Can you make a diode just by connecting a piece of N-type material like phosphorus to a P-type material like gallium? Why are they added to silicon?
First, silicon is not necessary for making semiconductors, nor even working semiconductor devices. The first commercially successful transistors were Germanium, and other semiconductor materials (e.g. selenium, copper oxide, etc.) were used to make rectifiers before transistors came along. Beyond that, there's a large number of other semiconducting materials (such as gallium arsenide) that work -- some even commercially.
Second, there are specific constraints on what makes a semiconductor work in a semiconductor device. A full treatment gets down into the quantum physics of it, which I'm neither inclined, nor entirely qualified, nor have the space to fully describe here. Basically, however, semiconductor devices work because the available states for electrons to exist in within the material are separated. This leads to the terms "valence band", "conduction band", and "band gap". You need a material that supports distinct valence and conduction bands, and that can be tuned to be P-type or N-type. All of the semiconductor materials do this.
Elemental gallium has overlapping conduction and valence bands, so it's a metal. It can be part of a semiconductor, but it's not a semiconductor. Elemental phosphorus has conduction and valence bands that are so far apart that it's an insulator. It can be part of a semiconductor, but it's not a semiconductor.
A PN junction needs a semiconductor. This is a material that is 'halfway' between a conductor and an insulator. It's not just that it conducts less well than a metal; it's that carriers (electrons and holes) in the material are constrained to different energy levels (think of floors in a building). You also need two sets of distinctly different levels for a P-N junction.
In addition, it's not 'just connecting' that is needed. The crystal structure needs to be continuous across the junction or else the energy levels will 'spread out', and the desired behaviour (conduction in only one direction) won't occur.
Note that not only silicon is a practical semiconductor -- germanium is. Semiconductors are also made of gallium arsenide (as in LEDs); gallium nitride (GaN -- used in high frequency RF circuits like 5G cellphone transmitters) silicon carbide (SiC - often used in electric vehicles to drive the motor) and even diamond for extremely high temperature circuits.
Fundamentally you need a crystalline non-metal (which therefore has distinct energy levels), a separation between these levels which is not too large (so that practical levels of current can flow); not too small (or too much current will flow -- germanium is in this range); and is practical for doping (which changes the energy levels). Silicon is extremely suitable for this -- it is plentiful (so low cost); easily purified; robust (strong); straightforward to dope, yet chemically quite inert.
Semiconductor rectifiers have been used since the 1920s. Typically using Copper Oxide or Selenium
Metal rectifiers consist of washer-like discs of different metals, either copper (with an oxide layer to provide the rectification) or steel or aluminium, plated with selenium. The discs are often separated by spacer sleeves to provide cooling.
Black phosphorus (BP), a novel two-dimensional (2D) layered semiconductor material, has attracted tremendous attention since 2014 due to its prominent carrier mobility, thickness-dependent direct bandgap and in-plane anisotropic physical properties. BP has been considered as a promising material for many applications, such as in transistors, photonics, optoelectronics, sensors, batteries and catalysis. However, the development of BP was hampered by its instability under ambient conditions, as well as by the lack of methods to synthesize large-area and high quality 2D nanofilms.
Gallium Nitride is a III/V (three-five) semiconductor. It is best know as the Blue LED that is the basis for making the billions and billions of white LEDs that are on the market. The .white LEDs are GalliumNitride (GaN) diodes with some phosphors to make the green and red emissions to make a "white" LED.
There is growing interest in GaN semiconductors for transistors and companies like CUI are using commercially made GAN transistors for high frequency power supplies (smaller indicators, less ripple, less heat loss, ...). Also, GaN can be used to make circuits efficiently run at frequencies well beyond where silicon starts to get inefficient (very inefficient). Microwave frequencies in excess of 70GHz used for automotive radar is not a problem for GaN.