When did the US stop making silver half dollars?

Half dollars were never pure silver - they contained at least 10% copper.

The last year for 90% silver halves was 1964.

From 1965 to 1969 the coins were made of 40% silver and 60% copper.

In 1971 the composition was changed to the same copper-nickel clad metal used for dimes and quarters.

Half dollars circulated regularly up till 1963. The Mint - foolishly as it turned out - tried to keep the half as a "prestige" coin with a reduced but non-zero silver content when other denominations were switched to copper-nickel.

Unfortunately the price of silver went so high that even the 40% coins were worth far more than 50¢ to precious-metal dealers so the denomination essentially disappeared from circulation. The Mint churned out more and more of them in an attempt to stop hoarding but all they did was make a lot of metal dealers wealthy.

By the time the Mint admitted their mistake, most people had gotten out of the habit of using halves and the coin is rarely seen today.