Abstract:
A method for determining the structure of a transistor having at least one first layer including GaN, one second layer including Al x Ga 1-x N disposed on the first layer, and one fourth layer including a metal or an alloy disposed on the second layer. The method includes setting the layer thickness of the second layer, setting the aluminum content x of the second layer, producing at least the second layer and the first layer, determining the surface potential of formula (I) and/or the charge carrier density n, and/or the charge carrier motility μ after producing the second layer and the first layer, and selecting the material of the fourth layer as a function of the at least one measurement result.

Description:
BACKGROUND  
       [0001]    The invention relates to a method for determining the structure of a transistor which comprises at least a first layer, a second layer and a fourth layer, wherein the fourth layer is arranged on the second layer and the second layer is arranged on the first layer, the first layer comprises GaN, the second layer comprises Al x Ga 1-x N and the fourth layer comprises a metal or an alloy. The invention also relates to a semiconductor element which comprises a transistor. 
         [0002]    U.S. Pat. No. 7,224,004 B2 discloses a transistor of the type cited at the outset. This transistor has at least a first and a third layer which comprise GaN. Arranged in between these is a second layer which comprises AlGaN. On account of the various lattice constants, mechanical tensions are produced within the semiconductor structure which cause piezoelectric polarization. This produces band bending at the interface between the first and second layers, and a two-dimensional electron gas is produced at said band bending. This can produce a zone of increased electrical conductivity within the transistor, without having to introduce dopants into the material. By applying an electrical voltage to a fourth layer, comprising a metal, that has been applied to the third layer, it is possible to influence the charge carrier density during operation of the transistor. 
         [0003]    Transistors of the type cited at the outset are used particularly as power amplifiers for radio-frequency signals, for example signals at a frequency of more than 0.9 GHz. Such transistors can be fabricated by means of metalorganic vapor phase epitaxy or molecular beam epitaxy, for example. 
         [0004]    The charge carrier density of the two-dimensional electron gas (2-DEG) influences the layer resistance within the 2-DEG and also the threshold value of the gate voltage of a transistor comprising said semiconductor structure, from which the flow of current within the 2-DEG starts. In addition, the threshold voltage is influenced by the surface potential of the semiconductor structure, the thickness thereof and the magnitude of the band bending which occurs at the interface. O. Ambacher et al.: Pyroelectric properties of AlGaN/GaN hetero- an quantum-well structures, J. Phys.: Condens. Matter 14 (2002) 3399 discloses an approximation formula which can be used to model the charge carrier concentration within the two-dimensional electron gas as a function of the aluminum content of the second layer. 
         [0005]    The technical application of the transistors cited at the outset requires the provision of such a transistor having a prescribable threshold voltage U th  and having a prescribable layer resistance. It is therefore an object of the present invention to specify a method for determining the structure of a transistor which has a prescribable threshold voltage U th  and/or a prescribable charge carrier density n s . 
       SUMMARY  
       [0006]    The threshold voltage U th  of a transistor of the type cited at the outset and the layer resistance thereof can be set using three degrees of freedom, namely the layer thickness and the aluminum content of the second layer and the metal used for the gate contact. According to the invention, it has been recognized that an alteration in the layer thickness cannot be compensated for by an alteration in the aluminum content. This opens up the opportunity to vary the layer thickness and the aluminum content in order to provide a transistor having prescribable properties, particularly having a prescribable charge carrier concentration n s  and threshold voltage U th . 
         [0007]    It has also been recognized in accordance with the invention that the charge carrier concentration is dependent on the surface potential of the outermost layer. Therefore, measuring the charge carrier concentration n s , for example using a Hall effect measurement, results in a first value so long as the semiconductor structure is not yet provided with a gate metalization. Following the application of the gate metalization as a fourth layer, measurement of the charge carrier concentration n s , for example using a Hall effect measurement, results in a second value, which is different than the first value. A reliable prediction of the second value, which is essential for the operation of the transistor, has not been possible to date. According to the invention, it has been recognized that the second value can be predicted from the first value if the surface potential is known. 
         [0008]    The invention proposes determining the influence of the gate metalization on the charge carrier concentration and the threshold voltage by determining the potential difference between the Schottky barrier and the surface potential of the outermost layer of the semiconductor structure. To this end, the method according to the invention may comprise measurement of the surface potential of the outermost layer of the semiconductor structure. In other embodiments of the invention, the surface potential can be determined from comparison experiments which, by way of example, take account of the fabrication method and/or the ambient conditions of the semiconductor wafer at the time of the measurement and/or during a prescribable time frame prior to the measurement. This allows a transistor having prescribable properties to be provided by customizing the Schottky barrier by selecting the gate metalization, by selecting the layer thickness d of the second and/or third layer and also the aluminum content x of the second layer. 
         [0009]    Totally surprisingly, it has been found that a transistor which has been optimized to such an extent has increased resistance toward a mismatch with subsequent components, which means that the failure rate of such a transistor or of an appliance equipped therewith decreases. 
         [0010]    In some embodiments of the invention, the material of the fourth layer may be selected from silver and/or palladium and/or nickel and/or platinum. This allows the Schottky barrier and hence the threshold voltage and the charge carrier density of the transistor to be set within broad ranges. Thus, by way of example, the barrier height of the Schottky barrier for a silver contact is 0.5 eV, that for a palladium contact is 0.9 eV, that for a nickel contact is 1.0 eV and the Schottky barrier for a platinum contact is 1.1 eV. 
         [0011]    In some embodiments of the invention, the total thickness d of the second and third layers can be chosen to be between approximately 15 nm and approximately 50 nm. In some embodiments of the invention, the aluminum content x of the second layer may be chosen to be between 0.08 and 0.18. In the parameter range indicated, it is possible to provide transistors which achieve a charge carrier density of between 2×10 12  and 9×10 12  cm −2  and a threshold voltage U th  of between 0 and −3 V. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0012]    The invention will be explained in more detail below with reference to exemplary embodiments and figures without limiting the general inventive concept. In the figures: 
           [0013]      FIG. 1  shows a cross section through a transistor proposed according to the invention. 
           [0014]      FIG. 2  shows a flowchart of the disclosed method. 
           [0015]      FIG. 3  illustrates the charge carrier density of a transistor proposed according to the invention as a function of the aluminum content of the second layer. 
           [0016]      FIG. 4  discloses the surface potential Φ B  of the semiconductor structure as a function of the fabrication method and the aluminum content x of the second layer. 
           [0017]      FIG. 5  shows the threshold voltage U th  as a function of the aluminum content x for various embodiments of the present invention. 
           [0018]      FIG. 6  shows the power density p of a plurality of transistors arranged in parallel for three different embodiments of the invention. 
           [0019]      FIG. 7  schematically illustrates the alteration in the threshold voltage U th  and in the charge carrier density n s  in the event of an alteration in the aluminum content x and/or the layer thickness d. 
           [0020]      FIG. 8  schematically illustrates how by varying the layer thickness d and the aluminum content x it is possible to fabricate a transistor having prescribable values for the threshold voltage U th  and the charge carrier density n s . 
           [0021]      FIG. 9  shows the power loss p v  for a plurality of transistors arranged in parallel according to the present invention for different voltage standing wave ratios VSWR. 
       
    
    
     DESCRIPTION OF PREFERRED EMBODIMENTS  
       [0022]      FIG. 1  shows a cross section through a transistor  1  according to the invention. The cross section shown in  FIG. 1  is limited to the channel region of the transistor. The fabrication of adjoining source and drain regions is familiar to a person skilled in the art and is therefore not explained further in  FIG. 1 . 
         [0023]    The transistor  1  is arranged on a substrate  10   a.  By way of example, the substrate  10   a  may comprise sapphire or silicon carbide. The substrate  10   a  may have an optional buffer layer  10   b  arranged on it. The buffer layer  10   b  may comprise AlN, for example. The buffer layer  10   b  may have a thickness of between 30 and 150 nm. The buffer layer  10   b  can be used to facilitate the nucleation of the first layer  11  situated above. Furthermore, the buffer layer  10   b  can bring about a prescribable electrical resistance between the first layer  11  and the substrate  10   a.    
         [0024]    The optional buffer layer  10   b  or the surface of the substrate  10   a  has the first layer  11  arranged on it. The first layer  11  comprises GaN. The first layer  11  may have a thickness of between 1 μm and 3 μm in some embodiments of the invention. 
         [0025]    The first layer  11  has a second layer  12  arranged on it. The second layer  12  comprises Al x Ga 1-x N. In some embodiments of the invention, the aluminum content x may be chosen to be between 0.08 and 0.18. In other embodiments of the invention, the aluminum content x can also rise to x=0.3. In some embodiments of the invention, the second layer  12  may have a thickness of between 10 nm and 45 nm. 
         [0026]    The second layer  12  may have an optional third layer  13  arranged on it. The third layer  13  may again comprise GaN or Al y Ga 1-y N. In this case, the aluminum content y is usually different than the aluminum content x of the second layer  12 , y being able to be chosen to be between 0.05 and 0.30. The third layer  13  may have a thickness of between 2 nm and 10 nm. In some embodiments of the invention, the second layer  12  and the third layer  13  together have a thickness of between 15 nm and approximately 50 nm. 
         [0027]    On account of the piezoelectric polarization of the first layer  11 , of the second layer  12  and of the optional third layer  13 , a two-dimensional electron gas (2-DEG) is produced at the interface  15  between the first layer  11  and the second layer  12 . The charge carrier density n s  may be determined by the aluminum content x in the second layer  12 . Furthermore, the charge carrier density n s  at the interface between the first layer  11  and the second layer  12  can be influenced by an influencing electrical field. Such an electrical field may be brought about by a surface potential on that surface  16  of the third layer  13  which is remote from the second layer  12 , for example, or, in the absence of the third layer  13 , by a surface potential on that surface of the second layer  12  which is remote from the first layer  11 . 
         [0028]    In order to bring about an electrical field which influences the charge carrier density n s  during operation of the transistor  1 , a fourth layer  14  comprising a metal or an alloy is applied to the surface  16  of the third layer or to the surface of the second layer  12 . In some embodiments of the invention, the fourth layer  14  may have a layer thickness of between 2 nm and 30 nm. In some embodiments of the invention, the fourth layer comprises silver and/or palladium and/or nickel and/or platinum. The presence of the fourth layer  14  can influence the surface potential on the surface  16  of the outermost layer  13  or  12 , as a result of which the presence of the fourth layer  14  also influences the charge carrier concentration at the interface  15  when there is no electrical voltage applied to the fourth layer. Furthermore, a prescribable electrical potential can be applied to the fourth layer  14  during operation of the transistor  1  in order to influence the charge carrier concentration by means of an electrical field in a prescribable manner. 
         [0029]    Beyond the cited constituents, the buffer layer  10   b,  the first layer  11 , the second layer  12 , the third layer  13  and/or the fourth layer  14  may comprise further elements, either as a dopant, in order to set a particular prescribable conductivity for the buffer layer  10   b,  the first layer  11 , the second layer  12 , the third layer  13  and/or the fourth layer  14 , or in order to influence the lattice constants of the buffer layer  10   b,  the first layer  11 , the second layer  12 , the third layer  13  and/or the fourth layer  14 , or as inevitable impurities. 
         [0030]      FIG. 2  shows a flowchart of the method proposed according to the invention for determining the structure of at least one transistor  1 . In the first method step  51 , the layer thickness of the second layer  12  and the layer thickness of the optional third layer  13  are first of all stipulated, these having an influence on the charge carrier concentration n s  at the interface  15  and on the threshold voltage U th . 
         [0031]    In the second method step  52 , the aluminum content x of the second layer  12  is stipulated, as a result of which a prescribable charge carrier concentration n s  and hence a prescribable layer resistance for the 2-DEG within the interface  15  are achieved. 
         [0032]    In the third method step  53 , at least the first layer  11 , the second layer  12  and, if present, the third layer  13  are fabricated. In some embodiments of the invention, method step  53  may also comprise the fabrication of the buffer layer  10   b  on the substrate  10   a  as well. Said layers can be fabricated using molecular beam epitaxy, for example. In other embodiments of the invention, the layers can be deposited from an activated vapor phase, for example using MOCVD or MOVPE. 
         [0033]    In the fourth method step  54 , at least the charge carrier density n s  and optionally the mobility μ of the 2-DEG is/are determined. This can be done using a Hall effect measurement in a manner which is known per se, for example. Since the charge carrier density is influenced by the surface potential of the surface  16  of the outermost layer  13 , interpreting the results of the Hall effect measurement may require the surface potential Φ B  of the outermost layer, for example the third layer  13 , to be measured too. This can be accomplished by photoreflection in a manner which is known per se, for example. 
         [0034]    Since the surface potential Φ B  may be dependent on the fabrication method which was used in step  53  and/or on the aluminum content stipulated in step  52 , the charge carrier density n s  following the fabrication of the first layer  11 , the second layer  12  and the third layer  13  may also be dependent on the fabrication method and/or on the aluminum content. In this case, the dependency of the charge carrier density n s  on the aluminum content x results particularly from parasitic aluminum impurities which are incorporated into the third layer  13 . It is therefore possible for the surface potential Φ B  or the influence thereof on the measured charge carrier density n s  also to be ascertained by interpolating comparison experiments on semiconductor structures of the same kind. 
         [0035]    In other embodiments of the invention, the determination of the surface potential and/or of the charge carrier density in the fourth method step  54  can also be effected by means of calculation and/or by resorting to tabulated measured values for semiconductor structures of the same kind. In this case, the charge carrier density and/or the surface potential for a multiplicity of transistors  1  of the same kind can be determined in a particularly simple manner. 
         [0036]    In the fifth method step  55 , a gate material is selected for the fourth layer  14 . In particular, the material of the fourth layer may comprise silver and/or palladium and/or nickel and/or platinum. By selecting the material of the fourth layer  14 , the surface potential at the interface between the outermost layer, for example the third layer  13 , and the fourth layer  14  is confined to the barrier height of the respective Schottky barrier. In some embodiments of the invention, the Schottky barrier is 0.5 eV when the fourth layer comprises essentially silver. The Schottky barrier is 0.9 eV when the fourth layer comprises essentially palladium. In another embodiment of the invention, the Schottky barrier is 1.0 eV when the material of the fourth layer comprises essentially nickel. If the material of the fourth layer comprises essentially platinum, the Schottky barrier is 1.1 eV. Self-evidently, the material of the fourth layer is not limited to the cited materials. In some embodiments of the invention, it is also possible to use alloys or multilayer systems for the fourth layer  14 . 
         [0037]    In the sixth method step  56 , it is now possible to determine the influence of the fourth layer on the charge carrier density n s  of the 2-DEG. The charge carrier density n s  is determined by the aluminum content x of the second layer, the thickness of the second layer  12  and of the third layer  13  and also the applied electrical field. Since the aluminum content x and the layer thicknesses d have already been stipulated in the first three method steps  51 ,  52  and  53 , the sixth method step  56  now only needs to take account of the change ΔΦ B  in the surface potential Φ B . This change ΔΦ B  in the surface potential corresponds to the difference between the Schottky barrier from the selected gate metal and the surface potential of the outermost layer, as determined in method step  54 . In this case, in some embodiments of the invention, the change in the charge carrier density Δn s  can be modeled by the following equation: 
         [0000]    
       
         
           
             
               ΔΦ 
               B 
             
             = 
             
               
                 
                   
                     e 
                     2 
                   
                   · 
                   d 
                 
                 
                   ɛ 
                   · 
                   
                     ɛ 
                     0 
                   
                 
               
                
               Δ 
                
               
                   
               
                
               
                 
                   n 
                   s 
                 
                 . 
               
             
           
         
       
     
         [0038]    In the subsequent method steps  57  and  58 , the threshold voltage U th  of the transistor  1  is determined. Usually, the single determination of the threshold voltage U th  will suffice, so that either method step  57  or method step  58  is performed. In some embodiments of the invention, however, both method steps  57  and  58  can be performed for the purpose of plausibility checking or in order to increase the precision. 
         [0039]    In method step  57 , provision is made for the threshold voltage U th  to be modeled on the basis of the surface potential Φ B  following the application of the fourth layer and the charge carrier density n s  following the application of the fourth layer  14  in accordance with the following equation: 
         [0000]    
       
         
           
             
               U 
               th 
             
             = 
             
               
                 
                   Φ 
                   B 
                 
                 e 
               
               - 
               
                 
                   Δ 
                    
                   
                       
                   
                    
                   
                     E 
                     c 
                   
                 
                 e 
               
               - 
               
                 
                   
                     e 
                     · 
                     
                       n 
                       s 
                     
                     · 
                     d 
                   
                   
                     ɛ 
                     · 
                     
                       ɛ 
                       0 
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0040]    In this case, e denotes the elementary charge, ΔE c  denotes the band bending at the interface  15 , d denotes the layer thickness of the second layer  12  and of the third layer  13 , and ε and ε 0  denote the relative and vacuum permittivities. 
         [0041]    In method step  58 , a simplified method for ascertaining the threshold voltage U th  of the transistor on the basis of the change in the surface potential ΔΦ B  following the application of the fourth layer  14  can be carried out: 
         [0000]    
       
         
           
             
               U 
               th 
             
             = 
             
               
                 U 
                 th 
                 0 
               
               + 
               
                 
                   
                     2 
                     · 
                     
                       ΔΦ 
                       B 
                     
                   
                   e 
                 
                 . 
               
             
           
         
       
     
         [0042]    In this case, e again denotes the elementary charge and U th   0  denotes the threshold voltage calculated from the measured values ascertained in method step  54 . U th   0  can be ascertained in a manner which is known per se, for example as described in O. Ambacher et al.: Pyroelectric properties of AlGaN/GaN hetero- and quantum-well structures, J. Phys.: Condens. Matter 14 (2002) 3399. 
         [0043]    If the values for the charge carrier concentration n s  and the threshold voltage U th  which have been ascertained in method step  56 ,  57  and/or  58  correspond to the desired target values, method step  59  may involve the gate material selected in method step  55  being applied as a fourth layer  14  to the surface of the outermost layer  12  or  13 . If the ascertained values of the threshold voltage and of the charge carrier concentration differ from the desired target values, method steps  55 ,  56 ,  57  and/or  58  can be repeated until a gate material has been found which is able to provide the desired properties of the transistor. 
         [0044]      FIGS. 3 ,  4 ,  5 ,  6  and  9  show measured values which have been obtained on transistors with the layer structure shown in  FIG. 1 .  FIG. 3  shows the influence of the fabrication method for the first layer  11 , the second layer and the third layer  13  on the charge carrier concentration n s .  FIG. 3  shows the charge carrier concentration n s  on the ordinate and the aluminum content x in atom % on the abscissa. The measured values shown in  FIG. 3  have been recorded by means of a Hall effect measurement without a fourth layer  15  having been arranged on the surface  16  of the third layer  13 . In this case, the values for semiconductor structures fabricated by means of molecular beam epitaxy are shown as circles. The filled squares represent measured values which have been recorded on semiconductor structures fabricated using MOVPE. 
         [0045]    In addition, theoretically determined curves for the charge carrier density are plotted, which have been calculated on the basis of O. Ambacher et al.: Pyroelectric properties of AlGaN/GaN hetero- and quantum-well structures, J. Phys.: Condens. Matter 14 (2002) 3399. In this context, it is found that the charge carrier density n s  within the 2-DEG is lower for semiconductor structures which have been obtained by means of MBE than charge carrier densities n s  from comparable semiconductor structures which have been fabricated by means of MOVPE. In accordance with the invention, it has been recognized that this difference in the charge carrier density n s  can be attributed to a difference in the surface potential of the third layer  13 . The surface potential Φ B  is accordingly 0.61 eV for the semiconductor structures fabricated by means of MBE and 0.26 eV for semiconductor structures fabricated by means of MOVPE. 
         [0046]    The influence of the fabrication method on the surface potential Φ B  is also shown in  FIG. 4 .  FIG. 4  again shows the aluminum content x on the abscissa and the surface potential Φ B  on the ordinate. In this case, the filled squares again denote measured values which have been obtained by means of Hall effect measurements on semiconductor structures which have been fabricated by means of MOVPE. The circles denote measured values for the surface potential which have been obtained by means of Hall effect measurements on semiconductor structures which have been fabricated by means of MBE. Values of the surface potential which have been ascertained by means of CV profiles for semiconductor structures which have been fabricated by means of MOVPE are shown by unfilled squares. In addition,  FIG. 4  shows results from measurements for the photoreflection as crosses. In this case, the values ascertained on semiconductor structures which have been fabricated by means of MOVPE are shown by rotated crosses. 
         [0047]    Hence, the measured values shown in  FIG. 4  also confirm that the surface potential Φ B  of the semiconductor structures is dependent on the fabrication method used. As a result, the fabrication method influences the measured charge carrier concentration n s  so long as no gate material has been applied as a fourth layer  14  to the surface  16 . 
         [0048]      FIG. 5  shows the threshold voltage U th  of a transistor which comprises the semiconductor structure shown in  FIG. 1 . As a departure from the semiconductor structures which have been used for the measurements shown in  FIGS. 3 and 4 , the semiconductor structures used for the measurements in  FIG. 5  have the fourth layer  14 —shown in FIG.  1 —as a gate electrode. 
         [0049]      FIG. 5  again shows the measured values obtained on semiconductor structures which have been fabricated by means of MOVPE as squares and the measured values from the semiconductor structures which have been fabricated by means of MBE as circles.  FIG. 5  reveals that the threshold voltage U th  and hence also the charge carrier density n s  becomes independent of the fabrication method used for fabricating the semiconductor structure by virtue of the application of the fourth layer  14 . This means that the surface potential Φ B  on the surface  16  is confined to the value of the Schottky barrier between the fourth layer  14  and the third layer  13  by virtue of the application of the fourth layer  14 . According to the invention, it has been recognized that the threshold voltage U th  of a transistor  1  and also the charge carrier density n s  thereof are stipulated only by the combination of the first layer  11 , the second layer  12 , the third layer  13  and the fourth layer  14 . If the process control involves measurements for the charge carrier density n s  and/or for the threshold voltage U th  being performed on the first layer  11 , the second layer  12  and the third layer  13 , these results are dependent on the surface potential of the third layer  13  and hence on the fabrication method, however. 
         [0050]    In one exemplary embodiment, a power amplifier for a radio-frequency signal needs to be produced. Such a power amplifier can be used in a base station in a mobile radio network, for example. The power amplifier needs to be operated at an operating voltage of 50 V. The power amplifier comprises a plurality of transistors arranged in parallel which each have a design as shown in  FIG. 1 . The plurality of transistors may comprise between 80 and 120 transistors, for example. In one embodiment of the invention, the plurality of transistors may be monolithically integrated on a single substrate  10   a.  The distance between the individual transistors may be between approximately 50 μm and approximately 150 μm. This results in a gate width for the arrangement of between approximately 25 mm and approximately 40 mm. The gate length of the transistors may be between approximately 0.4 μm and approximately 1.0 μm. 
         [0051]    The power amplifier takes the normal direct current and produces a radio-frequency output signal. In this context, a radio-frequency output signal within the meaning of the present invention is understood to be an output signal which has a frequency of more than 0.85 GHz, or a frequency of more than 2 GHz, or a frequency of more than 5 GHz. The portion of the direct current drawn which is not converted into the radio-frequency output signal by the power amplifier is converted into power loss p v  and must be dissipated as heat. In some exemplary embodiments, the electrical power converted by the power amplifier may be more than 5 W, more than 10 W or more than 30 W. 
         [0052]    As  FIG. 6  shows, the charge carrier concentration n s  and hence the power density p rise as the aluminum content x of the second layer increases. In order to limit the thermal load, a low aluminum content x is therefore advantageous. On the basis of the relationship explained in method step  57  as shown in  FIG. 2 , this causes the threshold voltage U th  to rise, however, and said threshold voltage can assume undesirably high values. 
         [0053]      FIG. 9  shows the power loss p v  in a plurality of transistors arranged in parallel according to the present invention for different voltage standing wave ratios VSWR. The plurality of transistors may have a design as described in connection with  FIG. 5 . In this case, the voltage standing wave ratio VSWR is a measure of the mismatch between the transistors and the component connected to the respective outputs. A VSWR of 1:1 means that there is no mismatch in this case and the power loss and hence the generation of heat are minimal. As the VSWR increases, for example to values of 2:1, 3:1, 4:1, 5:1, 6:1 or 10:1, the mismatch increases and an ever greater component of the electrical power drawn is converted into power loss p v , which can damage the semiconductor element. 
         [0054]    As can be seen from  FIG. 9 , a semiconductor element which has been optimized using the method according to the invention can be operated at an operating voltage V DS  of 50 V even with a large mismatch or with a VSWR of 10:1, although this causes the power loss to increase by a factor of between 1.5 and 2.0 in comparison with regular operation at a VSWR of 1:1. This allows an improvement in the reliability of the semiconductor element and of an appliance equipped with the element. In this case, the measured values shown in  FIG. 9  have been recorded at an operating point of the transistor at which the transistor is in gain compression. In some embodiments of the invention, this may be the P3db point. In another embodiment of the invention, the transistor may be at the P1db point. 
         [0055]      FIG. 7  schematically explains the relationship between the threshold voltage U th  on the ordinate and the charge carrier density n s  on the abscissa for various values of the aluminum content and of the layer thickness d of the second layer  12  and of the third layer  13 . By way of example, the point A in  FIG. 7  represents a semiconductor structure with an aluminum content of x=0.14 and a layer thickness d=22 nm. 
         [0056]    Starting at A, the point B can be reached by increasing the aluminum content to x=0.18 for an unaltered layer thickness d=22 nm. The semiconductor structure B is distinguished in that the threshold voltage U th  falls, with a simultaneous rise in the charge carrier concentration n s . 
         [0057]    By increasing the thickness to d=30 nm for an unaltered aluminum content x=0.18, the point C is reached. The point C describes a semiconductor structure with a further-reduced threshold voltage U th  for a slightly higher charge carrier density n s . In particular, it should be pointed out that an alteration in the layer thickness d cannot be compensated for by an alteration in the aluminum content x. In this way, appropriate choice of the layer thickness d and the aluminum content x allow the simultaneous implementation of a low charge carrier density n s  and a sufficiently low threshold voltage U th , as explained in more detail in connection with  FIG. 8 . 
         [0058]    According to  FIG. 8 , a transistor  1  having a threshold voltage U th  of −1.8 V and a charge carrier density n s  of 5×10 12  cm −2  is intended to be fabricated. The transistor  1  having the desired, properties is denoted by the point D in  FIG. 8 . 
         [0059]    Starting at point B with an aluminum content of x=18% and a layer thickness d=22 nm, the layer thickness is first of all increased to d=30 nm for an unaltered aluminum content of x=0.18. In this way, a semiconductor structure is obtained, the properties of which are denoted by C in  FIG. 8 . 
         [0060]    The semiconductor structure C has a threshold voltage which is below the threshold voltage of −1.8 V. In addition, the charge carrier density n s  and hence the power loss of the transistor are greater than the desired target value. 
         [0061]    By reducing the aluminum content to x=0.12 for an unaltered layer thickness of d=30 nm, a severe reduction in the charge carrier density n s  is obtained for a simultaneously only moderate rise in the threshold voltage U th . In this way, it is possible to obtain the desired properties of the transistor  1  which are shown at point D. In this case, it should be pointed out that the threshold voltage U th  is first stipulated by the selection of the gate metal, as described above in connection with  FIG. 2 . According to the invention, it is accordingly possible to fabricate transistors having the desired properties by selecting the layer thickness d, the aluminum content x and the gate material. 
         [0062]    Self-evidently, the solution principle disclosed in the description above can also be used to optimize other parameters of a transistor. The above description should therefore be regarded not as limiting but rather as explanatory. The claims which follow should be understood to mean that a cited feature is present in at least one embodiment of the invention. This does not preclude the presence of further features. Where the claims and the description define “first”, “second”, “third” and “fourth” features, this designation is used to distinguish features of the same kind, without stipulating an order of rank.