Tabel of contents

A MESFET (Metal-Semiconductor Field Effect Transistor) is considered a unipoloar transistor to distinguish it from the bipolar junction transistor where electrons and holes participate in the current flow. The rectifying contact is formed by bringing the semiconductor material in contact with the gate metal to form a Schottky barrier. As illustrated in Figure 3.0-1, the MESFET is composed of a narrow semiconducting channel with two ohmic and one rectifying metalcontacts, because each material has its own Fermi energy, the Fermi levels adapt themselves to a common level at thermal equilibrium. Through this effect, a depletion layer under the gate is formed. The thickness of this depletion layer can be controlled by an applied gate voltage, therefore a MESFET can be regarded as a voltage controlled resistor. Sine the depletion layer is significant in dimension even without an applied gate or drain voltage, the type of transistor can be defined by controlling either the channel thickness or the donor impurity concentration N_D in the channel region. The H-GaAs-III process of Vitesse defines the kind of transistor by controlling the donor concentration. The process follows these steps: an undoped buffer layer of GaAs is epitaxially grown on top of a semi-insulation substrate. After depositing a dielectric cap, the active region with a donor concentration of about N_D = 10¹⁶ cm^-3 is defined by ion implantation. This implantation doping is equal to that needed for the enhancement device. To get depletion devices a additional implantation with a different mask is done. After this the gate metal is deposited and defined. An implant with impurity concentration of about N_D = 10¹⁸ cm^-3 will define the source and drain areas. After a activation anneal the source and drain metal can be deposited and sintered. The Vitesse GaAs MESFET is shown in Figure 5.0-1.

GaAs MESFETs have many advantages. Above all the electron velocity at low field is sufficiently high so that high switching speed and therefore high cutoff frequencies can be achieved. An additional high input impedance, negative temperature coefficient at high current level and thermal conductivity, low resistance and low I× R drop along the channel height so that GaAs MESFETs are especially useful for low-noise amplification, high-efficiency power generation and high-speed logic application.

1.1 An analytic model of a MESFET based on physical principles

By describing the physics of the MESFET analytically, it becomes rapidly obvious that this model is far from usable for both circuit design and circuit simulation. By changing one of the node voltages the whole model has to be recalculated. This means solving nonlinear algebraic equations by iteration. For a transient simulation this must be done on every for every time point again and again. That includes currents and charges. By developing bigger circuits normally every computer rapidly reaches its limitations by using this model.

On the other side there are empirical models. An empirical model describes the device characteristics by using a simple analytic equation. These equations may not have any physical origin, rather they describe the device behavior in a mathematical sense. An important advantage of an empirical models is their flexibility. The equations can be fitted over given parameters. However the disadvantage of such models is that the equations may lead to non-physical results because they are often too simple. The empirical model however makes it possible to use it in simulation software like HSPICE.

The physical models are based on the device materials, geometry and processing parameters such as doping concentration, carrier mobility and carrier velocity, doping profile and doping concentration, barrier height, layer thickness and device dimensions. These models are useful for basic research on a transistor. One famous physical FET model is the classical Shockley model which is derived from the Poisson and current-continuity equations.

1.1.1 MESFET Device Characteristics

Figure -1 shows a schematic model of a depletion MESFET with the basic device dimensions. L is the term for the gate length and W the term for the gate width. The parameter "a" is the channel height and "h" is the height of the depletion zone at a specific point under the gate. The MESFET consists of an active channel, contacted with higher doped areas called source and drain. The third electrode is the gate. The applied drain voltage V_D and gate voltage V_G are referenced to the source which is normally grounded. When a positive voltage V_D is applied, a physical current from source to drain will flow. In this way the terms drain and source explain themselves. The gate controls the current under the gate by varying the depletion height h so that a MESFET can be basically regarded as voltage controlled resistor.

Figure -3 shows a typical I-V diagram of a depletion MESFET. The drain current I_D is plotted against the drain voltage for various gate voltages. The linear region corresponds to the model of a voltage controlled transistor. Above all, for voltages << V_{D sat}, V_SD is direct proportional to the drain current I_D. At the saturation region I_D remains nearly constant because the drift velocity has reached its saturation. I_D can so be regarded as independent of V_SD. At the break down region, a small increase of V_SD causes a rapid increase of the drain current I_D because the electron energy is big enough to ionize other atoms provide additional electrons. This is called the avalanche effect.

When a metal and a semiconductor with therefore different Fermi energy levels are brought in contact with each other, the Fermi energy of each material will line up to a common level. In case on a n-type semiconductor the Fermi energy of the semiconductor is larger than the Fermi energy of the metal. Because there are many empty energy states in the metal, electrons with sufficient energy will flow into the metal taking the energy state. By leaving the semiconductor there will be unfilled energy states left in the semiconductor. Moreover, because a contact between the two different materials can never be ideal, the presence of chemical defects or broken bonds will cause large numbers of unfilled so called surface states. These free states will be filled by electrons from the semiconductor bulk. This provide a positive charge under the junction and a depletion region is formed. During this process the valence and conduction band is bent by the positive depletion charge preventing more electrons to take metal or surface states. Because the number of unfilled surface states is high, the metal side is assumed to be constant. From Figure -5, the following equation can be found:

-q× V_B = -q× V_bi + E_C - E_F

Once the relationship between valence and conduction band of the semiconductor with the Fermi level of the metal are known, it can be used as a boundary condition on the solution of Poisson’s equation in semiconductors. Under the abrupt assumptions that r » q× N_D for x smaller than depletion height, and r » 0 and dV/dx» 0 for x larger than depletion height. The results are summarized below

where V_G is the applied gate voltage, V_B is the barrier height and V_bi the built in voltage given by , n_i is the intrinsic carrier concentration. It should be noted that for GaAs the barrier height V_B>>k× T/q for all cases of metals to be contacted. Table 1 shows the barrier heights for different materials. For the n-type of GaAs, the barrier height is nearly the same value.

The Schottky junction represents a diode. The current transport in this contact is mainly due to majority carriers. In the Schottky diode there are four kind of carrier transport where the thermionic emission current is the dominant one. The four processes are

The active region is moderately doped with a concentration between 10¹⁶-10¹⁷ cm^-3, the tunneling effect (2) can be neglected. Effects (3) and (4) can also be neglected because this are recombination processes. The fact that the minority concentration is limiting the recombination process gives an idea about the magnitude of these currents.

The thermic emission current, also called gate current, is given by

Equation -6,

where A is the area of the junction, R the "Richardson’s constant", m^* the effective mass, h the Planck’s constant and k the Boltzmann’s constant. For the Richardson’s constant, usually experimental values are used, not the calculated one. The equation for the gate current gives a good idea how the built in voltage is related to the applied gate voltage.

The following points will derive a expression for the drain current as a function of the terminal voltages. Firstly the Schottky junction and effects like the carrier mobility and overshoot will be discussed. To get a mathematical description assumptions and approximations have to be made. The n⁺-regions are regarded as perfect conductors and the built-in potential at the n⁺/n-interface is zero. The substrate is regarded as perfect insulator so that the surface charge at the n-GaAs/GaAs is zero. Due to convention V_D is greater than and equal to V_S.

1.1.3.1 Field dependent mobility and velocity

The conduction band minimum in GaAs, the G -valley, is made up of a single valley with a very light effective mass m*=0.067m₀ and at a energy of 0.3eV higher there is a heavy mass valley called the L-valley with an effective mass of m*=0.22m₀.

Walukiewicz et [8] gives the following empirical description of the mobility of carriers in a semiconductor dependent of doping concentration and temperature. The average error is about 4% however it is longer at lower fields.

Q =N_A/N_D range [0, 0.9]

n range [10¹⁵, 5× 10¹⁸] cm^-3

a =0.53+0.03× Q

m _max =8200_[300K]

for Q <0.2: n_ref=9.63× 10^{18× Q 3}-4.2× 10^{18× Q 2}+3.7× 10^{17× Q} +9.85× 10¹⁶ cm^-3

for Q >0.2: n_ref=10^{17.07× (1-Q )^0.04} cm^-3

for Q <0.2: m _min=-16500× Q ³+17450× Q ²-8080× Q +2750

for Q >0.2: m _min=2350× (1-Q )^1.45

Equation -1

Moreover he wrote the dependence of mobility of the temperature in format as a power law. The problem in doing this is that the optical scattering rates do not scale as the power law so that for large doping concentrations n>10¹⁷ cm^-3, the Fermi function does not have an exponential behavior. The thermal dependence is given by

Equation -2.

The parameter s for n [10¹⁷, 10¹⁸] cm^-3 is in the range of s=0.5± 0.1 in a modern MESFET.

There are many ways to analytically describe the velocity of carriers. All of them have in common that they are empirical models. Many models for the drain current support the so called linear model. In this model it is assumed that the carrier mobility is constant so that the drift velocity of carriers becomes to . This linear approach can only be made because the scattering of electron at fields lower than 2.5 kV/cm is low so that the drift velocity increases linearly with field. Since this model is a simple linear equation it makes further mathematical description much easier. This model is limited to a electric field lower than 2.5 kV/cm. This not a very big restriction since the main application of GaAs takes place in low fields to achieve high carrier velocity. Moreover such electric fields can only be found at very short channel devices L<0.25m with a source-drain voltage of 1V when the channel is entirely open. This means that the electric field is not constant in the channel region. Because of carrier distribution under the pinch-off point electric fields can be achieved of 10 kV/cm so that models based on the linear model will lead more and more to incorrect results.

Another way do describe the drift velocity is that the electric field vs. v_D is described empirically so that to each value of the electric field a good value of v_D is available. This has advantages for models that are supporting fields around 2.5 kV/cm. Moreover the field dependent mobility of carriers is also included by describing the curve. A disadvantage of this model is that the equation only delivers the saturation velocity. The transport of electrons is a dynamic process over time and effects like the overshoot will be disregarded. At this point it should be mentioned that electrons never reach steady state in a normal MESFET. Moreover because the equation is not linear the mathematical handling of further models including this model will be intractable. The drift velocity v_D can be described after myself as

Equation -3.

 

This empirical description only depends on the extreme points of the curve. Corrections for a better fit in specific areas can be made by changing the exponent of e ². For low fields the equation becomes to . This confirms the linear model. Furthermore could be replace by m , however the equation would become again dependent of the mobility of carriers. If a dynamic temperature model should be done this could be a interesting option. For the drain current a third model will be chosen, as discussed later, that assumes that the electrons never reach steady state and stay at the peak velocity. Figure 5.1-9 shows the graph of Equation 5.1.3-3.

Figure -11 shows a channel cross section of a GaAs MESFET with a applied gate voltage of V_G=-1V so that the channel is driven towards pinch off. The source-drain voltage is 3V. The drain current is in saturation. The narrowest channel opening is under the drain end of the gate. Under this point a statistic capacity is built up through velocity behavior of electrons in GaAs in high field environment. Because the resistance under the end of the gate is the highest most of the voltage will fall off at this point and therefore there will be the highest electric field. If an electron is flowing in this area it will be accelerated through this field. Since the behavior of electron shown in Figure 5.1-9 electrons reaches saturated velocity which is lower in this region than the peak velocity. Because more electrons are coming in than out a heavy electron accumulation takes place. After passing the peak of the electric field the opposite occurs. After the peak the channel is opening very fast and therefore the electric field is decreasing. Because the electric field is becoming smaller the electrons can again reach there peak velocity causing a strong depletion layer. In this way a statistic capacitance is built up causing a new electric field that reinforce the present electric field.

Figure -13 shows the behavior of electrons in a high electric field. Before the carriers enter the high field area they remain in equilibrium. By entering the high field area which is much higher than the electric field for the peak velocity, the electrons are accelerated to a velocity which is more than two times higher than the peak velocity. In this area the electrons are in nonequilibrium. After a distance of 1 m m the electrons reach equilibrium again. In order to use this effect as an advantage the gate has to be short. In this way the overshoot effect shortens the transit time of the carriers in a direct ratio and therefore improves the high-frequency response. This effect also takes place in the carrier accumulation under the pinch off point of the gate as previously mentioned and therefore reinforces the accumulation effect. Because the effect is not abrupt as described here, the carriers will reach not such high velocity.

1.1.3.4 Analytic Drain current description

First of all a uniform charge distribution under the gate is assumed. For the depletion layer width under the gate, the abrupt junction expression formerly used for the Schottky contact is given by

, where V_bi is the built in potential given by . V_(x) is the applied drain voltage at x with respect to the source. The gate voltage V_G is negative in n-channel devices with respect to the source. The depletion height at the source side of the gate is given by

, and Equation -5.

Equation -6.

The last term can be defined as the pinch-off voltage so that V_GD becomes . The pinch-off voltage V_P is proportional to the second power of the thickness of the channel height so that this value is very sensitive to dimensional variations. The value of V_GS at cut-off is called the threshold voltage, V_T. The threshold voltage depends on doping concentration, channel height and built-in voltage. By controlling the device parameters, the device can be determined as enhancement or depletion.

Figure -14: assumed velocity-field

Equation -9.