen that whereas the L = 200nm HEMT is below threshold, the L = 30nm HEMT is above. Obviously these two HEMTs have different threshold voltages and this brings us to the first of the short channel effects mentioned in the previous chapter, that is the lowering of the threshold voltage with decreasing L. This is shown for the HEMT under investigation in figure 8.8. The symbol AVT instead of AV0ff is used on the vertical axis. A very low drain voltage, VDS = 0.05, is applied. It can be seen that the change in the threshold voltage is substantial, of the order of half an eV. The effect of the change in V0g on the saturation current is shown in figure 8.9 where the calculated ID—VD characteristics are plotted for three values of the gate length L, i.e. at L = 30, 60,100nm. We observe that irrespective of whether a classical or a quantum (in the sense of including the Schroedinger equation) calculation is performed, there is a monotonic increase of the saturation current when L is decreased from L = 100nm to L = 30nm. The above phenomena occur in all HEMTs, not only in the one presented above.

Lowering of the threshold voltage V (V in the diagram) with gate length L. The values of this calculation were compared with experimental values and with other simulations; see [2] for more details

FIGURE 8.8 Lowering of the threshold voltage Voff (VT in the diagram) with gate length Lc. The values of this calculation were compared with experimental values and with other simulations; see [2] for more details.

Increase of the current Iwith drain voltage Vas the gate length L decreases in both classical and quantum calculations; see text for details

FIGURE 8.9 Increase of the current IDwith drain voltage VDSas the gate length LG decreases in both classical and quantum calculations; see text for details.

Comparison of the leakage current of HEMT and III-V MOSFETS. Figure reproduced from R. Chau et al Microelectronic Eng. 80, 1 (2005)

FIGURE 8.10 Comparison of the leakage current of HEMT and III-V MOSFETS. Figure reproduced from R. Chau et al Microelectronic Eng. 80, 1 (2005)

 
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